Welcome to the homepage of the MAXIMALS algebra seminar, at the University of Edinburgh. The seminar represents the interests of all Edinburgh faculty working in algebra and number theory, and is currently being coordinated by Sjoerd Beentjes, Cheuk Yu Mak, Brian Williams and Fei Xie.
COVID-19 Arrangements. Due to the COVID-19 crisis, the Edinburgh MAXIMALS and EDGE seminars are combined into one weekly joint Hodge seminar. The future events listed below are therefore for the joint Edinburgh seminar, and also for Heriot-Watt's MAXIMALS seminar which for now is running separately.
Mailing Lists. Since most of our seminars will occur together with EDGE in a doubleheader format, we have a joint mailing list, that we will use to send unified announcements. To subscribe to the maxi-edge mailing list, send an email to sympa at mlist.is.ed.ac.uk, with subject line: subscribe maxi-edge your_first_name your_last_name. To unsubscribe at any time, send an email to the same with subject unsubscribe maxi-edge.
To subscribe to the original maximals mailing list, where there will be Algebra focused announcements, send an email to sympa at mlist.is.ed.ac.uk, with subject line: subscribe maximals your_first_name your_last_name. To unsubscribe at any time, send an email to the same with subject unsubscribe maximals.
Upcoming Hodge seminar events at University of Edinburgh
Upcoming MAXIMALS events at Heriot-Watt
Past talks in Hodge seminar:
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Hodge Seminar: Antonella Grassi
30th June 2022, 1:45pm to 2:45pm JCMB 5323 -- Show/hide abstractAbstract: Title: Stringy Kodaira and applications. Abstract:Kodaira classified fibres on relatively minimal complex elliptic surfaces. I will discuss the classification problem for complex threefolds and its application, in particular to questions from string theory. -
Hodge Seminar:Ruijie Yang (Universität zu Berlin)
30th June 2022, 12:30pm to 1:30pm -- Show/hide abstractAbstract:
Title: Hodge theory of singularities of divisors
Abstract: Given a holomorphic function on a complex manifold, the vanishing cycle complex of the function encodes cohomology of Milnor fibers. In projective algebraic geometry, we often encounter divisors rather than functions. In this talk, I will explain how to glue vanishing cycle complexes of local defining functions of a divisor and endow each associated graded part of the monodromy weight filtration with a structure of twisted pure Hodge module. This construction brings tools from birational geometry such as vanishing theorems to the study of vanishing cycles. Furthermore, we define a sequence of ideal sheaves, which encode Hodge-theoretic information of singularities of divisors. It turns out that these ideal sheaves are closely related to the Hodge ideals introduced by Mustata-Popa. This is based on the joint work (partially in progress) with Christian Schnell. -
Hodge Seminar: Maria Chlouveraki (Versailles & Athens)
27th June 2022, 3:30pm to 4:30pm JCMB 5323 -- Show/hide abstractAbstract:
Title: The block defect of Hecke algebras
Abstract: The complexity of a block of a symmetric algebra can be measured by the notion of defect, a numerical datum associated with each of the simple modules contained in the block. Geck showed that the defect is a block invariant for Iwahori–Hecke algebras of finite Coxeter groups in the equal parameter case, and speculated that a similar result should hold in the unequal parameter case. In a joint work with Nicolas Jacon, we have proved that the defect is a block invariant for Hecke algebras associated with the complex reflection groups of type G(r,1,n), which include the Weyl groups of type Bn in the unequal parameter case, by showing that the defect corresponds to the notion of weight in the sense of Fayers. We conjecture that the defect is a block invariant for Hecke algebras in general. -
Hodge Seminar: Andrew Hanlon (Stony Brook)
16th June 2022, 2:00pm to 3:00pm -- Show/hide abstractAbstract:Title: On GIT stability in mirrors to toric varieties
Abstract: Using geometric invariant theory, Collins and Yau have produced obstructions to the existence of solutions to deformed Hermitian Yang-Mills on line bundles on a Kahler variety. Passing to the mirror in the toric setting, we will re-interpret their obstructions as integrals over certain Lagrangians and ponder their categorical interpretation. This talk is based on joint work with Tristan Collins and Jeff Hicks.https://ed-ac-uk.zoom.us/rec/share/Dc0jyIqlE9wmNkw7tAhV2LTJ5l3pLHBjtvQ-z5-mPQdmQ0yQGUIsxWmoUNAsfAN4.dvH4oC3nRMg5WsdQ -
Hodge Seminar Pretalk: Andrew Hanlon
16th June 2022, 1:15pm to 1:45pm -- Show/hide abstractAbstract: Title: Deformed Hermitian Yang-Mills and special Lagrangians Abstract: We will see how the deformed Hermitian Yang-Mills equation can be viewed as the mirror to the special Lagrangian equation and give a brief overview of obstructions to solutions found by Collins and Yau. -
Hodge Seminar: Noah Snyder (Indiana)
26th May 2022, 3:00pm to 4:00pm JCMB 5323 and Zoom -- Show/hide abstractAbstract:
Title: String diagrams and explicit descriptions of homotopy groups
Abstract If X is an n-connected space, then \Omega^n X can be thought of as a (grouplike) E_n-monoidal \infty-groupoid, and then one can study it using string diagrams and their generalizations the way one studies tensor categories or braided tensor categories via string diagrams. Although the language in the last sentence is quite modern, one can think of this approach as just a generalization of the Pontryagin-Thom construction, and so it goes back to the earliest days of calculating homotopy groups. This suggests as a natural goal not merely computing homotopy groups abstractly, but instead giving explicit string diagram descriptions of the elements and of the relations. For example, following Pontryagin, the generator of \pi_3(S^2) is given by a "figure-8" framed unknot. My main motivation for asking these kinds of questions is to be able to compute explicitly the SO(3)-action on the space finite tensor categories guaranteed by the cobordism hypothesis. The main examples I'll discuss are \Omega^n S^n for small n (i.e. the homotopy groups of spheres) and \Omega^2 \Sigma RP^2 which comes up in studying BSO(3). The former examples will touch on work of Pontryagin, a MO answer of Schommer-Pries, and a calculation by Henriques, while the latter example is relevant to SO(3) actions on tensor categories, as mentioned above, and will appear in joint work in progress with Douglas, Reutter, and Schommer-Pries.
Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/gMmVxYoxc8I3prC1t3UskpNHQRJLF5A_bw_OpNtZ0Sbd8N6CXShHeuKH3ebtCwk._pcT5ErLeaFJFn78?startTime=1653573924000 -
Hodge Seminar: Dinakar Muthiah (Glasgow)
26th May 2022, 1:45pm to 2:45pm JCMB 5323 and Zoom -- Show/hide abstractAbstract:
Title: Fundamental monopole operators and affine Grassmannian slices
Abstract:Affine Grassmannians are objects of central interest in geometric representation theory. For example, the geometric Satake correspondence tells us that their singularities carry representation theoretic information. In fact, it suffices to work with affine Grassmannian slices, which retain all of this information.Recently, Braverman, Finkelberg, and Nakajima showed that affine Grassmannian slices arise as Coulomb branches of certain quiver gauge theories. Remarkably, their construction works in Kac-Moody type as well. Their work opens the door to studying affine Grassmannians for Kac-Moody groups. Unfortunately, it is difficult at present to do any explicit geometry with the Coulomb branch definition. For example, a basic feature is that affine Grassmannian slices embed into one another. However, this is not apparent from the Coulomb branch definition. In this talk, I will explain why these embeddings are necessarily subtle. Nonetheless, I will show a way to construct the embeddings using fundamental monopole operators.This is joint work in progress with Alex Weekes.Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/gMmVxYoxc8I3prC1t3UskpNHQRJLF5A_bw_OpNtZ0Sbd8N6CXShHeuKH3ebtCwk._pcT5ErLeaFJFn78?startTime=1653569487000 -
Hodge Seminar: Pretalk
26th May 2022, 1:00pm to 1:30pm -
Hodge Seminar: Tobias Dyckerhoff (Hamburg)
19th May 2022, 2:00pm to 3:00pm Zoom -- Show/hide abstractAbstract:Title: Complexes of derived categories
Abstract: Derived categories have come to play adecisive role in a wide range of topics within algebra, geometry, and topology. Several recent developments, in particular inthe context of topological Fukaya categories, arouse the desire to study not just singlecategories, but rather complexes of categories. In this talk, we will discuss examples of suchcomplexes in algebra, topology, algebraic geometry, and symplectic geometry, along with someresults involving them.Zoom recording: https://ed-ac-uk.zoom.us/rec/share/TUNWdGdeJ70pbaRWqaTqDiphVV7BN2aloh82FlosJDMqDGLaU87SQygTntmBwIl3.EGvJqRTyAs8WKeKU -
Hodge Seminar pretalk: Tobias Dyckerhoff (Hamburg)
19th May 2022, 1:15pm to 1:45pm -
Hodge Seminar: Ursula Whitcher (University of Michigan)
17th May 2022, 2:00pm to 3:00pm JCMB 5323 -- Show/hide abstractAbstract:Talk Title: Strong arithmetic mirror and Calabi-Yau pencils
Abstract: Mirror symmetry predicts surprising geometric correspondences between distinct families of algebraic varieties. In some cases, these correspondences have arithmetic consequences. For example, one can use mirror symmetry to explore the structure of the zeta function, which encapsulates information about the number of points on a variety over a finite field. We describe a factor of the zeta functions of invertible Calabi-Yau pencils and suggest strategies for predicting similar arithmetic structures in other Calabi-Yau pencils.Recording: https://ed-ac-uk.zoom.us/rec/share/kMg5t2ykr2OlGw_ZTZ5Saoh8GHFfK9mvmH37B6-Slt_466HXdeaHCaBEOXLd04Oc.qjn9V6mw0W_Aj3wN -
Hodge Seminar pretalk: Ursula Whitcher
17th May 2022, 1:15pm to 1:45pm -- Show/hide abstractAbstract:A gentle introduction to classical mirror symmetry- Hodge Seminar: Yin Li (Edinburgh)
5th May 2022, 2:00pm to 3:00pm JCMB 5323 and Zoom -- Show/hide abstractAbstract:Title: Cyclic dilations and closed Lagrangian submanifolds
Abstract: Given a closed, possibly non-exact Lagrangian submanifold L in a Liouville manifold M, one can define a Maurer-Cartan element in the (completed) string homology of L with respect to an L_\infty structure. When the first Gutt-Hutchings capacity of M is finite, and L is topologically a K(\pi,1) space, the existence of such a Maurer-Cartan element implies that L bounds a non-constant pseudoholomorphic disc of Maslov index 2. When M is 6-dimensional, this leads to a topological classification of closed, prime Lagrangian 3-fold in M, generalizing a theorem of Fukaya and Irie.Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/W-Y-otJBED4pQKP8jUcNE619zxxZmiTMoKivXVwHuVnpoWm1nz0fKSm-zPdXfu-v.K1-CMIv55pmeXOPr- Whittaker Lecture: Claire Voisin (IMJ-PRG), Evgeny Shinder (Sheffield)
29th April 2022, 12:30pm to 4:00pm Lecture Theatre C, JCMB -- Show/hide abstractAbstract: Schedule: 12:30-13:30 Lunch (room 5214) 13:30-14:30 Evgeny Shinder 14:30-15:00 Coffee break (room 5214) 15:00-16:00 Claire Voisin (third lecture) See https://www.maxwell.ac.uk/events/ and https://www.maxwell.ac.uk/events/distinguished_lectures/- Whittaker Lecture: Claire Voisin (IMJ-PRG), Egor Yasinsky (Paris)
28th April 2022, 1:00pm to 5:00pm Lecture Theatre C, JCMB -- Show/hide abstractAbstract:Schedule:
13:00-14:00 Lunch (room 5214)
14:00-15:00 Egor Yasinsky
15:00-15:30 Coffee break (room 5214)
15:30-16:30 Claire Voisin (second lecture)
16:30-17:00 Wine reception (room 5214)
See https://www.maxwell.ac.uk/events/distinguished_lectures/
for more details.- Whittaker Lecture: Claire Voisin (IMJ-PRG)
27th April 2022, 1:00pm to 2:30pm Lecture Theatre C, JCMB -- Show/hide abstractAbstract:Schedule: 13:00-14:00 Lunch (room 5214)
14:00-15:00 Claire Voisin (first lecture)
See https://www.maxwell.ac.uk/events/distinguished_lectures/
for more details.- Hodge Seminar: Abigail Ward (MIT)
7th April 2022, 2:00pm to 3:00pm JCMB 5323 and Zoom -- Show/hide abstractAbstract:Symplectomorphisms mirror to birational transformations of the complex plane
We construct a non-finite type four-dimensional Weinstein domain M_{univ} and describe a HMS-type correspondence between certain birational transformations of the complex plane preserving a standard holomorphic volume form and symplectomorphisms of M_{univ}. The space M_{univ} is universal in the sense it admits every Liouville four-manifold mirror to a log Calabi-Yau surface as a Weinstein subdomain; our construction recovers a mirror correspondence between the automorphism group of any open log Calabi-Yau surface and the symplectomorphism group of its mirror by restriction to these subdomains. This is joint work in progress with Ailsa Keating.Zoom recording: https://ed-ac-uk.zoom.us/rec/share/oB8W1uKHfN7b6Rk4FLtOpQ2Mnm6xo3dPrP5d9zpFdc6Oc-BCVaH4Vrjcol_2_R8p.D9etHGaJTIVA0qQO - Hodge Seminar Pretalk: Abigail Ward (MIT)
7th April 2022, 1:15pm to 1:45pm JCMB 5323 and Zoom -- Show/hide abstractAbstract: Pretalk Title: Homological mirror symmetry for log Calabi-Yau surfaces
Abstract:
I'll discuss work of Hacking and Keating proving homological mirror symmetry for log Calabi-Yau surfaces, as well as an alternate approach, in progress, joint with Andrew Hanlon.- Hodge Seminar: Quoc Ho (HKUST)
31st March 2022, 2:00pm to 3:00pm JCMB 5323 and Zoom -- Show/hide abstractAbstract:Title: Revisiting mixed geometry
I will present joint work with Penghui Li on our theory of graded sheaves which provides a uniform construction of "mixed versions" or "graded lifts" in the sense of Beilinson--Ginzburg--Soergel which works for arbitrary Artin stacks. In particular, we obtain a general construction of graded lifts of many categories arising in geometric representation theory and categorified knot invariants. Our sheaf theory comes with a six-functor formalism, a perverse t-structure in the sense of Beilinson--Bernstein--Deligne--Gabber, and a weight (or co-t-)structure in the sense of Bondarko and Pauksztello, all compatible, in a precise sense, with the six-functor formalism, perverse t-structures, and Frobenius weights on ell-adic sheaves. Historically, constructions of graded lifts were done on a case-by-case basis and were technically subtle, due to Frobenius' non-semisimplicity. Our construction sidesteps this issue by semi-simplifying the Frobenius action itself. However, our categories agree with those previously constructed when they are available. For example, the monoidal DG-category of chain complexes of Soergel bimodules is equivalent to the category of constructible graded sheaves on B\G/B.Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/tj9yD-9_brz7_8XhSAxyLoZqGdBlncmeHvxphVCC-1eJmmPyKchYgyunI7R08Gqw.QtWE2At3kFeblzFr?startTime=1648731717000 - Hodge Seminar: Adeel Khan (Academia Sinica)
24th March 2022, 2:00pm to 3:00pm Bayes 5.10 and Zoom -- Show/hide abstractAbstract:Title: Virtual fundamental classes and categorification
Abstract: I will discuss a derived analogue of specialization and microlocalization of sheaves (in the sense of Verdier and Kashiwara-Schapira), and explain how these can be regarded as categorifications of Kontsevich's virtual fundamental class. I will also discuss an interesting relationship with Joyce's categorification of Donaldson-Thomas theory.Zoom links:
https://ed-ac-uk.zoom.us/rec/share/nT_V1MD18zmZm8RnO_Zv1GOk4VFzSGX5hqdd2ChFKiZqEy8C5cXKIIpdyOd-hT-q.s5NL8ZkJaQqqxXDK- Hodge Seminar: Emily Norton (Kent)
17th March 2022, 2:00pm to 3:00pm JCMB seminar room and Zoom -- Show/hide abstractAbstract:Title: Calibrated representations of cyclotomic Hecke algebras at roots of unity
Abstract: The cyclotomic Hecke algebra is a "higher level" version of the Iwahori-Hecke algebra of the symmetric group. It depends on a collection of parameters, and its combinatorics involves multipartitions instead of partitions. We are interested in the case when the parameters are roots of unity. In general, we cannot hope for closed-form character formulas of the irreducible representations. However, a certain type of representation called "calibrated" is more tractable: those representations on which the Jucys-Murphy elements act semisimply. We classify the calibrated representations in terms of their Young diagrams, give a multiplicity-free formula for their characters, and homologically construct them via BGG resolutions. This is joint work with Chris Bowman and José Simental.Zoom Recording:
https://ed-ac-uk.zoom.us/rec/share/VFg5IOEpI7MKgLD5GW8xYgJgxsruHkuK9vpnu93peFSCJPbdILzraPH_mt_1Aeeg.C6iGIodvAv5mSr0T?startTime=1647525985000
- Hodge Seminar: Ryo Fujita (Université de Paris)
10th March 2022, 2:00pm to 3:00pm Bayes 5.10 and Zoom -- Show/hide abstractAbstract:Title: Deformed Cartan matrices and generalized preprojective algebras
Abstract: In their study of the deformed W-algebras associated with complex simple Lie algebras, E. Frenkel-Reshetikhin (1998) introduced certain two parameter deformations of the Cartan matrices. They play an important role in the representation theory of quantum affine algebras. In this talk, we explain a representation-theoretic interpretation of these deformed Cartan matrices and their inverses in terms of the generalized preprojective algebras recently introduced by Geiss-Leclerc-Schröer (2017). We also discuss its application to the representation theory of quantum affine algebras in connection with the theory of cluster algebras. This is a joint work with Kota Murakami (arXiv: 2109.07985 ).Zoom recording: https://ed-ac-uk.zoom.us/rec/share/ku6Y00dIOuXM5pIZLzZPqd3bzDUyYMmoWKUfloXIddvWm8Q8GpklEtXCzm3nK7KO.gRCWQQ_JP6EaYDIr - Hodge Seminar pretalk: Ryo Fujita (Université de Paris)
10th March 2022, 1:15pm to 1:45pm- Hodge Seminar: Eirini Chavli (University of Stuttgart)
3rd March 2022, 2:00pm to 3:00pm JCMB seminar room and Zoom -- Show/hide abstractAbstract:Title: The center of the generic Hecke algebra
Abstract: In 1997 M. Geck and R. Rouquier described the center of the Heckealgebra for the case of Coxeter groups. Such a description for the center
of the Hecke algebra in the complex case is still unknown, apart from
the cases of the reflection groups $G_4$ and $G(4,1,2)$, thanks to A.
Francis. In this talk, we give a new description of the center of the
Hecke algebra in the real case, a description one can generalize to the
complex case. Using GAP, we also give a basis of the center for some
exceptional groups of rank 2 (common work with G. Pfeiffer).Zoom recording: https://ed-ac-uk.zoom.us/rec/share/hCdwQq5mZO3fynPL6W-CSZ0_QPx1z0PRo3fUPAvfBNEgNFGRB2z9L0XaSfjgqsb2.ZwspSpIoRY2tGUl7 - Hodge Seminar: Tomasz Przezdziecki (Edinburgh)
24th February 2022, 2:00pm to 3:00pm JCMB seminar room and Zoom -- Show/hide abstractAbstract:Title: KLR algebras and quantum symmetric pairs - categorification, combinatorics and Schur-Weyl duality
Abstract: We consider a family of KLR-type algebras associated to the infinite linear quiver with an involution and a framing. We approach their representation theory through categorification and combinatorics. Orientifold KLR algebras are known to categorify highest weight modules over the Enomoto-Kashiwara algebra and their canonical bases. Using a new shuffle realization of these modules, together with a combinatorial construction of their PBW and canonical bases in terms of Lyndon words, we are able to classify irreducibles and prove finite global dimension.
The second part of the talk is devoted to the connection between orientifold KLR algebras and quantum Kac-Moody symmetric pairs. The key role is played by a Schur-Weyl duality type functor, generalizing a construction due to Kang, Kashiwara and Kim. We explain the connection between the framing in the definition of the orientifold KLR algebra, parameters of a coideal subalgebra, and the poles of meromorphic K-matrices. We present some properties of this functor and formulate a conjecture that, under appropriate assumptions, it induces an equivalence of categories.
Zoom recording:
- Hodge Seminar: Vladimir Fock (IRMA)
17th February 2022, 2:00pm to 3:00pm Bayes 5.10 and Zoom -- Show/hide abstractAbstract:Title: Hecke algebras and TQFT.
Abstract: An open-closed two-dimensional topological field theory associatesvector spaces to surfaces with marked points on the boundary. The simplest examples of such theories can be constructed from finite dimensional algebras with trace. We will describe the TQFT related to Hecke algebra in details and explain its relation to counting of coverings of surfaces. In the introduction I'll try to explain basics about Hecke algebras, what is Satake correspondence and why it is interesting to generalize it to surfaces.Zoom recordings: https://ed-ac-uk.zoom.us/rec/share/Q4SU_hhV9v4-c9JBfwe79ebzEhbxrZ_8cd4DWwcVZD84NxY8bITSigqrfgRYnCSL.Fg7nMiUUYTqyi_0r?startTime=1645106572000- Hodge Seminar pretalk: Vladimir Fock (IRMA)
17th February 2022, 1:15pm to 1:45pm -- Show/hide abstract- Hodge Seminar: Hamid Abban (Loughborough)
10th February 2022, 2:00pm to 3:00pm Bayes 5.10 and Zoom -- Show/hide abstractAbstract:
Title: An inductive approach to K-stability via linear algebra
Abstract: K-stability is an algebraic condition that detects the existence of Kähler-Einstein metrics on Fano manifolds. However, it has many other fruitful properties. In this talk, I will give a shortcut introduction to K-stability from an algebraic perspective. Then I discuss various techniques for checking K-stability of a given Fano variety. I will explain the main ideas behind a new method for verifying K-stability which relies on (complicated) induction and (easy) linear algebra. The method will be illustrated by several examples and results. This is a joint work with Ziquan Zhuang.zoom recording: - Hodge Seminar pretalk: Hamid Abban (Loughborough)
10th February 2022, 1:00pm to 2:00pm- Hodge Seminar: Tristan Bozec (IMAG)
3rd February 2022, 2:00pm to 3:00pm Bayes 5.10 and Zoom -- Show/hide abstractAbstract:Title: Quiver moduli and Calabi--Yau structures.
Abstract: In this talk I will describe a procedure, based on constrained or relative critical loci, that allows to construct lagrangian subvarieties inside symplectic quiver varieties. We will see how this works on the example of the Hilbert scheme of points on the plane. I will explain using derived geometry that the noncommutative counterpart of lagrangian strucutres is given by so-called relative Calabi--Yau ones, involving for instance generalizations of Ginzburg dg-algebras. If time permits I will give applications to multiplicative variants linked to Poisson and Hamiltonian geometries. This is a joint work with Damien Calaque and Sarah Scherotzke.
Zoom recording:
https://ed-ac-uk.zoom.us/rec/share/9avliEQhJzmOQsrqSUS5n30doWgC4hCFb7gdSu0OzNdiYyimVIAP1gc9z1P3oL1K.QILC326V3hziTyeR- Hodge Seminar pretalk: Tristan Bozec
3rd February 2022, 1:00pm to 2:00pm- Hodge Seminar: Sukjoo Lee (Edinburgh)
27th January 2022, 2:00pm to 3:00pm Bayes 5.10 and Zoom -- Show/hide abstractAbstract:
Title: The mirror P=W conjecture from Homological Mirror Symmetry
Abstract: The mirror P=W conjecture, recently formulated by A.Harder, L.Katzarkov and V.Przyjalkowski, is the refined Hodge number symmetry between a log Calabi-Yau mirror pair $(U, U^\vee)$. It predicts that the weight filtration on the cohomology $H^\bullet(U)$ is equivalent to the perverse filtration on the cohomology $H^\bullet(U^\vee)$ associated to the mirror Landau-Ginzburg potentials. In this talk, I will introduce the precise formulation of the conjecture and discuss how to see this from the categorical viewpoint when $U$ admits a Fano compactification.Zoom links:
https://ed-ac-uk.zoom.us/rec/share/e4PdRGMyWroaRVRVXckvRqBMJ2zJxa56DydNCaVHk51kS8fqfJZNMsDg-PHkgIH7.ULFc9n30Zn2ekBBg- Hodge Seminar: Hongdi Huang (Rice University)
8th December 2021, 3:00pm to 4:00pm Bayes Centre 5.10 and Zoom -- Show/hide abstractAbstract:Title: Twisting of graded quantum groups and solutions to the quantum Yang-Baxter equation
Let $H$ be a Hopf algebra over a field $k$ such that $H$ is $\mathbb Z$-graded as an algebra. In this talk, we introduce the notion of a twisting pair for $H$ and show that the Zhang twist of $H$ by such a pair can be realized as a 2-cocycle twist. As an application of twisting pairs, we discuss an algorithm to produce a family of solutions to the quantum Yang-Baxter equation from a given solution via the Faddeev-Reshetikhin-Takhtajan construction.
Recording: Hodge Seminar 8 Dec - Zoom- Hodge Seminar: Matthew Pressland (University of Leeds)
8th December 2021, 1:45pm to 2:45pm Bayes Centre 5.10 and Zoom -- Show/hide abstractAbstract:
Title: Categorification of positroids and positroid varieties
Abstract: The Grassmannian and its totally positive part have a very rich combinatorial structure, studied by many people. In particular, Postnikov has shown how the totally positive Grassmannian is stratified by positroid varieties. Recent work of Galashin and Lam shows that the coordinate ring of each (open) positroid stratum is a cluster algebra, with this structure determined by a combinatorial object called a Postnikov diagram. In this talk I will explain how the same diagram also gives rise to representation theoretic objects which can be used to (additively) categorify this cluster algebra. This is partly joint work with İlke Çanakçı and Alastair King.
Recording : Hodge Seminar 8 Dec - Zoom- Pretalk: Matthew Pressland
8th December 2021, 1:00pm to 1:30pm -- Show/hide abstractAbstract:
Title: Grassmannian cluster algebras
Abstract: For this pretalk, I will give an overview of the Grassmannian, its ring of functions and its positroid stratification. I will review the definition of a cluster algebra, and hint at its relationship to the Grassmannian: the main talk will make this connection more precise.- Hodge Seminar: Cancelled
2nd December 2021, 1:00pm to 2:00pm JCMB 5323 and Zoom- Hodge Seminar: Orsola Capovilla-Searle (UC Davis)
24th November 2021, 3:00pm to 4:00pm Online talk, live on Bayes Centre 5.10 and Zoom -- Show/hide abstractAbstract:
Title: Infinitely many Lagrangian Tori in Milnor fibers constructed via Lagrangian Fillings of Legendrian links
Abstract: One approach to studying symplectic manifolds with contact boundary is to consider Lagrangian submanifolds with Legendrian boundary; in particular, one can study exact Lagrangian fillings of Legendrian links. There are still many open questions on the spaces of exact Lagrangian fillings of Legendrian links in the standard contact 3-sphere, and one can use Floer theoretic invariants to study such fillings. We show that a family of oriented Legendrian links has infinitely many distinct exact orientable Lagrangian fillings up to Hamiltonian isotopy. Within this family, we provide some of the first examples of a Legendrian link that admits infinitely many planar exact Lagrangian fillings. Weinstein domains are examples of symplectic manifolds with contact boundary that have a handle decomposition compatible with the symplectic structure of the manifold. Weinstein handlebody diagrams are given by projections of Legendrian submanifolds. We provide Weinstein handlebody diagrams of the 4-dimensional Milnor fibers of T_{p,q,r} singularities, which we then use to construct infinitely many Lagrangian tori and spheres in these spaces.
Recording:https://ed-ac-uk.zoom.us/rec/share/RBxuXV8H9c-ItJJxbKaWofYEFXgSz1HSOpR-vtd7C8rP_asJsOMjUbxMC2qGehc.Vbw9fQ9YwF8GCtV_?startTime=1637768023000 - Hodge Seminar: Junliang Shen (Yale)
24th November 2021, 1:45pm to 2:45pm Online talk, live on Bayes Centre 5.10 and Zoom -- Show/hide abstractAbstract:
Title: Cohomology of the moduli of Higgs bundles via positive characteristic
Abstract: In this talk, I will explain how techniques arising from the non-abelian Hodge theory in positive characteristic provide "consistency checks" of the P=W conjecture, where the latter concerns the cohomological aspect of the non-abelian Hodge theory over the complex numbers. We will focus on two aspects: (1) the Galois conjugation, and (2) the Hodge-Tate decomposition. Based on joint work with Mark de Cataldo, Davesh Maulik, and Siqing Zhang.
Recording:https://ed-ac-uk.zoom.us/rec/share/RBxuXV8H9c-ItJJxbKaWofYEFXgSz1HSOpR-vtd7C8rP_asJsOMjUbxMC2qGehc.Vbw9fQ9YwF8GCtV_?startTime=1637762003000 - Hodge Seminar pretalk: Junliang Shen (Yale)
24th November 2021, 1:00pm to 1:30pm Online talk, live on Bayes Centre 5th floor and Zoom -- Show/hide abstractAbstract:
Pretalk title: "Non-perverse" perverse filtrations
Abstract: This is an introduction to perverse filtrations associated with projective morphisms. These are main characters in the P=W conjecture as we will see in the main talk.- Hodge Seminar: Wille Liu (Max Plank)
18th November 2021, 3:45pm to 4:45pm Bayes Centre 5.10 and Zoom -- Show/hide abstractAbstract: Recording:https://ed-ac-uk.zoom.us/rec/share/Kc2N8FurJOKmSs3vIZpIquzVMAKz0V2fw5-AglrIVyWmNXW1O9FpA7ebu2ENhAhK.yEETtkii1Hhy-omj?startTime=1637245328000 - Hodge Seminar: Elie Casbi (Max Plank)
18th November 2021, 2:15pm to 3:15pm Bayes Centre 5.10 and Zoom -- Show/hide abstractAbstract: Recording :https://ed-ac-uk.zoom.us/rec/share/Kc2N8FurJOKmSs3vIZpIquzVMAKz0V2fw5-AglrIVyWmNXW1O9FpA7ebu2ENhAhK.yEETtkii1Hhy-omj?startTime=1637240823000 - Hodge Seminar: Vyjayanthi Chari (UC Riverside)
18th November 2021, 1:00pm to 2:00pm Bayes Centre 5.10 and Zoom- Hodge Seminar: Umut Varolgunes (Edinburgh)
4th November 2021, 1:00pm to 2:00pm Bayes Centre 5.10 and Zoom -- Show/hide abstractAbstract: Title: Trying to quantify Gromov's non-squeezing theorem
Abstract: Gromov's celebrated result says (colloquially) that one cannot symplectically embed a ball of radius 1.1 into a cylinder of radius 1. I will show that in 4d if one removes from this ball a Lagrangian plane passing through the origin, then such an embedding becomes possible. I will also show that this gives the smallest Minkowski dimension of a closed subset with this property. I will end with many questions. This is based on joint work with K. Sackel, A. Song and J. Zhu.
Recording :https://ed-ac-uk.zoom.us/rec/share/tHyPq8xqDLl_7JCFMqq01YCd1qmjaPUwHYeJiMf8I6TIQX5GM39nfg8BHbzZnYrk.kHes3x5Xj7eD2ryp?startTime=1636031090000 - Hodge Seminar: Dougal Davis (Edinburgh)
27th October 2021, 3:00pm to 4:00pm Bayes Centre 5.10 and Zoom -- Show/hide abstractAbstract:Title: Mixed Hodge modules, Lusztig-Vogan polynomials, and unitary representations of real reductive groups
Abstract: Let (G, K) be the symmetric pair associated with a real reductive group G_R. In this talk, I will explain joint work in progress with Kari Vilonen concerning K-equivariant twisted mixed Hodge modules on the flag variety of G, and an application to the representation theory of G_R. The Grothendieck group of mixed Hodge modules, which enhances the Grothendieck group of G_R-modules, has two bases consisting of standard and irreducible objects. At the level of weights, the change of basis matrix was computed algorithmically by Kazhdan-Lusztig and Lusztig-Vogan in terms of Hecke algebra combinatorics. Our first main theorem upgrades this to the full Grothendieck group by adding an extra Hodge parameter to the Lusztig-Vogan polynomials. Our second main theorem is a "polarised" version of the Jantzen conjecture; following ideas of Schmid and Vilonen, it allows the signature multiplicity polynomial of Adams-van Leeuwen-Trapa-Vogan to be read off from our Hodge polynomial. These two results combined recover a key formula in the ALTV algorithm for identifying the unitary representations of G_R.
- Hodge Seminar: Franco Rota (Glasgow)
27th October 2021, 1:45pm to 2:45pm Bayes Centre 5.10 and Zoom -- Show/hide abstractAbstract:Title: Motivic semi-orthogonal decompositions for abelian varieties
Abstract: A motivic semiorthogonal decomposition is the decomposition of the derived category of a quotient stack [X/G] into components related to the "fixed-point data". They represent a categorical analog of the Atiyah-Bott localization formula in equivariant cohomology, and their existence is conjectured for finite G (and an additional smoothess assumption) by Polishchuk and Van den Bergh. I will present joint work with Bronson Lim, in which we construct a motivic semiorthogonal decomposition for a wide class of smooth quotients of abelian varieties by finite groups, using the classification by Auffarth, Lucchini Arteche, and Quezada.
- Hodge Seminar pretalk: Franco Rota (Glasgow)
27th October 2021, 1:00pm to 1:30pm- Hodge Seminar: Sasha Minets (Edinburgh)
21st October 2021, 1:00pm to 2:00pm Bayes Centre 5.10 and Zoom -- Show/hide abstractAbstract: KLR algebras in positive characteristic and their stratifications Abstract:For a given quiver, modules over KLR algebras are used to categorify the corresponding quantum group. While for Dynkin quivers representation theory of KLR algebras is fairly well understood, it becomes much more intricate for affine quivers, especially in positive characteristic. In this talk, I will explain how to obtain some structural results in this case by studying analogues of KLR algebras associated to curves and surfaces. This is based on ongoing work with Ruslan Maksimau.- Hodge Seminar: Wahei Hara (Glasgow)
13th October 2021, 3:00pm to 4:00pm Bayes Centre 5.02 and Zoom -- Show/hide abstractAbstract:Global generation of instanton bundles of charge 3 on del Pezzo threefolds of degree 4
Abstract: We show that any del Pezzo threefold of degree 4 admits an instanton bundle E of charge 3 such that E(1) is globally generated. This question is the most important part in our classification of weak Fano bundles on del Pezzo threefolds of degree 4. We study elliptic curves of degree 7 and show that any del Pezzo threefold of degree 4 contains such curves that are generated by quadratic equations using the deformation theory, and then we construct the desired instanton bundles by Hartshorne-Serre correspondence. This talk depends on a joint works with T. Fukuoka and D. Ishikawa.- Hodge Seminar: Shengxuan Liu (Warwick)
13th October 2021, 1:45pm to 2:45pm Bayes Centre 5.02 and Zoom -- Show/hide abstractAbstract:Stability condition on Calabi-Yau threefold of complete intersection of quadratic and quartic hypersurfaces
Abstract: In this talk, I will first introduce the background of Bridgeland stability condition. Then I will mention some existence result of Bridgeland stability. Next I will prove the Bogomolov-Gieseker type inequality of X_(2,4), Calabi-Yau threefold of complete intersection of quadratic and quartic hypersufaces, by proving the Clifford type inequality of the curve X_(2,2,2,4). Then this will provide the existence of Bridgeland stability condition of X_(2,4).- Hodge seminar pretalk: Shengxuan Liu (Warwick)
13th October 2021, 1:00pm to 1:30pm -- Show/hide abstractAbstract:pre-talk title: A short introduction to stability condition
- Hodge Seminar: Lucien Hennecart (Edinburgh)
7th October 2021, 1:00pm to 2:00pm Bayes Centre 5.10 and Zoom -- Show/hide abstractAbstract: Title: The number of isoclasses of absolutely indecomposable representations of the modular group is a polynomial Abstract: We will explain how to prove that the number of isomorphism classes of absolutely indecomposable representations of the modular group over a finite field is a polynomial (with integer coefficients) in the cardinality of the finite field. For this, we consider the stack of representations, its inertia stack and the nilpotent version of the inertia stack. By standard techniques, we reduce the question to the calculation of the number of points of the nilpotent inertia stack. This is done using a Jordan stratification and favourable homological properties. We will sketch the ideas to prove the positivity of the coefficients of the counting polynomial, which include a purity property of the representation stack. This is ongoing joint work with Fabian Korthauer.- Hodge Seminar: Kostya Tolmachov (Edinburgh)
29th September 2021, 3:00pm to 4:00pm Bayes Centre 5.10 and Zoom -- Show/hide abstractAbstract:Monodromic model for Khovanov-Rozansky homology - Hodge Seminar: Léa Bittmann (Edinburgh)
29th September 2021, 1:45pm to 2:45pm Bayes Centre 5.10 and Zoom -- Show/hide abstractAbstract:A Schur-Weyl duality between Double Afffine Hecke Algebras and quantum groups - Jeff Hicks: Realizability and Unobstructedness
8th September 2021, 2:00pm to 3:00pm Bayes Centre, The University of Edinburgh, 47 Potterrow, Edinburgh EH8 9BT, UK -- Show/hide abstractAbstract:The complex-to-tropical correspondence is the observation that, in good circumstances, complex curves in the algebraic torus can be degenerated to piecewise linear curves (tropical curves) in Euclidean space. The realization problem asks if this process can be reversed. That is, starting with the data of a tropical curve, produce a complex curve which realizes it under the log-norm map. When the tropical curve is in the real plane, then it is always possible to find a complex curve in (C*)^2 which degenerates to it; however, this is generally not the case.
https://ed-ac-uk.zoom.us/j/82494257502
Meeting ID: 824 9425 7502
A similar tropical-to-symplectic correspondence exists. In contrast to the setting above, every tropical curve can be lifted to a Lagrangian submanifold inside a symplectic space. In this talk, I'll discuss some work-in-progress showing that the Lagrangian realization problem gives us insight into the complex realization problem, and look at some specific examples highlighting how homological mirror symmetry can be used to transfer results from symplectic geometry (unobstructedness) to the realization problem.
- Jeff Hicks: Symplectic to Tropical to Complex Geometry
8th September 2021, 1:15pm to 1:45pm International Centre for Mathematical Sciences (ICMS), 47 Potterrow, Edinburgh EH8 9BT, UK -- Show/hide abstractAbstract: Mirror symmetry is a proposed relation between symplectic geometry and complex geometry; namely, there exists pairs of symplectic and complex spaces (mirror pairs) and a dictionary between their symplectic/complex invariants. The Strominger-Yau-Zaslow (SYZ) conjecture proposes that these mirror pairs should be constructed as dual torus fibrations. In this pre-talk, we'll go over some first examples of SYZ fibrations, and match up some symplectic objects with complex ones.https://ed-ac-uk.zoom.us/j/82494257502
Meeting ID: 824 9425 7502
- Web Hodge: Si Li (Tsinguha University) - Regularized integrals and quantum master equation
12th May 2021, 2:00pm to 3:00pm -- Show/hide abstractAbstract: We present an effective BV quantization theory for chiral deformation of two dimensional conformal field theories. We introduce a simple procedure (which we call regularized integral) to integrate differential forms with arbitrary holomorphic poles on Riemann surfaces. This gives a geometric renormalization method for 2d chiral quantum field theories. As an application, we construct a vertex algebra analogue of the canonical trace map in algebraic index theory.- Web Hodge: Dustin Clausen (University of Copenhagen)-Solid quasicoherent sheaves
5th May 2021, 1:15pm to 3:00pm Online -- Show/hide abstractAbstract: I will discuss an enlargement, called solid R-modules, of the usual category of R-modules for a commutative ring R. I'll also try to explain how solid R-modules can be used to give a clean proof of Serre duality. This is joint work with Peter Scholze.- Web Hodge: Yusuf Baris Kartal (Princeton) Algebraic torus actions on Fukaya categories
28th April 2021, 1:15pm to 3:00pm -- Show/hide abstractAbstract: Given symplectic manifold M, its continuous dynamics (modulo
Hamiltonians) is governed by its first cohomology. In particular, one
expects an "action" of the first cohomology on invariants such as the
Fukaya category of M. The goal of this talk is to show that under some
hypotheses (such as negative monotonicity), this action is tame, in
the sense that it can be extended to an algebraic action of an
algebraic torus. We plan to show how to use this to deduce tameness
results on the change of Lagrangian Floer homology groups under
symplectic isotopies. Time remaining, we will discuss applications to
mirror symmetry. Notes- Web Hodge: Asher Auel (Dartmouth College)- Brill-Noether theory for cubic fourfolds
21st April 2021, 1:25pm to 3:00pm Online -- Show/hide abstractAbstract: It is well-known that the study of special cubic fourfolds leads to beautiful connections between algebraic cycles, the rationality problem, K3 surfaces, hyperkaeler manifolds, and derived categories. In this talk, I’ll explain a connection to the theory of algebraic curves, via a notion of Brill-Noether theory for cubic fourfolds. Specifically, I’ll discuss several open problems related to Brill-Noether special K3 surfaces Hodge-theoretically associated to smooth cubic fourfolds. Talk notes- Web Hodge: Dougal Davis (Edinburgh)-Elliptic quantum groups as shuffle algebras
14th April 2021, 10:30am to 12:00pm Online -- Show/hide abstractAbstract: Elliptic quantum groups are the algebraic gadgets that arise from quantising functions on an elliptic curve valued in a Lie algebra, and are closely related to quantum moduli of principal bundles. They have something of the flavour of Yangians and quantum loop algebras, but due to the rich geometry of elliptic curves are much more subtle to define. In this talk, I will discuss joint work in progress with Yaping Yang and Gufang Zhao in which we give a "shuffle" realisation of the elliptic quantum group as an algebra object in an appropriate category of sheaves on a coloured Hilbert scheme. This combines a previous construction of Yang and Zhao for the positive part with a new doubling procedure to produce the full algebra. Time permitting, I will also discuss how to recover previously studied dynamical elliptic quantum groups from our sheafified one via a Fourier transform.
Pre-talk: In the pre-talk, I will give a quick introduction to the theory of Yangians (where the elliptic curve is replaced with the additive group in the above story). This should serve as a useful down-to-earth example of the more abstract constructions in the main talk.
Slides for pre-talk, Slides for main talk- Web Hodge: Benedict Morrissey (Chicago)-Nonabelianization for Reductive groups
7th April 2021, 1:15pm to 3:00pm -- Show/hide abstractAbstract: Nonabelianization provides coordinate charts on the moduli space of local systems on a curve. In the work of Gaiotto, Moore, and Neitzke this is done by giving a map from the moduli space of one dimensional local systems on a spectral curve to the moduli space of n-dimensional local systems on the original curve. I will describe joint work with M. Ionita which extends this construction from GL(n) to an arbitrary reductive algebraic group. Time permitting I will describe work in progress towards proving that these maps are symplectomorphisms (under certain conditions).- Web Hodge: Maxim Jeffs (Harvard)-Mirror symmetry and Fukaya categories of singular varieties
31st March 2021, 1:15pm to 3:00pm -- Show/hide abstractAbstract: One problem that arises when studying mirror symmetry is that even mirrors of smooth varieties are often singular, and it is difficult to make sense of the A-model for these mirrors. In this talk I will explain Auroux' definition of the Fukaya category of a singular hypersurface and two results about this definition, illustrated with some fundamental examples. The first result is analogous to Orlov's derived Knorrer periodicity theorem, relating Auroux' category to a Landau-Ginzburg model on a higher-dimensional variety; the second result is a homological mirror symmetry equivalence at certain large complex structure limits. Slides- Web Hodge: Yalong Cao (Kavli IPMU, Tokyo) - Gopakumar-Vafa type invariants for Calabi-Yau 4-folds
24th March 2021, 10:30am to 12:00pm Online -- Show/hide abstractAbstract: Gopakumar-Vafa type invariants on Calabi-Yau 4-folds (which are non-trivial only for genus zero and one) are defined by Klemm-Pandharipande from Gromov-Witten theory, and their integrality is conjectured. In this talk, I will explain how to give a sheaf theoretical interpretation of them using counting invariants on moduli spaces of one dimensional stable sheaves. Based on joint works with D. Maulik and Y. Toda.- Web Hodge: Ben Davison (Edinburgh) - Finite-dimensional Jacobi algebras, flopping curves, and BPS invariants
17th March 2021, 1:15pm to 3:00pm Online -- Show/hide abstractAbstract: Floppable curves in threefolds are a fundamental feature of the minimal model programme in 3 dimensions. Not only do they provide the stepping stones between different minimal models, but their local geometry itself turns out to be tremendously rich. The list of examples starts with Atiyah's flop, then through the infinite class in Reid's "Pagoda" into the zoo of (-3,1) curves. Associated to these curves are a number of enumerative invariants, amongst them the Gopakumar-Vafa invariants, which can be reinterpreted as modified counts of coherent sheaves on the threefold containing the curve. The contraction algebra, controlling the deformation theory of the structure sheaf of the curve, turns out to be a strictly more refined invariant. This is a finite-dimensional Jacobi algebra introduced by Donovan and Wemyss. In this talk I will explain how the Brown-Wemyss conjecture that all finite-dimensional Jacobi algebras arise as contraction algebras implies a strong positivity statement for BPS invariants of these algebras (while defining what this all means), and explain how to prove this positivity statement, while also explaining how the classical Gopakumar-Vafa invariants of floppable curves can be "categorified" using cohomological DT theory.- Web Hodge: Sam Raskin (UT Austin)- Geometric Langlands for l-adic sheaves
10th March 2021, 1:15pm to 3:00pm Online -- Show/hide abstractAbstract: In celebrated work, Beilinson-Drinfeld formulated a categorical analogue of the Langlands program for unramified automorphic forms. Their conjecture has appeared specialized to the setting of algebraic D-modules: non-holonomic D-modules play a prominent role in known constructions. In this talk, we will translate their work back into a statement suitable for (certain) automorphic functions, refining the Langlands conjectures in this setting. We will deduce this result from a categorical conjecture suitable in other geometric settings, including l-adic sheaves. One of the main constructions is a suitable moduli space of local systems. This is joint work with Arinkin, Gaitsgory, Kazhdan, Rozenblyum, and Varshavsky. In the pre-talk, we will give background on the relevant parts of the arithmetic Langlands correspondence, setting the stage for the problem considered in the main talk.- Web Hodge: Lenny Taelman (University of Amsterdam)-Derived equivalences between hyperkähler varieties
3rd March 2021, 2:00pm to 3:00pm Online -- Show/hide abstractAbstract: We study equivalences D(X)-->D(Y) between the derived categories of coherent sheaves on complex hyperkähler varieties X and Y. An important tool is the Looijenga--Lunts--Verbitsky Lie algebra acting on the total cohomology of X. We show that this Lie algebra is preserved
by derived equivalences, and deduce various consequences from this. Talk notes- Web Hodge: Alexei Oblomkov (Amherst) - Khovanov-Rozansky homology and sheaves on Hilbert scheme of points on the plane.
24th February 2021, 2:00pm to 3:00pm Online -- Show/hide abstractAbstract: Talk is based on the joint work with Lev Rozansky. I will explain a construction that attaches to a $n$-stranded braid $\beta$ a two-periodic complex $S_\beta$ of $\mathbb{C}^*\times \mathbb{C}^*$-equivariant sheaves on $Hilb_n(\mathbb{C}^2)$ such that the $H^*(S_\beta)$ is a categorification of the Oceanu-Jones trace. We show the corresponding link homology coincide with the triply graded Khovanov-Rozansky link homology which categorifies of Jones construction of HOMFLYPT polynomial. We also show that $S_{\beta FT}=S_\beta \otimes L$ where $FT$ is a full twist and $L$ is a generator of the Picard group of $Hilb_n(\mathbb{C}^2)$. The natural involution of $\mathcal{C}^2$ results in Poincare duality of the Khovanov-Rozansky homology (conjectured in 2005). As an application we obtain explicit an combinatorial formula for Khovanov-Rozansky homology of torus knots.- Pre-talk: Alexei Oblomkov (Amherst) - Matrix factorizations: derived algebraic geometry without tears.
24th February 2021, 1:15pm to 1:45pm Online -- Show/hide abstractAbstract: I will give a brief introduction to the theory of matrix factorizations (due to Eisenbud, Kontsevich and Orlov). In particular, I will explain how matrix factorizations help us to understand derived categories of some important classes of singular varieties.- Web Hodge: Owen Gwilliam (UMass Amherst). Title: Spontaneous symmetry breaking, a view from derived geometry
10th February 2021, 2:00pm to 3:00pm -- Show/hide abstractAbstract: We will give an overview of how physics and homological algebra have met in the setting of gauge theory, with an emphasis on how the new subject of derived geometry provides a clarifying framework. The talk's concrete aim is to explain the Higgs mechanism as a case study. Our approach will be low-tech and will emphasize the motivations; anyone familiar with notions like vector bundle and cochain complex should be able to follow.- Web Hodge: Younghan Bae (ETH)-The Chow ring of the moduli stack of prestable curves
3rd February 2021, 1:15pm to 3:00pm -- Show/hide abstractAbstract: I will discuss the Chow ring of the moduli stack of prestable (without stability condition) curves. This stack contains Deligne-Mumford's moduli space of stable curves as an open substack. When the genus is zero, we will compute the Chow ring explicitly. We will see how higher Chow groups play a role. This is a joint work with Johannes Schmitt. Notes for the pre-talk. Notes for the main talk.- Web Hodge: Yagna Dutta (Bonn) Holomorphic 1-forms and geometry
27th January 2021, 1:15pm to 3:00pm Zoom -- Show/hide abstractAbstract: In this talk I will discuss various topological and geometric consequences of the existence of zeros of global holomorphic 1-forms on smooth projective varieties. Such consequences have been indicated by a plethora of results. I will present some old and new results in this direction. One of the key points is an interesting connection between two sets of such 1-forms, one that arises out of the generic vanishing theory and the other that falls out of the decomposition theorem of the albanese morphism. This is an on-going joint work with Feng Hao and Yongqiang Liu. Notes for the pre-talk. Notes for the main talk.- Web Hodge: Jeremy Lane (McMaster)-Gradient-Hamiltonian vector fields and applications in symplectic geometry
20th January 2021, 2:00pm to 3:00pm -- Show/hide abstractAbstract: Gradient-Hamiltonian vector fields have emerged in recent years as an important bridge between algebraic geometry and symplectic geometry. As shown in the work of Nishinou-Nohara-Ueda and Harada-Kaveh, gradient-Hamiltonian vector fields allow one to construct integrable systems on smooth projective varieties from toric degenerations.
In this talk I will describe how this construction can be extended to degenerations of quasi-projective varieties which are not necessarily smooth. Such degenerations are common and arise in many important contexts, such as toric degenerations of the base affine space G//N of a reductive algebraic group G associated to Lusztig's dual canonical basis. As a consequence, we are able to construct integrable systems with nice properties such as convexity on arbitrary Hamiltonian K-manifolds, for K a compact connected Lie group. Slides- Web Hodge: Brian Williams (Edinburgh) - Dualities, vertex algebras, and holomorphic strings
13th January 2021, 1:15pm to 3:00pm -- Show/hide abstractAbstract: Abstract: Physical reasoning has been instrumental in predicting and guiding various mathematical dualities, perhaps most famously in the context of mirror symmetry. In this talk I will discuss progress towards a mathematical proof of a duality between a topological string theory, called the B-model, and a physical string theory, called the heterotic string. The duality turns out to have an elegant mathematical formulation in terms of topological vertex algebras and we describe variants and generalizations that can be understood in a similar fashion. Slides for pre talk. Slides for main talk.- Web Hodge: Eugene Gorsky (UC Davis)-Parabolic Hilbert schemes on singular curves and representation theory
16th December 2020, 1:15pm to 3:00pm -- Show/hide abstractAbstract: I will construct representations of various interesting algebras (such as rational Cherednik algebras and quantized Gieseker varieties) using the geometry of parabolic Hilbert schemes of points on plane curve singularities. A connection to Coulomb branch algebras of Braverman, Finkelberg and Nakajima will be also outlined. The talk is based on a joint work with Jose Simental and Monica Vazirani. Talk notes- Web Hodge: Fei Xie (Edinburgh)-Derived categories of quintic del Pezzo fibrations
9th December 2020, 1:15pm to 3:00pm -- Show/hide abstractAbstract: I will discuss the quintic del Pezzo surfaces with rational Gorenstein singularities over both algebraically closed and non-closed fields. I will describe a semiorthogonal decomposition (SOD) of their derived categories and generalise it to a fibration. The SOD of the fibration can be obtained via the Homological Projective Duality (HPD) method and has a moduli space interpretation. In the pre-talk, I will give an introduction to the HPD and its relation to classical projective duality. Pre-talk notes Main talk notes- Web Hodge: Yu Zhao (MIT)-A Weak Categorical Quantum Toroidal Action on the Derived Categories of Hilbert Schemes
2nd December 2020, 2:00pm to 3:00pm -- Show/hide abstractAbstract: The quantum toroidal algebra is the affinization of the quantum Heisenberg algebra. Schiffmann-Vasserot, Feigin-Tsymbaliuk and Negut studied the quantum toroidal algebra action on the Grothendieck group of Hilbert schemes of points on surfaces, which generalized the action by Nakajima and Grojnowski in cohomology. In this talk, we will categorify the above quantum toroidal algebra action. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semi-divisorial log terminal singularities. Main talk slides- Web Hodge: Yu Zhao (MIT)-Hilbert Schemes of Points on Surfaces and Heisenberg Algebra
2nd December 2020, 1:15pm to 1:45pm -- Show/hide abstractAbstract: We will introduce Nakajima and Grojownoski's construction of the Heisenberg algebra action on the cohomology of Hilbert schemes of points on surfaces. If time permits, we will also introduce the generalizations like quantum toroidal algebra action by Schiffmann-Vasserot and Feigin-Tsymbaliuk and the K-theoretic Hall algebra action by Kapranov-Vasserot and the speaker on the Grothendieck groups of Hilbert schemes. Pre-talk slides- Web Hodge: Cheuk Yu Mak (Edinburgh)-Symplectic annular Khovanov homology
25th November 2020, 1:15pm to 3:00pm -- Show/hide abstractAbstract: Annular Khovanov homology is an invariant of annular links (links in a solid torus) introduced by Asaeda-Przytycki-Sikora as an analogue of Khovanov homology for links. Auroux-Grigsby-Wehrli showed that the first non-trivial piece of the annular Khovanov homology can be identified with the Hochschild homology of the Fukaya-Seidel category of A_n Milnor fibers with coefficients in braid bimodules. In this talk, we will introduce a symplectic version of annular Khovanov homology using Hochschild homology of the Fukaya-Seidel category of more general type A nilpotent slices. Building on the work of Abouzaid-Smith and Beliakova-Putyra-Wehrli, we show that the symplectic version is isomorphic to the ordinary version. Finally, we will explain how to derive a spectral sequence from the symplectic annular Khovanov homology to the symplectic Khovanov homology directly using symplectic geometry. This is based on a joint work with Ivan Smith.
In the pre-talk, we will talk about the Fukaya-Seidel category of A_n Milnor fibers. Pre-talk notes Main talk notes- Web Hodge: Pieter Belmans (Bonn/Antwerp)-Moduli of semiorthogonal decompositions in families
18th November 2020, 2:00pm to 3:00pm -- Show/hide abstractAbstract: In a joint work with Shinnosuke Okawa and Andrea Ricolfi we have constructed a moduli space of semiorthogonal decompositions, and described some of its geometric properties. I will introduce semiorthogonal decompositions, and explain how they behave in families of smooth projective varieties. As an application I will discuss how its geometry can be used to show how certain derived categories of smooth projective varieties are indecomposable by studying indecomposability in families. This is joint work with the previous co-authors and Francesco Bastianelli. Talk notes- Web Hodge: Pieter Belmans (Bonn/Antwerp)-An introduction to semiorthogonal decompositions
18th November 2020, 1:15pm to 1:45pm -- Show/hide abstractAbstract: I will give an overview to the history of semiorthogonal decompositions. Such decompositions are an important method to understand the structure of derived categories in algebraic geometry (and neighbouring fields), and often reflect interesting geometric properties of the varieties. This overview takes us from Beilinson's collection on projective space (from 1978) and the Bondal--Orlov semiorthogonal decomposition of the intersection of quadrics (from 1995) all the way to the interesting state of the field in 2020. Talk notes- Web Hodge: Giulia Gugiatti (Edinburgh)-Hypergeometric functions and new mirrors of Fano varieties
11th November 2020, 1:15pm to 3:00pm -- Show/hide abstractAbstract: I will outline a strategy to exhibit the hypergeometric function of a Fano weighted complete intersection X of dimension n as a period of a pencil of (n-1)-dimensional varieties. Since conjecturally this function encodes the quantum period of X, these ideas can be viewed as the basis of a new method to find Landau-Ginzburg (LG) mirrors for all Fano weighted complete intersections. I will show how to use the strategy to produce LG mirrors for the Johnson-Kollar surfaces X_{8k+4} in P(2,2k+1,2k+1,4k+1). The main feature of these surfaces, which makes the construction especially interesting, is that they have empty anticanonical linear system, and therefore fall out of any known mirror construction. This is joint work with A. Corti. Talk notes- Web Hodge: Georg Oberdieck (University of Bonn)-Moduli spaces in equivariant categories
4th November 2020, 1:15pm to 2:45pm -- Show/hide abstractAbstract: I will discuss joint work with Thorsten Beckmann in which we construct moduli spaces of Bridgeland semistable objects in equivariant categories. This can be used to determine equivariant categories of symplectic surfaces,but also to define invariants of non-commutative CHL manifolds.- Web Hodge: Tudor Dimofte (Edinburgh/UC Davis)-3d mirror symmetry and HOMFLY-PT homology
28th October 2020, 1:15pm to 3:00pm -- Show/hide abstractAbstract: Since the original physical prediction of triply-graded HOMFLY-PT link homology by Gukov-Schwarz-Vafa, and its mathematical definition by Khovanov-Rozansky, many other (conjectural) constructions of HOMFLY-PT link homology have appeared --- with different algebraic and geometric origins, and manifesting different properties. One recent proposal of Oblomkov-Rozansky (closely related to work of Gorsky-Neguț-Rasmussen) associated to a link L a coherent sheaf E_L on a Hilbert scheme, whose cohomology reproduces HOMFLY-PT homology. Another proposal, by Gorsky-Oblomkov-Rasmussen-Shende, computes HOMFLY-PT homology of algebraic knots via Borel-Moore homology of affine Springer fibers. I will explain how the first (Hilbert scheme) construction is realized in the "B" twist of a 3d supersymmetric gauge theory, and then carefully apply 3d mirror symmetry to discover a variant of the second (Springer fiber) construction. I will also indicate how both 3d gauge theory setups are related to the original work of Gukov-Schwarz-Vafa based using M-theory on the resolved conifold. (Preprint soon to appear, with N. Garner, J. Hilburn, A. Oblomkov, and L. Rozansky). In the pre-talk, I will review some features of 3d gauge theories and their A and B twists. Talk notes- Web Hodge: Hiro Tanaka (Texas State University)-Enriching Fukaya categories over stable homotopy theory using broken techniques
21st October 2020, 1:15pm to 3:00pm -- Show/hide abstractAbstract: It has been a mission (since at least the 25-year-old work of Cohen-Jones-Segal) to enrich various Lagrangian Floer theories over spectra in the sense of stable homotopy theory. While the topological obstructions to such a construction are well understood, and various proposals have been made to assign a stable homotopy type to a pair of Lagrangians, to date there has been no success in creating a "spectral Fukaya category" (to define not only the spectra between two Lagrangians, but to define the composition maps coherently). In the thirty-minute pre-talk, I will discuss some background on spectra and their relation to usual (homological) algebra. In the main talk, I will discuss joint work with Jacob Lurie to finally construct spectral Fukaya categories for a large class of exact symplectic manifolds. The main innovative input is a stack of broken holomorphic disks (the moduli stack of holomorphic disks in a point), which serves as a conduit to encode both the geometry of moduli of disks and the algebra of Koszul duality. We will focus on the test case of Morse theory to illustrate most of the new principles.- Web Hodge: Rekha Biswal (MPIM Bonn)-Macdonald polynomials and level two Demazure modules for affine sl_{n+1}
14th October 2020, 1:15pm to 3:00pm -- Show/hide abstractAbstract:
An important result due to Sanderson and Ion says that characters of level one Demazure modules are Macdonald polynomials specialized at t=0. In this talk, I will introduce a new class of symmetric polynomials expressed as linear combination of Macdonald polynomials. Using representation theory, we will see that these new class of polynomials are graded characters of an interesting class of modules of the current algebra sl_{n+1}[t], interpolate between characters of level one and level two Demazure modules and gives rise to new results in representation theory of current algebras as a corollary.- Web Hodge: Be eri Greenfeld (Bar Ilan University)-Combinatorics of words, symbolic dynamics and growth of algebras
7th October 2020, 1:15pm to 3:00pm -- Show/hide abstractAbstract: The most important invariant of a finite dimensional algebra is its dimension. Let $A$ be a finitely generated, infinite dimensional associative or Lie algebra over some base field $F$. A useful way to 'measure its infinitude' is to study its growth rate, namely, the asymptotic behavior of the dimensions of the spaces spanned by (at most $n$)-fold products of some fixed generators. Up to a natural asymptotic equivalence relation, this function becomes a well-defined invariant of the algebra itself, independent of the specification of generators.
The question of 'how do algebras grow?', or, which functions can be realized as growth rates of algebras (perhaps with additional algebraic properties, as grading, simplicity etc.) plays an important role in classifying infinite dimensional algebras of certain classes, and is thus connected to ring theory, noncommutative projective geometry, quantum algebra, arithmetic geometry, combinatorics of infinite words, symbolic dynamics and more.
We present new results on possible and impossible growth rates of important classes of associative and Lie algebras, by combining novel techniques and constructions from noncommutative algebra, combinatorics of (infinite trees of) infinite words and convolution algebras of étale groupoids attached to them. Talk slides- Web Hodge: Pavel Safronov (Edinburgh)-String topology, Euler structures and topological field theories
30th September 2020, 1:15pm to 3:00pm -- Show/hide abstractAbstract: Chas and Sullivan have introduced interesting algebraic operations on the homology of the free loop space of a manifold which go under the name of the string topology operations. For surfaces they reduce to the Lie bialgebra introduced by Goldman and Turaev. In this talk I will explain that string topology forms a part of an extended two-dimensional topological field theory. I will also discuss homotopy invariance of the string topology operations and the conjecture that the string coproduct changes by the Whitehead torsion. This is a report on work in progress joint with Florian Naef. Pre-talk slides Main talk slides- Web Hodge: Filip Zivanovic (Edinburgh)-Symplectic geometry of Conical Symplectic Resolutions
23rd September 2020, 1:45pm to 3:30pm -- Show/hide abstractAbstract: Conical symplectic resolutions (CSRs) are a big family of spaces that are of central importance in Geometric Representation Theory, such as quiver varieties, Slodowy nilpotent slices, and slices in affine Grassmanians. They tend to come in pairs related by certain duality (called symplectic duality or 3d-mirror symmetry). CSRs have very interesting geometry. In particular, all the existing examples are known to be hyperkahler. In this talk, I will focus on the symplectic geometry of CSRs. I will explain how one can obtain a family of compact exact Lagrangian submanifolds (called minimal) in a CSR, whose Floer-theoretic invariants are of purely-topological nature. I will show how one can explicitly count the number of minimal Lagrangians in the case of quiver varieties of type A. Time permitting, I will show how one can get more Lagrangians, generated from the minimal ones by certain crystal operators. Talk slides- EDGE: Andrew Dancer (Oxford University)- Symplectic duality and implosion
16th June 2020, 3:30pm to 4:30pm https://msri.zoom.us/j/99482769816?pwd=YjY0ZldQYjIzaDRvc2M4L0xVQnB3UT09 Meeting ID: 994 8276 9816 Password: EDGEJune16 -- Show/hide abstractAbstract: We discuss hyperkahler implosion spaces. their relevance to group actions, and why they should fit into the symplectic duality picture. For certain groups we present candidates for the symplectic duals of the associated implosion spaces and provide computational evidence. This is joint work with Amihay Hanany and Frances Kirwan.- Special EDGE: Swarnava Mukhopadhyay (TIFR Mumbai) - Graph potentials and the moduli space of vector bundles of rank two on a curve.
4th June 2020, 3:30pm to 4:30pm http://www.math.tifr.res.in/~swarnava/edinburgh.pdf -- Show/hide abstractAbstract:In this talk, we will discuss a conjectural decomposition of the derived category of the moduli space $M$ of stable vector bundles rank two bundles and fixed determinant on a smooth algebraic curve $\Sigma$ into derived categories of symmetric products of the original curve. We will also consider natural potentials $W$ associated with a decomposition of a surface $\Sigma$ into pairs of pants. Using TQFT gluing axioms, we will show how to compute the respective periods of $W$ very fast and simultaneously for all genera.
Finally, we will explain how periods of $W$ relate to mirrors of the Fano variety $M$ and discuss mirror heuristics as well as algebro-geometric evidence towards the conjectural decomposition of $D^b(M)$. This is a joint work in progress with Sergey Galkin and Pieter Belmans.
- EDGE: Johannes Nicaise-(Imperial College London) -Stable rationality of complete intersections
2nd June 2020, 3:30pm to 4:30pm Zoom: https://us02web.zoom.us/j/9918493831?pwd=OUR4TFB6R3BGWlpnSmMxV3NUQlg0UT09 ID: 991 849 3831 Password: summer -- Show/hide abstractAbstract:
I will explain an ongoing project with John Christian Ottem to establish several new classes of stably irrational complete intersections. Our results are based on degeneration techniques and a birational version of the nearby cycles functor that was developed in collaboration with Evgeny Shinder.- EDGE: Pierrick Bousseau-(ETH Zurich)-Quasimodular forms from Betti numbers
12th May 2020, 3:30pm to 4:30pm Zoom: https://zoom.us/j/91829737031?pwd=SERqQzJXbitVVTRFa3h6L1MyZDdWZz09 ID: 918 2973 7031 Password: 003229 -- Show/hide abstractAbstract: This talk will be about refined curve counting on local P^2, the noncompact Calabi-Yau 3-fold total space of the canonical line bundle of the projective plane. I will explain how to construct quasimodular forms starting from Betti numbers of moduli spaces of one-dimensional coherent sheaves on P^2. This gives a proof of some stringy predictions about the refined topological string theory of local P^2 in the Nekrasov-Shatashvili limit. This work is in part joint with Honglu Fan, Shuai Guo, and Longting Wu.
- Web EDGE: Victoria Hoskins (Radboud University Nijmegen)-Motives of moduli spaces of bundles over a curve
28th April 2020, 3:30pm to 4:30pm Zoom: https://zoom.us/j/97727241417?pwd=WnpxdmZRdVJmZHZFTkg3YW9kenB0Zz09 ID: 977 2724 1417 Password: 014589 -- Show/hide abstractAbstract: Following Grothendieck’s vision that a motive of an algebraic variety should capture many of its cohomological invariants, Voevodsky introduced a triangulated category of motives which partially realises this idea. After describing some properties of this category, I will explain how to define the motive of certain algebraic stacks. I will then state and sketch a proof for a formula the motive of the moduli stack of vector bundles on a smooth projective curve; this formula is compatible with classical computations of invariants of this stack due to Harder, Atiyah--Bott and Behrend--Dhillon. The proof involves rigidifying this stack using Quot and Flag-Quot schemes parametrising Hecke modifications as well as a motivic version of an argument of Laumon and Heinloth on the cohomology of small maps. If there is time, I will discuss how this can be used to study motives of other related moduli spaces such as the moduli space of Higgs bundles. This is joint work with Simon Pepin Lehalleur.- Web EDGE: Gregory Sankaran (University of Bath)-Blowups with log canonical singularities
21st April 2020, 3:30pm to 4:30pm Zoom: https://zoom.us/j/93972891364?pwd=TVkyV0c2VkhuSGlpaDYyb3dCejlxQT09 ID: 939 7289 1364 Password: 031550 -- Show/hide abstractAbstract: We show that the minimum weight of a weighted blow-up of ${\mathbb A}^d$
with $\varepsilon$-log canonical singularities is bounded by a constant
dependin only on $\varepsilon$ and $d$. This was conjectured by Birkar.
Using the recent classification of 4-dimensional empty simplices by
Iglesias-Vali\~no and Santos, we work out an explicit bound for blowups of
${\mathbb A}^4$ with terminal singularities: the smallest weight is always at
most 32, and at most 6 in all but finitely many cases.- Web EDGE:Chenyang Xu (MIT)- K-moduli of Fano varieties
14th April 2020, 4:00pm to 5:00pm https://zoom.us/j/699366190?pwd=Tm95emxrdzMrdGlzNmxROGtGSHh6Zz09 Conference ID: 699 366 190 Password: 030418 -- Show/hide abstractAbstract: One main theme of the algebraic K-stability theory of Fano varieties is to use it to construct moduli spaces of K-(semi,polystable) Fano varieties. Several main ingredients have been established, based on the recent development of our understanding of K-stability theory and other inputs. In this talk, we will discuss the current status of the construction.- web EDGE: Grisha Mikhalkin (Geneva, Switzerland)- Area in real K3-surfaces
7th April 2020, 4:00pm to 5:00pm Conference ID: 870 554 816 Password: 014504, https://zoom.us/j/870554816?pwd=bERmR0ZQTitYNXJ1aFZLckxzeXZJZz09 -- Show/hide abstractAbstract: Real locus of a K3-surfaces is a multicomponent topological surface. The canonical class provides an area form on these components (well defined up to multiplication by a scalar). In the talk we'll explore inequalities on total areas of different components as well a link between such inequalities and a class of real algebraic curves called simple Harnack curves. Based on a joint work with Ilia Itenberg.- Web EDGE: Piotr Pokora (Krakow, Poland)-- Conic-line arrangements in the complex projective plane.
31st March 2020, 12:10pm to 1:10pm https://vimeo.com/403295855 -- Show/hide abstractAbstract: In my talk I would like to present an introduction to complex conic-line arrangements in the projective plane. We will start with an intriguing construction of the Chilean configuration of conics (or the Hesse arrangement of conics) which has some unexpected properties. This is going to be our Letimotif example. Then I will discuss some positivity and negativity properties related to conic-line arrangements, mostly in the context of Harbourne indices and (time permitting) Seshadri constants. At the end of my talk I would like to put conic-line arrangements into the perspective of log-surface.
Based on a joint works: with Tomasz Szemberg arXiv:2002.01760 and Marek Janasz (preprint soon on arXiv)- WebEDGE: Konstantin Shramov ( Steklov Institute and HSE) - Automorphisms of elliptic surfaces
26th March 2020, 11:15am to 12:15pm https://vimeo.com/401138481 -- Show/hide abstractAbstract: I will discuss automorphism groups acting on compact complex surfaces that have a structure of an elliptic fibration, and stabilizers of points therein. In particular, we will see that the image of an automorphism group of a surface of Kodaira dimension 1 in the automorphism group of the base of its pluricanonical fibration is finite. I will also speculate on possible
higher dimensional generalizations.- EDGE: Yanki Lekili (King’s College London)-Homological mirror symmetry for Milnor fibers via moduli of A_infty structures
18th February 2020, 3:30pm to 4:30pm Bayes Centre 5.10 -- Show/hide abstractAbstract: We show that the base of the semiuniversal unfoldings of an invertible quasi-homogeneous singularity (appearing in Arnold’s strange duality and its generalizations due to Berglund–H\"ubsch–Krawitz) can be identified with moduli spaces of A∞-structures on particular finite-dimensional graded algebras. The same algebras also appear in the Fukaya category of the mirror dual family. Based on these identifications, we give applications to homological mirror symmetry for Milnor fibers, and prove homological mirror symmetry for the affine quartic surface, the affine double plane and their higher dimensional analogues. This is joint work with Kazushi Ueda based on a preprint on arXiv.- EDGE: Giulia Sacca' (Columbia University)-Birational geometry of the intermediate Jacobian fibration
18th February 2020, 2:00pm to 3:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: A few year ago with Laza and Voisin we constructed a hyperkähler
compactification of the intermediate Jacobian fibration associated to
a general cubic fourfold. In this talk I will first show how such a
compactification J(X) exists for any smooth cubic fourfold X and then
discuss how the birational geometry of the fibration J(X) is governed
by any extra algebraic cohomology classes on J(X).- EDGE: Shizhuo Zhang (University of Edinburgh) - Exceptional collection of line bundles and Brill-Noether
11th February 2020, 3:30pm to 4:30pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Let X be a smooth projective surface over complex numbers and E be a full exceptional collection of line bundles on X. A famous conjecture of Orlov says the surfaces with such collection must be a rational surface. I will talk about several results in this direction. Then I will talk about a systematical way to construct exceptional collection of line bundles on all smooth rational surfaces introduced by Lutz Hille and Markus Perling and solutions of several conjectures of theirs. If time allows, I will talk about the application of such collections to the Brill-Noether problems on moduli space of sheaves on del Pezzo surfaces.- EDGE: Naoki Koseki (University of Edinburgh)-Stability conditions on CY double/triple covers
4th February 2020, 3:30pm to 4:30pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Recently, Chunyi Li proved the existence of Bridgeland stability conditions on quintic threefolds. In my talk, I will treat CY threefolds obtained as double/triple covers over the projective space in a similar way, and construct stability conditions on them.- EDGE: Abigail Ward (Harvard University)-Homological mirror symmetry for the Hopf surface
21st January 2020, 3:30pm to 4:30pm Bayes Centre 5.10 -- Show/hide abstractAbstract: We show evidence that homological mirror symmetry is a phenomenon that exists beyond the world of Kähler manifolds by exhibiting HMS-type results for a family of complex surfaces which includes the classical Hopf surface (S^1 x S^3). Each surface S we consider can be obtained by performing logarithmic transformations to the product of P^1 with an elliptic curve. We use this fact to associate to each S a mirror "non-algebraic Landau-Ginzburg model" and an associated Fukaya category, and then relate this Fukaya category and the derived category of coherent analytic sheaves on S.- EDGE: Francois Greer (Stony Brook University) - Lagrangian spheres which are not vanishing cycles
21st January 2020, 11:00am to 12:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Let X be a smooth projective variety with its induced Kahler structure. If X admits a degeneration to a nodal variety, then X contains a Lagrangian sphere as the vanishing cycle. Donaldson asked whether the converse holds. We answer this question in the negative for Calabi-Yau threefolds and discuss related examples from Teichmuller theory.- EDGE: Sara Tukachinsky (Institute for Advanced Study) - Counts of pseudoholomorphic curves: Definition, calculations, and more
14th January 2020, 2:00pm to 3:00pm Bayes Center 5.10 -- Show/hide abstractAbstract: Genus zero open Gromov-Witten (OGW) invariants should count pseudoholomorphic disks in a symplectic manifold X with boundary conditions in a Lagrangian submanifold L, satisfying various constraints at boundary and interior marked points. In a joint work with Jake Solomon we developed an approach for defining OGW invariants using machinery from Fukaya A_\infty algebras. In a recent work, also joint with Solomon, we find that the generating function of OGW satisfies a system of PDE called open WDVV equations. For projective spaces, open WDVV give rise to recursions that, together with other properties, allow the computation of all OGW invariants.- Special EDGE: Graeme Wilkin (University of York)-Morse theory on singular spaces
13th January 2020, 4:00pm to 5:00pm JCMB 6206 -- Show/hide abstractAbstract: Morse theory has a long history with many spectacular applications in different areas of mathematics. In this talk I will explain an extension of the main theorem of Morse theory that works for a large class of functions on singular spaces. The main example to keep in mind is that of moment maps on varieties, and I will present some applications to the topology of symplectic quotients of singular spaces.- EDGE Navid Nabijou (University of Glasgow)-Tangent curves, degenerations, and blowups.
5th December 2019, 3:30pm to 4:30pm JCMB 6201 -- Show/hide abstractAbstract: It is well-known that every smooth plane cubic E supports precisely 9 flex lines. By analogy, we may ask: "How many degree d curves intersect E in a single point?" The problem of calculating such numbers of tangent curves has fascinated enumerative geometers for decades. Despite being an extremely classical and concrete problem, it was not until the advent of Gromov-Witten invariants in the 1990s that a general method was discovered. The resulting theory is incredibly rich, and the curve counts satisfy a suite of remarkable properties, some proven and some still conjectures.
In this talk, I will discuss two distinct projects which take inspiration from this geometry. In the first, we study the behaviour of tangent curves as the cubic E degenerates to a cycle of lines. Using the machinery of logarithmic Gromov-Witten theory, we obtain detailed information concerning how these tangent curves degenerate along with E. The resulting theorems are purely classical, with no reference to Gromov-Witten theory, but they do not appear to admit a classical proof. This is joint work with Lawrence Barrott. In a separate project, joint with Dhruv Ranganathan, we perform iterated blowups of moduli spaces to prove the so-called local-logarithmic conjecture for hyperplane sections; this gives access to a large number of previously unknown enumerative theories.
No prior knowledge of Gromov-Witten theory will be assumed.- EDGE: Tara Holm (Cornell University) - Rigidity and flexibility of Hamiltonian 4-manifolds
12th November 2019, 3:30pm to 4:30pm Bayes Center 5.10 -- Show/hide abstractAbstract: Hamiltonian S^1-manifolds of dimension four are classified by decorated graphs. We give a generators and relations description of the equivariant cohomology of a Hamiltonian S^1-manifold. As a first consequence, we show that the equivariant cohomology does not determine the space: it does not satisfy the cohomological rigidity phenomenon found in toric varieties. A second consequence is a proof of the finiteness of Hamiltonian circle actions on a closed symplectic four-manifold that does not use pseudo-holomorphic tools. Extensive examples will be included throughout.This is based on joint work with Liat Kessler.- EDGE Days 2019
8th November 2019, 9:00am to 5:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Speakers are Grisha Belousov, Caucher Birkar, Thibaut Delcroix, Ruadhai Dervan, Adrien Dubouloz, Eleonore Faber, Liana Heuberger, Jesus Martinez Garcia, Takashi Kishimoto, Dimitra Kosta, Eveline Legendre, Jihun Park, Anya Pratoussevich, Yura Prokhorov, Carl Tipler, Alessandro Tomassiello, Nathan Broomhead, Jennya Shinder, Kaori Suzuki and Michael Wemyss. https://www.maths.ed.ac.uk/cheltsov/edge2019/index.html- EDGE Days 2019
7th November 2019, 9:00am to 5:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Speakers are Grisha Belousov, Caucher Birkar, Thibaut Delcroix, Ruadhai Dervan, Adrien Dubouloz, Eleonore Faber, Liana Heuberger, Jesus Martinez Garcia, Takashi Kishimoto, Dimitra Kosta, Eveline Legendre, Jihun Park, Anya Pratoussevich, Yura Prokhorov, Carl Tipler, Alessandro Tomassiello, Nathan Broomhead, Jennya Shinder, Kaori Suzuki and Michael Wemyss. https://www.maths.ed.ac.uk/cheltsov/edge2019/index.html- EDGE Days 2019
6th November 2019, 9:00am to 5:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Speakers are Grisha Belousov, Caucher Birkar, Thibaut Delcroix, Ruadhai Dervan, Adrien Dubouloz, Eleonore Faber, Liana Heuberger, Jesus Martinez Garcia, Takashi Kishimoto, Dimitra Kosta, Eveline Legendre, Jihun Park, Anya Pratoussevich, Yura Prokhorov, Carl Tipler, Alessandro Tomassiello, Nathan Broomhead, Jennya Shinder, Kaori Suzuki and Michael Wemyss. https://www.maths.ed.ac.uk/cheltsov/edge2019/index.html- EDGE Days 2019
5th November 2019, 9:00am to 5:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Speakers are Grisha Belousov, Caucher Birkar, Thibaut Delcroix, Ruadhai Dervan, Adrien Dubouloz, Eleonore Faber, Liana Heuberger, Jesus Martinez Garcia, Takashi Kishimoto, Dimitra Kosta, Eveline Legendre, Jihun Park, Anya Pratoussevich, Yura Prokhorov, Carl Tipler, Alessandro Tomassiello, Nathan Broomhead, Jennya Shinder, Kaori Suzuki and Michael Wemyss. https://www.maths.ed.ac.uk/cheltsov/edge2019/index.html- EDGE Days 2019
4th November 2019, 9:00am to 5:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Speakers are Grisha Belousov, Caucher Birkar, Thibaut Delcroix, Ruadhai Dervan, Adrien Dubouloz, Eleonore Faber, Liana Heuberger, Jesus Martinez Garcia, Takashi Kishimoto, Dimitra Kosta, Eveline Legendre, Jihun Park, Anya Pratoussevich, Yura Prokhorov, Carl Tipler, Alessandro Tomassiello, Nathan Broomhead, Jennya Shinder, Kaori Suzuki and Michael Wemyss. https://www.maths.ed.ac.uk/cheltsov/edge2019/index.html- EDGE: Yin Li (King's College London)-Exact Calabi-Yau structures on wrapped Fukaya categories
29th October 2019, 3:30pm to 4:30pm -- Show/hide abstractAbstract: An exact CY structure on the wrapped Fukaya category of a Weinstein manifold induces a distinguished class B in the degree one equivariant symplectic cohomology. For exact Lagrangian submanifolds which are infinitesimally equivariant with respect to B, we construct a derivation on their Floer cohomology, whose supertrace is an refinement of the topological intersection number. As an application, we prove that for any Weinstein manifold whose wrapped Fukaya category is exact CY, there is a bound on the maximal number of pairwise disjoint Lagrangian spheres, which generalizes a theorem of Seidel. On the other hand, I will explain how to use Koszul duality to show that the wrapped Fukaya category of the Milnor fiber of a 3-fold triple point admits an exact CY structure. This work is motivated by an old paper of Davison: https://arxiv.org/abs/1010.3564.- EDGE: Qingyuan Jiang (University of Edinburgh) - The geometry of (resolutions of) degeneracy loci
22nd October 2019, 2:00pm to 3:00pm Bayes Center 5.10 -- Show/hide abstractAbstract: In this talk we will discuss the geometry -- the derived categories, Chow groups and Hodge structures -- on certain canonical resolution spaces of degeneracy loci of maps between vector bundles. We will mainly focus on the classical examples of symmetric powers of curves, and then explain the general results and the proofs through these examples.- Special EDGE: Victor Przyjalkowski-On mirror P=W conjecture
11th October 2019, 1:00pm to 2:00pm JCMB, 5323 -- Show/hide abstractAbstract: We discuss Katzarkov-Kontsevich-Pantev conjectures which relate Hodge numbers of Fano varieties with Hodge-type numbers of their Landau--Ginzburg models. We observe their proofs in dimensions 2 and 3. We also discuss mirror P=W conjecture which claims more deep relation of mixed Hodge structures of log Calabi--Yau varieties and perverse Leray filtrations of Landau--Ginzburg models.- EDGE: Johan Martens (University of Edinburgh) - The Hitchin connection in (almost) arbitrary characteristic
8th October 2019, 3:30pm to 4:30pm Bayes Centre 5.10 -- Show/hide abstractAbstract: The Hitchin connection is a flat projective connection on bundles of non-abelian theta-functions over the moduli space of curves, originally introduced by Hitchin in a Kahler context. We will describe a purely algebra-geometric construction of this connection that also works in (most) positive characteristics. A key ingredient is an alternative to the Narasimhan-Atiyah-Bott Kahler form on the moduli space of bundles on a curve. We will comment on the connection with some related topics, such as the Grothendieck-Katz p-curvature conjecture. This is joint work with Baier, Bolognesi and Pauly.- Special EDGE: Costya Loginov(Steklov Mathematical Institute)-Semistable degenerations of Fano varieties
3rd October 2019, 2:00pm to 3:00pm JCMB, Seminar Room 5323 -- Show/hide abstractAbstract: Let us consider a family of projective varieties over a curve germ. We will focus on so called semistable families. By the semistable reduction theorem any family whose generic fiber is smooth is birational to a semistable family after a finite base change. We say that the special fiber of such family is a semistable degeneration of its generic fiber. A natural invariant of the special fiber is its dual complex. For example, due to Kulikov there is a characterization of semistable degenerations of K3 surfaces in terms of its dual complexes. Analogous result was obtained by Fujita for del Pezzo surfaces. We prove that if every fiber of a semistable family is Fano then the dual complex of the special fiber is a simplex of bounded dimension. We show that, in contrast to the case of K3 surfaces, the monodromy around the special fiber is always trivial for semistable degenerations of del Pezzo surfaces. We also discuss maximal degenerations of Fano varieties and prove that they are unique in dimensions not greater than 3.- EDGE: Richard Thomas (Imperial College) - Vafa-Witten invariants for projective surfaces
1st October 2019, 3:30pm to 4:30pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Vafa and Witten proposed new gauge-theoretic invariants of 4-manifolds in 1994. I’ll explain how to define them, refine them, and calculate some of them, when the 4-manifold is a complex projective surface.- Structure and Symmetry Theme Day
27th September 2019, 10:00am to 5:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Afternoon Schedule: @14:30 Speaker: Nick Sheridan (University of Edinburgh) Title: The Gamma and SYZ conjectures: a tropical approach to periods. Abstract: I'll start by explaining a new method of computing asymptotics of period integrals using tropical geometry, via some concrete examples. Then I'll use this method to give a geometric explanation for a strange phenomenon in mirror symmetry, called the Gamma Conjecture, which says that mirror symmetry does not respect integral cycles: rather, the integral cycles on a complex manifold correspond to integral cycles on the mirror multiplied by a certain transcendental characteristic class called the Gamma class. We find that the appearance of zeta(k) in the asymptotics of period integrals arises from the codimension-k singular locus of the SYZ fibration. * @16:00 Speaker: Roberto Volpato (University of Padova) Title: Strings on K3: the maths and the physics. Abstract: I will give an overview of recent (and less recent) ideas that have been developed in the study of string theory on K3 manifolds, including topics in conformal field theory, moonshine, black hole physics, automorphic forms, hyperkahler geometry. *- EDGE: Luca Battistella (Max Planck Institute Bonn) - Reduced relative Gromov-Witten theory in genus one
24th September 2019, 3:30pm to 4:30pm Bayes Centre 5.10 -- Show/hide abstractAbstract: In joint work with N. Nabijou and D. Ranganathan, we produce a logarithmic desingularisation of the main component of the moduli space of genus one stable maps to projective space having prescribed contact order with a hyperplane. Recasting ideas of A. Gathmann in more modern language, we observe that these spaces are nested: requiring a contact order higher by one identifies a divisor, which together with some boundary corrections corresponds to a line bundle of tropical origin and tautological class. A splitting principle governing these boundary terms allows us to reconstruct the restricted reduced Gromov-Witten theory of the divisor (together with the rubber and relative theories) inductively from that of the ambient space and genus zero data.- EDGE: Laura Pertusi (Università di Milano) - Stability conditions on Gushel-Mukai fourfolds
17th September 2019, 3:30pm to 4:30pm Bayes Centre 5.02 -- Show/hide abstractAbstract: An ordinary Gushel-Mukai fourfold X is a smooth quadric section of a linear section of the Grassmannian G(2,5). Kuznetsov and Perry proved that the bounded derived category of X admits a semiorthogonal decomposition whose non-trivial component is a subcategory of K3 type. In this talk I will report on a joint work in progress with Alex Perry and Xiaolei Zhao, where we construct Bridgeland stability conditions on the K3 subcategory of X. Then I will explain some applications concerning the existence of a homological associated K3 surface and hyperkaehler geometry.- EDGE: JongHae Keum (Korean institute for advanced study)-Algebraic surfaces with minimal Betti numbers
10th September 2019, 3:30pm to 4:30pm JCMB room 5323 -- Show/hide abstractAbstract: These are algebraic surfaces with the Betti numbers of the complex projective plane, and are called $Q$-homology projective planes. We describe recent progress in the study of such surfaces including smooth examples, the fake projective planes. We also discuss open questions on Montgomery-Yang problem.- EDGE: Michel van Garrel (Warwick University) Log BPS numbers as Euler characteristics
21st May 2019, 2:00pm to 3:00pm JCMB Seminar Room 5323 -- Show/hide abstractAbstract: Let (S,E) be the pair of a del Pezzo surface and a smooth anticanonical divisor on it. It is an example of a log K3 surface. We are interested in studying suitably defined rational curves in (S,E), whose behavior is parallel to the theory of rational curves on a K3 surface. I will define a moduli space of sheaves on (S,E), whose Euler characteristic gives the desired number of rational curves. Time permitting, I will comment on how this relates to mirror symmetry. This is joint work with Jinwon Choi, Sheldon Katz and Nobuyoshi Takahashi.- EDGE: Nikolaos Tziolas (University of Cyprus) Vector fields and moduli of canonically polarized surfaces in positive characteristic
20th May 2019, 10:30am to 11:30am JCMB 5323 -- Show/hide abstractAbstract: The purpose of this talk is to present some results about the geometry of smooth canonically polarized surfaces defined over a field of positive characteristic which have a nontrivial global vector field, equivalently non reduced automorphism scheme, and the implications that the existence of such surfaces has in the moduli problem of canonically polarized surfaces.- EDGE: John Pardon (Princeton University) Structural results in wrapped Floer theory
13th May 2019, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: I will discuss results relating different partially wrapped Fukaya categories. These include a K\"unneth formula, a `stop removal' result relating partially wrapped Fukaya categories relative to different stops, and a gluing formula for wrapped Fukaya categories. The techniques also lead to generation results for Weinstein manifolds and for Lefschetz fibrations. The methods are mainly geometric, and the key underlying Floer theoretic fact is an exact triangle in the Fukaya category associated to Lagrangian surgery along a short Reeb chord at infinity. This is joint work with Sheel Ganatra and Vivek Shende.- EDGE: Dylan Allegretti (University of Sheffield) The monodromy of meromorphic projective structures
7th May 2019, 2:00pm to 3:00pm JCMB Seminar room 5323 -- Show/hide abstractAbstract: The notion of a complex projective structure is fundamental in low-dimensional geometry and topology. The space of projective structures on a surface has the structure of a complex manifold, and there is a holomorphic map from this space to the character variety of the surface, sending a projective structure to its monodromy representation. In this talk, I will describe joint work with Tom Bridgeland in which we introduced the notion of a "meromorphic projective structure" with poles at a discrete set of points. In the case of a meromorphic projective structure, the monodromy can be viewed as a point in a moduli space introduced by Fock and Goncharov in their work on cluster varieties. This appears to be a manifestation of a general relationship between cluster varieties and spaces of stability conditions on 3-Calabi-Yau triangulated categories.- EDGE: Ana Rita Pires (University of Edinburgh) Symplectic embeddings and infinite staircases
16th April 2019, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: McDuff and Schlenk determined when a four-dimensional symplectic ellipsoid can be symplectically embedded into a four-dimensional ball, and found that if the ellipsoid is close to round, the answer is given by an "infinite staircase" determined by the odd index Fibonacci numbers, while if the ellipsoid is sufficiently stretched, all obstructions vanish except for the volume obstruction. Infinite staircases have also been found when embedding ellipsoids into certain specific polydisks and ellipsoids (Frenkel - Muller, Cristofaro-Gardiner - Kleinman, Usher), but it seems to be a rare behaviour. In this talk, we will see how lattice point counts, Ehrhart theory, and the sharpness of ECH capacities can be used to investigate for what other domains infinite staircases exist, and discuss the relationship with toric varieties. This is joint work with Dan Cristofaro-Gardiner, Tara Holm, and Alessia Mandini.- EDGE: Hongjie Yu (IST Austria) Counting $\ell$-adic local systems over a curve
2nd April 2019, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: Let $X_{1}$ be a projective, smooth and geometrically connected curve over $\mathbb{F}_{q}$ with $q=p^{n}$ elements where $p$ is a prime number, and let $X$ be its base change to an algebraic closure of $\mathbb{F}_{q}$. The Frobenius element permutes the set of isomorphism classes of irreducible $\ell$-adic local systems ($\ell \neq p$) with a fixed rank on $X$. In 1981, Drinfeld has calculated the number of fixed points of this permutation in the rank 2 case. Curiously, it looks like the number of $\mathbb{F}_q$-points of a variety defined over $\mathbb{F}_q$. In this talk, we generalize Drinfeld's result to higher rank case. Our method is purely automorphic, in fact we do that by using Arthur-Lafforgue's trace formula.- EDGE: Rafe Mazzeo (Stanford) Prospects for the Kapustin-Witten equations
26th March 2019, 3:30pm to 4:30pm -- Show/hide abstractAbstract: Gaiotto and Witten have conjectured a relationship between some count of solutions of the Kapustin-Witten equations on a 4-manifold containing a knot in its boundary and the Jones polynomial of that knot. This is far from established and there are some considerable analytic difficulties ahead. I will describe progress toward this goal, including work by Taubes, joint work with Witten and other joint work with S. He.- EDGE: Jochen Heinloth (University of Essen) Existence of good moduli spaces for algebraic stacks
19th March 2019, 3:30pm to 4:30pm Bayes Centre, Seminar Room 5.10 -- Show/hide abstractAbstract: In this talk we explain how two very basic models of actions of one parameter subgroups allow to formulate local criteria for the existence of separated moduli spaces which can be verified rather easily for an interesting class of moduli problems. (This is joint work with Jarod Alper and Daniel Halpern-Leistner). As a byproduct we find a short proof for one of the basic decomposition results for reductive groups.- EDGE: Costya Shramov (Steklov Institute and HSE) Automorphisms of Severi-Brauer surfaces
12th March 2019, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: I will discuss finite groups acting by automorphisms and birational automorphisms of Severi-Brauer surfaces. We will see that they are bounded provided that the base field is a function field, and also make some observations on their structure in the general case.- Special EDGE: Roberto Svaldi (Cambridge University) On the boundedness of elliptic Calabi-Yau varieties
7th March 2019, 2:30pm to 3:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: One of the main goals in Algebraic Geometry is to classify varieties. The minimal model program (MMP) is an ambitious program that aims to realize this goal, from the point of view of birational geometry, that is, we are free to modify the structure of a given variety along closed subsets to improve its geometric features. According to the MMP, there are 3 building blocks in the birational classification of algebraic varieties: Fano varieties, Calabi-Yau varieties, and varieties of general type. One important question, that is needed to further investigate the classification process, is whether or not varieties in these 3 classes have finitely many deformation types (a property called boundedness). Our understanding of the boundedness of Fano varieties and varieties of general type is quite solid but Calabi-Yau varieties are still quite elusive. In this talk, I will discuss recent results on the boundedness of elliptic Calabi-Yau varieties, which are the most relevant in physics. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. This is joint work with C. Birkar and G. Di Cerbo.- EDGE: Sven Meinhardt (University of Sheffield) Field and Vertex algebras from geometry and topology
29th January 2019, 3:30pm to 4:30pm -- Show/hide abstractAbstract: I will propose a definition of (generalized) field and vertex algebras in any symmetric monoidal category and provide a way to construct nontrivial examples. To keep the talk accessible, all of this will be illustrated for the category of modules over a commutative ring. As we shall see, every oriented cohomology theory provides examples from (non-commutative) geometry. More specifically, every (small) dg-category gives rise to a field algebra through its moduli stack of representations. Another class of examples can be constructed for twisted perfect complexes on schemes. This work was motivated by a recent construction of vertex algebras by Dominic Joyce and all credits go to him.- EDGE: Tarig Abdelgadir (University of New South Wales) Moduli of tensor stable points and refined representations
29th January 2019, 2:00pm to 3:00pm -- Show/hide abstractAbstract: Moduli spaces are a fruitful way of studying a given abelian category, take for example the moduli of point-like objects in non-commutative projective spaces or the moduli of representations of the McKay quiver. These examples yield moduli varieties, elliptic curves and minimal resolutions of Kleinian singularities. Sometimes, however, the problem lends itself better to algebraic stacks as in the case of Ringel's canonical algebras and weighted projective lines. Our guiding example in this talk will be a generalisation of these, namely Geigle-Lenzing (G-L) projective spaces as defned by Herschend, Iyama, Minamoto, Opperman. We will start with a stack and a direct sum of line bundles that form a tilting bundle and recover the stack from the corresponding category of quiver representations before applying this technique to G-L spaces. This is joint work with Daniel Chan (UNSW Sydney).- EDGE: Tudor Padurariu (MIT) K-theoretic Hall algebras for quivers with potential
15th January 2019, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.02 -- Show/hide abstractAbstract: For a quiver with potential, Kontsevich and Soibelman defined an algebra called the cohomological Hall algebra. After work of Davison and Meinhardt, the CoHA is actually a localized bialgebra and in many cases it satisfies a PBW theorem. We study a version of the Hall algebra defined using the K_0 group of certain categories of singularities associated to the quiver with potential, and explain that similar definitions as the ones used for CoHA put a localized bialgebra structure on this algebra. We also prove a PBW theorem for this algebra using semi-orthogonal decompositions inspired by the work of Spenko and Van den Bergh.- EDGE: Yongnam Lee (KAIST) On function fields and deformations of hypersurfaces
4th December 2018, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: In this talk, we consider dominant rational maps from very general hypersurfaces and smooth deformations of hypersurfaces under some suitable conditions. The first part is based on a joint work with Gian Pietro Pirola, and the second part is a joint work with Fabrizio Catanese.- EDGE: Francesco Sala (Kavli IPMU) 2D Cohomological Hall algebra of a curve
27th November 2018, 3:30pm to 4:30pm Bayes Centre, 5th floor -- Show/hide abstractAbstract: Given a curve X, one can associate with it the following abelian categories of homological dimension two: the category of Higgs sheaves on X, the category of vector bundles on X with flat connection, the category of finite-dimensional representations of the fundamental group of X. The corresponding moduli stacks of objects are stacks of coherent sheaves over different "forms" of the curve: the Dolbeaut, de Rham, and Betti form of X. In the present talk, I will introduce convolution algebras associated with such stacks and provide some characterization result and some conjectures relating them. This is based on a joint paper with Olivier Schiffmann and a work in progress with Mauro Porta.- EDGE: Jeff Hicks (UC Berkeley) Dimers and Tropical Lagrangians
20th November 2018, 3:30pm to 4:30pm Bayes Centre Room 5.02 -- Show/hide abstractAbstract: Mirror symmetry is a conjectured duality between the symplectic geometry of a space X and complex geometry on a mirror space Y. These spaces are conjectured to share dual torus fibrations over a common affine base Q. A proposed mechanism for mirror symmetry is that the symplectic geometry of X and complex geometry of Y can be mutually compared to tropical geometry on the base Q. Starting with the combinatorial data of a dimer on a torus, we construct a Lagrangian in X whose valuation projection is a tropical hypersurface in Q. We will then explore a mutation and wall-crossing story for these tropical Lagrangians. The construction of these tropical Lagrangians will also tell us why they are homologically mirror to sheaves supported on complex hypersurfaces with matching tropical valuation.- Special EDGE: Ivan Smith (Cambridge) Lagrangian pinwheels
15th November 2018, 3:30pm to 4:30pm JCMB Seminar Room 5323 -- Show/hide abstractAbstract: Pinwheels are certain simple cell complexes; Lagrangian pinwheels arise as vanishing cycles of cyclic quotient singularities. We give a symplectic analogue of a theorem of Hacking and Prokhorov classifying the Wahl-type degenerations of the projective plane in terms of Markov triples, and contrast this with a finiteness theorem for pinwheels which admit Lagrangian embeddings in surfaces of general type, partially answering a question of Kronheimer. This talk reports on joint work with Jonny Evans.- EDGE: Mauricio Corrêa (Oxford) Moduli spaces of reflexive sheaves and classification of distributions on P^3
13th November 2018, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: We describe the moduli space of distributions in terms of Grothendieck’s Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2. We show how the connectedness of the curves in the singular sets of foliations is a integrable phenomena. This parte of the talk is a work joint with M. Jardim (Unicamp) and O. Calvo-Andrade (Cimat). We also study foliations by curves via the investigation of their singular schemes and conormal sheaves and we provide a classification of foliations of degree at most 3 with conormal sheaves locally free. This parte of the talk is a work joint with M. Jardim (Unicamp) and S. Marchesi (Unicamp).- EDGE: Aaron Bertram (Utah) Stability Conditions on Projective Space
13th November 2018, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: I describe a one-parameter family of stability conditions on projective space of any dimension that seems to converge to Gieseker stability. In dimensions two and three this is known due to previous work of many people, but it is open in higher dimension. One feature of this family is that the ``walls'' are easily computed, as I will show with several examples. This is joint work with my students, Matteo Altavilla, Dapeng Mu and Marin Petkovic.- Special EDGE: Alexander Kuznetsov (Moscow) Residual categories
12th November 2018, 3:00pm to 4:00pm JCMB Seminar Room 5323 -- Show/hide abstractAbstract: From the homological projective duality perspective it is important to construct Lefschetz semiorthogonal decompositions which are as close as possible to being rectangular. I will explain an approach to this problem based on the notion of a residual category. Among other things it shows that the "out-of-rectangular" part of the derived category has an interesting structure. I will show some examples of how this part looks for homogeneous varieties and discuss the connection with quantum cohomology of these varieties. This is a joint work with Maxim Smirnov.- ICMS workshop: Constructions and obstructions in birational geometry (NO EDGE)
6th November 2018, 3:30pm to 4:30pm Bayes Centre, 5th floor- EDGE: Gregory G. Smith (Queen's University) Sums of squares on real projective varieties
30th October 2018, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: How does classical algebraic geometry help one recognize sums of squares? After sampling some of the unexpected connections between real and complex geometry, we will look at new bounds on the number of terms in a sum-of-squares expression for a quadratic form on a real projective variety. This talk is based on joint work with G. Blekherman, R. Sinn, and M. Velasco.- EDGE Special Seminar: Victor Przyjalkowski (HSE Moscow) Weighted complete intersections
25th October 2018, 3:00pm to 4:00pm JCMB, Seminar Room 5323 -- Show/hide abstractAbstract: We observe a classification and the main properties of one of the main class of examples of higher dimensional Fano varieties --- smooth complete intersections in weighted projective spaces. We discuss their main properties and boundness results. We also discuss extremal examples from Hodge theory point of view and their relations with derived categories structures and their semiorthogonal decompositions. If time permits, we discuss mirror symmetry for the complete intersections and invariants of their Landau--Ginzburg models related to ones of the complete intersections.- EDGE: Angela Ortega (HU Berlin) Generic injectivity of the Prym map for double ramified coverings
23rd October 2018, 3:00pm to 4:00pm JCMB Seminar Room 5323 -- Show/hide abstractAbstract: Given a finite morphism of smooth curves one can canonically associate it a polarised abelian variety, the Prym variety. This induces a map from the moduli space of coverings to the moduli space of polarised abelian varieties, known as the Prym map. In this talk we will consider the Prym map between the moduli space of double coverings over a genus g curve ramified at r points, and the moduli space of polarised abelian varieties of dimension (g-1+r)/2 with polarisation of type D. We will show the generic injectivity of the Prym map in the cases (a) g=2, r=6 and (b) g=5, r=2. In the first case the proof is constructive and can be extended to the range r > max{6, 2(g+2)/3}. This is a joint work with Juan Carlos Naranjo.- EDGE Special Seminar: Ziquan Zhuang (Princeton) Superrigidity and K-stability
18th October 2018, 2:10pm to 3:10pm JCMB Seminar Room 5323 -- Show/hide abstractAbstract: Superrigidity and K-stability are properties of Fano varieties that have many interesting geometric implications. For instance, birational superrigidity implies non-rationality and K-stability is related to the existence of Kähler-Einstein metrics. Nonetheless, both properties are hard to verify in general. In this talk, I will first explain how to relate birational superrigidity to K-stability using alpha invariants; I will then outline a method of proving birational superrigidity that works quite well with most families of index one Fano complete intersections and thereby also verify their K-stability. This is partly based on joint work with Charlie Stibitz and Yuchen Liu.- EDGE: Hiro Lee Tanaka (Harvard) Morse theory on a point, and associative algebras
16th October 2018, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: I'll talk about the first step in a project to enrich Morse theory and Lagrangian Floer theory over coefficient rings that are deeper than the integers (e.g., over ring spectra). Such a thing is useful because invariants contained in spectra are way more sensitive than invariants contained in chain complexes. Also, even if you don't care about Morse theory or symplectic geometry, this "first step" is fun in its own right: We construct a stack of broken lines (which one can think of as the appropriate moduli space of Morse trajectories on a point), and then we prove that a factorizable sheaf on this stack is the same thing as a possibly non-unital associative algebra. I'll try to explain what any of this has to do with Floer theory in the exact case if time allows. This is joint work with Jacob Lurie.- EDGE: Laura Pertusi (Milano) Geometry of fourfolds with an admissible K3 subcategory
16th October 2018, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: The derived category of a cubic fourfold admits a semiorthogonal decomposition whose non trivial component is a subcategory of K3 type by a result of Kuznetsov. This allowed to prove many properties on the geometry of the hyperkaehler manifolds associated to the cubic fourfold. More recently, Kuznetsov and Perry found a semiorthogonal decomposition with a K3 type component in the case of an other class of fourfolds, known as Gushel-Mukai fourfolds. The aim of this talk is to discuss a generalization of some results on lattice theory, proved for cubic fourfolds by Addington, Thomas and Huybrechts, in the setting of Gushel-Mukai fourfolds. In particular, we discuss the conditions under which their associated hyperkahler fourfold is birational to a moduli space of stable sheaves (resp. to the Hilbert square) on a K3 surface.- MAXIMALS Doubleheader
9th October 2018, 2:00pm to 4:30pm- EDGE: Helge Ruddat (Mainz) Smoothing toroidal crossing varieties
2nd October 2018, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: I explain the proof of a new result on smoothing toroidal and normal crossing varieties by constructing certain kinds of log structures that enjoy Hodge to de Rham degeneration with a suitable definition of differential forms. We generalize work by Friedman and work by Gross and Siebert. This is a joint project with Simon Felten and Matej Filip.- EDGE: Clark Barwick (Edinburgh) Inverse Galois Problems
25th September 2018, 3:30pm to 4:30pm JCMB, Seminar Room 5323 -- Show/hide abstractAbstract: There is a big class of problems that assert, under various circumstances, that certain kinds of Galois-theoretic data can be ‘realised’ at the level of fields, rings, or schemes. These problems go under various names – anabelian conjectures, Abhyankar conjectures, section conjectures, inverse Galois problems, etc. – but the idea is always the same. We survey these problems, and we discuss those that have actually been solved.- EDGE: Nick Sheridan (Edinburgh) Cubic fourfolds, K3 surfaces and mirror symmetry
18th September 2018, 3:05pm to 3:55pm JCMB, Seminar Room 5323 -- Show/hide abstractAbstract: While many cubic fourfolds are known to be rational, it is expected that the very general cubic fourfold is irrational (although none have been proven to be so). There is a conjecture for precisely which cubics are rational, which can be expressed in Hodge-theoretic terms (by work of Hassett) or in terms of derived categories (by work of Kuznetsov). The conjecture can be phrased as saying that one can associate a `noncommutative K3 surface’ to any cubic fourfold, and the rational ones are precisely those for which this noncommutative K3 is `geometric’, i.e., equivalent to an honest K3 surface. It turns out that the noncommutative K3 associated to a cubic fourfold has a conjectural symplectic mirror (due to Batyrev-Borisov). In contrast to the algebraic side of the story, the mirror is always `geometric’: i.e., it is always just an honest K3 surface equipped with an appropriate Kähler form. After explaining this background, I will state a theorem: homological mirror symmetry holds in this context. (joint work with Ivan Smith).- EDGE V. Balaji (Chennai Mathematical Institute) On semi-simplicity of tensor products in positive characteristics
5th July 2018, 2:05pm to 2:55pm JCMB, Seminar Room 5323 -- Show/hide abstractAbstract: We work over an algebraically closed field k of characteristic p > 0. In 1994, Serre showed that if semi-simple representations V_i of a group Γ are such that ∑(dim V_i − 1) < p, then their tensor product is semi-simple. In the late nineties, Serre generalized this theorem comprehensively to the case where Γ is a subgroup of G(k), for G a reductive group, and answered the question of “complete reducibility” of Γ in G, (Seminaire Bourbaki, 2003). In 2014, Deligne generalized the results of Serre (of 1994) to the case when the V_i are semi-simple representations of a group scheme G. In my talk I present the recent work of mine (2017) with Deligne and Parameswaran where we consider the case when G is a subgroup scheme of a reductive group G and generalize the results of Serre and Deligne. A key result is a structure theorem on “doubly saturated” subgroup schemes G of reductive groups G. As an application, we obtain an analogue of classical Luna’s étale slice theorem in positive characteristics.- EDGE Filippo Viviani (Rome) - On the first steps of the minimal model program for the moduli space of stable pointed curves
28th June 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstractAbstract: I will report on a joint work with G. Codogni and L. Tasin, in which we investigate the possible first steps of the minimal model program for the moduli space of stable pointed curves. We prove that these first steps have a modular interpretation and we relate them to the first steps of the so called Hassett-Keel program, which studies certain log canonical models of the moduli space of stable pointed curves and their modular interpretations.- EDGE Miles Reid (Warwick) - The Tate-Oort Group and Godeaux Surfaces in Mixed Characteristic
15th June 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstractAbstract: We construct Godeaux surfaces and Calabi-Yau 3-folds whose Pic^tau contains 5-torsion, in mixed characteristic at 5. The same ideas give Campedelli surfaces and Calabi-Yau 3-folds with 7-torsion, and (in progress) Godeaux surfaces with 3-torsion. The aim is to put varieties in characteristic p with action of Z/p, al_p and mu_p into a single family, together with the characteristic zero case with a Z/p action. Joint work with Kim Soonyoung.- GLEN seminar: Maxence Mayrand (Oxford)
15th June 2018, 11:00am to 12:00pm JCMB 5323 -- Show/hide abstractAbstract: Stratified hyperkähler spaces http://www.maths.ed.ac.uk/~jmartens/GLEN.html- GLEN seminar: Hendrik Suess (Manchester)
15th June 2018, 9:45am to 10:45am JCMB 5323 -- Show/hide abstractAbstract: On irregular Sasaki-Einstein manifolds in dimension 5. http://www.maths.ed.ac.uk/~jmartens/GLEN.html- GLEN seminar: Andrew Dancer (Oxford)
14th June 2018, 3:30pm to 4:30pm JCMB 5323 -- Show/hide abstractAbstract: Integrability in Riemannian Geometry http://www.maths.ed.ac.uk/~jmartens/GLEN.html- GLEN seminar: Leonardo Mihalcea (Virginia Tech)
14th June 2018, 2:00pm to 3:00pm JCMB 5323 -- Show/hide abstractAbstract: Chern classes of Schubert cells, Hecke algebras, and stable envelopes.
http://www.maths.ed.ac.uk/~jmartens/GLEN.html- EDGE Nicolò Sibilla (Kent) - Log schemes, root stacks and parabolic bundles
7th June 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstractAbstract: Log schemes are an enlargement of the category of schemes that was introduced by Deligne, Faltings, Illlusie and Kato, and has applications to moduli theory and deformation problems. Log schemes play a central role in the Gross-Siebert program in mirror symmetry. In this talk I will introduce log schemes and then explain recent work joint with D. Carchedi, S. Scherotzke, and M. Talpo on various aspects of their geometry. I will discuss a comparison result between two different ways of associating to a log scheme its etale homotopy type, respectively via root stacks and the Kato-Nakayama space. Our main result is a new categorified excision result for parabolic sheaves, which relies on the technology of root stacks.- EDGE: Daniel Halpern-Leistner (Cornell) - Wall crossing in moduli problems large and small
31st May 2018, 3:05pm to 3:55pm JCMB, Seminar room 5323 -- Show/hide abstractAbstract: Geometric invariant theory is an essential tool for constructing moduli spaces in algebraic geometry. Recently a theory has emerged in my work and the work of others which treats the results and structures of geometric invariant theory in a broader context. The theory of Theta-stability applies directly to moduli problems without the need to approximate a moduli problem as an orbit space for a reductive group on a quasi-projective scheme. I will give an overview of the picture that has emerged, including a discussion of Harder-Narasimhan theory and relatively simple criteria for the existence of good moduli spaces. Then I will discuss applications to wall crossing formulas of K-theoretic Donaldson invariants of algebraic surfaces.- EDGE: Xiaolei Zhao (Northeastern/UCSB) - Twisted cubics on cubic fourfolds and stability conditions
31st May 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstractAbstract: It is a classical result of Beauville and Donagi that Fano varieties of lines on cubic fourfolds are hyper-Kahler. More recently, Lehn, Lehn, Sorger and van Straten constructed a hyper-Kahler eightfold out of twisted cubics on cubic fourfolds. In this talk, I will explain a new approach to these hyper-Kahler varieties via moduli of stable objects on the Kuznetsov components. Along the way, we will derive several properties of cubic fourfolds as consequences. This is based on a joint work with Chunyi Li and Laura Pertusi.- EDGE: Anna Barbieri (Sheffield) - A Riemann-Hilbert problem for uncoupled BPS structures
24th May 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstractAbstract: Abstract: BPS structures locally describe the space of Bridgeland stability conditions of a CY3 category together with a generalised Donaldson-Thomas theory. On the other hand, Riemann-Hilbert problems are inverse problems in the theory of differential equations. After defining the notion of BPS structures I will introduce and motivate a Riemann-Hilbert problem naturally attached to BPS structures.- EDGE: Fabio Bernasconi (Imperial College London) - Pathologies for Fano varieties and singularities in positive characteristic
17th May 2018, 3:05pm to 3:55pm JCMB, Seminar room, 5323 -- Show/hide abstractAbstract: Abstract: In this talk, after motivating the study of singularities in the context of the Minimal Model Program, we will explain the connection between vanishing theorems on Fano-type varieties and cohomological properties of certain classes of singularities. In particular, we will explain how to construct del Pezzo surfaces in low characteristic violating Kodaira vanishing and how to deduce the existence of klt not Cohen-Macaulay threefold singularities. Time permitting, we will discuss some possible obstructions to the existence of flips.- EDGE: Davide Cesare Veniani (Mainz) - Recent advances about lines on quartic surfaces
17th May 2018, 2:05pm to 2:55pm JCMB, Seminar room, 5323 -- Show/hide abstractAbstract: Abstract: The number of lines on a smooth complex surface in projective space depends very much on the degree of the surface. Planes and conics contain infinitely many lines and cubics always have exactly 27. As for degree 4, a general quartic surface has no lines, but Schur's quartic contains as many as 64. This is indeed the maximal number, but a correct proof of this fact was only given quite recently. Can a quartic surface carry exactly 63 lines? How many can there be on a quartic which is not smooth, or which is defined over a field of positive characteristic? In the last few years many of these questions have been answered, thanks to the contribution of several mathematicians. I will survey the main results and ideas, culminating in the list of the explicit equations of the ten smooth complex quartics with most lines.- EDGE: Francesca Carocci (Imperial College London) - Reduced vs Cuspidal GW invariants for the quintic 3-fold
10th May 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstractAbstract: Abstract: Moduli spaces of stable maps of genus g>0 are highly singular and with many irreducible components which affect the enumerative meaning of the invariants arising from them . In this talk we will try to give a flavour of how bad these spaces can be, already in the simplest example in genus 1. We will then hint at two possible approaches to deal with the so called "degenerate contributions" , namely: Li-Zinger reduced invariants and Viscardi-Smyth cuspidal invariants. We will then explain in which sense these two approaches coincide for the quintic 3-fold.- EDGE: Alexander Kasprzyk (Nottingham) - Bounding the anticanonical degree of toric Gorenstein Fano varieties
26th April 2018, 3:05pm to 3:55pm JCMB, room 4325B -- Show/hide abstractAbstract: Abstract: In recent work with Balletti and Nill we proved a sharp bound on the volume of a reflexive polytope -- hence on the anticanonical degree of a toric Gorenstein Fano variety. In this talk I will review some of the known classifications of Fano polytopes and remind the audience of their connections to geometry. I will then explain how we proved a sharp bound in the case of reflexive polytopes. Finally, time permitting, I will point out an implication for the study of Fano manifolds via Mirror Symmetry.- EDGE: Thomas Baier (IST Lisbon) - Mabuchi Geodesic Rays on Hamiltonian K-Spaces and Mixed Polarizations
26th April 2018, 2:05pm to 2:55pm JCMB, room 4325B -- Show/hide abstractAbstract: The aim of this talk (based on joint work in progress with José Mourão and João Pimentel Nunes) is to link aspects of the metric geometry of manifolds with (non-abelian) Hamiltonian symmetry, and their geometric quantization: We provide a description of Mabuchi geodesic rays of K-invariant Kähler metrics, parametrized by convex functions on the moment map image (not unlike the well-known toric case). In the second part, we discuss K-invariant mixed polarizations (in the sense of geometric quantization), and how they arise in this setting.- EDGE: Chris Elliott (IHES) - The Multiplicative Hitchin System in Supersymmetric Gauge Theory
19th April 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstractAbstract: Abstract: Multiplicative Higgs bundles are an analogue of ordinary Higgs bundles where the Higgs field takes values in a Lie group instead of its Lie algebra. In this talk I'll discuss two contexts where multiplicative Higgs bundles appear in supersymmetric gauge theory. I'll explain how hyperkähler structures on these moduli spaces arise physically and mathematically and relate to the theory of Poisson Lie groups, and finally I'll introduce a speculative multiplicative analogue of the geometric Langlands conjecture. This is based on joint work in progress with Vasily Pestun.- EDGE: Swarnava Mukhopadhyay (Bonn) - Hyperplane arrangements and invariants of tensor products.
12th April 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstractAbstract:We study the map from compactly supported cohomology to the usual cohomology of a complement of a hyperplane arrangement and give an explicit formula for a generalized version of this map. Our motivation comes from work of Schechtman and Varchenko who connected invariant theoretic objects to the cohomology of local systems on complements of hyperplane arrangements. Our results allow to equip certain spaces of invariants with interesting mixed Hodge structures via the Knizhnik-Zamolodchikov connection. This is a joint work with P. Belkale and P. Brosnan.
- EDGE: Giulio Codogni (Ecole Polytechnique Fédérale de Lausanne) - Positivity of the Chow-Mumford line bundle for families of K-stable klt Fano varieties
5th April 2018, 2:05pm to 2:55pm JCMB, seminar room 5323 -- Show/hide abstractAbstract: The Chow-Mumford (CM) line bundle is a functorial line bundle defined on the base of any family of polarized varieties, in particular on the base of families of klt Fano varieties. It is conjectured that it yields a polarization on the conjectured moduli space of K-semi-stable klt Fano varieties. This boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-semi-stable/K-polystable fibers. In this talk, I will present a proof of the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming stability only for very general fibers. These results work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. I will also present an application to the classification of Fano varieties. This is a joint work with Zs. Patakfalvi.- EDGE: Nils Carqueville (Vienna) - Topological quantum field theories: state sums and defects
29th March 2018, 2:05pm to 2:55pm JCMB, Seminar room, 5323 -- Show/hide abstractAbstract: A general framework will be discussed which unifies group orbifolds and state sum
models, in the context of topological quantum field theory (TQFT) in arbitrary
dimension. After a review of the 2-dimensional case, I will outline general aspects
of the construction and and discuss examples in 3 dimensions, including Turaev-Viro
theory as and orbifold, as well as surface defects in quantised Chern-Simons theory.
(Based on joint work with I. Runkel and G. Schaumann.)- EDGE: Andrea Fanelli (Dusseldorf) - Del Pezzo fibrations in positive characteristic
15th March 2018, 2:05pm to 2:55pm JCMB Seminar Room 5323 -- Show/hide abstractAbstract:In this talk, I will discuss some pathologies for the generic fibre of del Pezzo fibrations in characteristic p>0, motivated by the recent developments of the MMP in positive characteristic. The main application of the joint work with Stefan Schröer concerns 3-dimensional Mori fibre
spaces.- EDGE: Naoki Koseki (Tokyo) - Perverse coherent sheaves on blow-ups at codimension two loci
9th March 2018, 2:05pm to 2:55pm JCMB, Room 4312 -- Show/hide abstractAbstract: Let X be the blow up of a smooth projective variety Y along codimension two
smooth closed subvariety. We will discuss how the moduli space of sheaves on X
and that of Y can be related. The key method is the notion of perverse coherent sheaves
introduced by T.Bridgeland. Our main result is a higher dimensional generalization of the
result of H.Nakajima and K.Yoshioka.
As an application of our main result, we will also discuss about the birational geometry of
Hilbert scheme of two points.- EDGE: Takero Fukuoka (Tokyo/Warwick) - Relative linear extensions of sextic del Pezzo fibrations
1st March 2018, 2:05pm to 2:55pm JCMB, Seminar Room 5325 -- Show/hide abstractAbstract: Abstract: An extremal contraction from a non-singular projective 3-fold onto a smooth curve is so-called a del Pezzo fibration. It is classically known that every del Pezzo surface $S$ is a (weighted) complete intersection of a certain Fano variety. In order to study del Pezzo fibrations, it is important to relativize such descriptions for those. The main result of this talk shows that the sextic del Pezzo fibrations are relative linear sections of $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$-fibrations and $\mathbb{P}^2 \times \mathbb{P}^2$- fibrations, which are constructed as Mori fiber spaces with smooth total space. As an application, we will classify the singular fibers of sextic del Pezzo fibrations.- EDGE: Sjoerd Beentjes (University of Edinburgh) - The crepant resolution conjecture for Donaldson-Thomas invariants
15th February 2018, 2:05pm to 2:55pm Seminar Room 5323 JCMB -- Show/hide abstractAbstract: Abstract: Donaldson-Thomas (DT) invariants are integers virtually enumerating curves on a Calabi-Yau threefold. These invariants are typically organised in a generating series. The crepant resolution conjecture predicts a transformation rule for the DT generating series of two related Calabi-Yau threefolds, in the setting of the McKay correspondence. In this talk, I will state this conjecture, explain why it is really a conjecture about expansions of rational functions, and sketch a proof using wall-crossing arguments and the motivic Hall algebra. This is joint work with John Calabrese and Jørgen Rennemo.- EDGE: Julio Andrade (University of Exeter) - Two Problems Involving the Divisor Functions
8th February 2018, 2:05pm to 2:55pm EDGE: JCMB, Seminar room, 5323 -- Show/hide abstractAbstract: Abstract: In this talk, I will discuss two problems involving the divisor functions. The first problem is about the auto-correlation of the values of the divisors functions and I will describe how we can completely solve the problem in the function field setting using a combination of analytic and algebraic techniques. The second problem is about the maxima pairwise of the divisors functions and I will present some recent results that improve some old results of Erdos and Hall.- EDGE: Andrea Ricolfi (Bonn) - The DT/PT correspondence for smooth curves
30th January 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstractAbstract: Abstract: Donaldson-Thomas (DT) and Pandharipande-Thomas (PT) invariants are enumerative invariants attached to Calabi-Yau 3-folds. They are related by a wall-crossing type formula called the “DT/PT correspondence”. After a general introduction, we will discuss a variation of this correspondence, relating the “local” DT and PT invariants, encoding the contribution of a single smoooth curve to the full (classical) invariants.- Joint EDGE-MAXIMALS: Peter Samuelson (University of Edinburgh) - The Hall algebra of the Fukaya category of a surface
16th January 2018, 3:05pm to 3:55pm JCMB, Seminar Room 5323 -- Show/hide abstractAbstract: Abstract: The construction of the Fukaya category of a symplectic manifold is inspired by string theory: roughly, objects are Lagrangians, morphisms are intersection points, and composition of morphisms comes from "holomorphic disks." For surfaces, a combinatorial construction of the (partially wrapped) Fukaya category was recently given by Haiden, Katzarkov, and Kontsevich. We will discuss this category and some recent results involving its (derived) Hall algebra (joint with B. Cooper).- EDGE: Tran Bach (Edinburgh) - On k-normality and Regularity of Normal Toric Varieties
7th December 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstractAbstract: We will give a bound for a very ample lattice polytope to be k-normal. Equivalently, we give a new combinatorial bound for the Castelnuovo-Mumford regularity of normal projective toric varieties.- EDGE: Alex Lemmens (University of Leuven) - Syzygies of toric varieties
30th November 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstractAbstract: Syzygies are a basic homological invariant for modules over a ring. One gathers the dimensions of the syzygy spaces in a table of integers called the graded Betti table. There are deep connections between the geometry of algebraic varieties (for instance curves) and their syzygies. In this talk we study the link between the graded Betti table of a projective toric variety and the combinatorics of the defining polytope.- EDGE: Stefan Schreieder (University of Munich) - The rationality problem for quadric bundles
23rd November 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstractAbstract: We study the rationality problem for quadric bundles X over rational bases S. By a theorem of Lang, such bundles are rational if r>2^n-2, where r denotes the fibre dimension and n=dim(S) denotes the dimension of the base. We show that this result is sharp. In fact, for any r at most 2^n-2, we show that many smooth r-fold quadric bundles over rational n-folds are not even stably rational. Our result is based on a generalization of the specialization method of Voisin and Colliot-Thélène—Pirutka.- EDGE: Soheyla Feyzbakhsh (University of Edinburgh) - Applications of Bridgeland stability conditions to classical algebraic geometry
16th November 2017, 2:05pm to 2:55pm Seminar room JMCB 5323 -- Show/hide abstractAbstract: I will explain how wall-crossing with respect to Bridgeland stability conditions provides a new upper bound for the number of global sections of sheaves on K3 surfaces. This, in particular, extends and completes a program proposed by Mukai to reconstruct a K3 surface from a curve on that. Furthermore, the upper bound characterizes special vector bundles on curves on K3 surfaces, which have the maximum number of global sections for the minimum degree. Therefore, it gives an explicit expression for Clifford indices of curves on K3 surfaces.- EDGE: Navid Nabijou (Imperial College London) - Relative Quasimaps and a Lefschetz-type Formula
9th November 2017, 2:05pm to 2:55pm JCMB Seminar room 5323 -- Show/hide abstractAbstract: The theory of stable quasimaps provides an alternative system of curve counts to the usual Gromov-Witten invariants. In joint work with Luca Battistella, we define moduli spaces of relative stable quasimaps to a pair (X,Y), where Y is a hyperplane section in X. Intuitively these spaces parametrise quasimaps in X with specified contact orders to Y, and can be used to define relative quasimap invariants. By investigating these moduli spaces we obtain a Lefschetz-type formula, expressing certain quasimap invariants of Y in terms of the invariants of X. Since the I-function from mirror symmetry is equal to a generating function for quasimap invariants, this result can be viewed as a “quantum Lefschetz theorem for I-functions.” It agrees with an earlier formula obtained by Ciocan-Fontanine and Kim.- EDGE: Andrea Petracci (Nottingham) - Deformations of toric singularities and Mirror Symmetry
2nd November 2017, 2:05pm to 2:55pm JCMB Seminar room 5323 -- Show/hide abstractAbstract: Abstract: Deformations of affine toric varieties have been studied by Klaus Altmann and Anvar Mavlyutov: they also give constructions of such deformations starting from Minkowski decompositions of polyhedra. On the other hand, deformations of projective toric varieties are not well understood. In this talk, I will present an approach, based on mirror symmetry, to construct deformations of Gorenstein affine toric 3-folds in the context of the Gross-Siebert program. This approach can be globalised and gives a combinatorial recipe to construct smoothings of Gorenstein toric Fano 3-folds. This talk is partly based on work in progress in collaboration with Alessio Corti, Paul Hacking, and Thomas Prince.- EDGE: Ben Davison (Glasgow) - The integrality conjecture for coherent sheaves on a Calabi-Yau 3-fold.
26th October 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstractAbstract: BPS numbers are certain invariants that "count" coherent sheaves on Calabi-Yau 3-folds. Because of subtleties in the definition, especially in the presence of strictly semistable sheaves, it is not a priori clear that the numbers are in fact integers. I will present a recent proof with Sven Meinhardt of this integrality conjecture. The conjecture follows from a stronger conjecture, namely that a certain constructible function on the coarse moduli space of semistable sheaves defined by Joyce and Song is integer valued. This conjecture in turn is implied by the stronger conjecture that their function is in fact the pointwise Euler characteristic of a perverse sheaf. We prove all of these conjectures by defining this perverse sheaf.- EDGE: Nicola Pagani (Liverpool) - The indeterminacy of the Abel-Jacobi maps
19th October 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstractAbstract: The Abel-Jacobi morphisms are the sections of the forgetful morphism from the universal Jacobian to the corresponding moduli space of smooth pointed curves. When the source and target moduli spaces are compactified, these morphisms can be reinterpreted as rational maps, and it is natural to ask for their locus of indeterminacy. We explicitly characterize the indeterminacy locus, which depends on the chosen compactification of the universal Jacobian (for the source we fix the Deligne-Mumford compactification \bar{M}_{g,n} by means of stable curves). In particular, we deduce that for evey Abel-Jacobi map there exists a compactification such that the map extends to a well-defined morphism on \bar{M}_{g,n}. This offers an approach to define and then compute the classes of several different extensions of the "Jacobian double ramification cycles" (= the pullbacks of the zero section via the Abel-Jacobi maps). This is a joint work with Jesse Kass.- EDGE: Adam Boocher (University of Utah) - Inequalities on Betti Numbers
12th October 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstractAbstract: If X is a projective variety then the Betti numbers of its coordinate ring are a sequence of numbers that encode the symmetry and complexity of the defining equations. They are a strict refinement of the Hilbert function and can detect subtle geometric information. In this talk I'll discuss two inequalities concerning the Betti numbers. One, the Buchsbaum-Eisenbud, Horrocks rank conjecture says roughly that the Koszul complex is the ``smallest'' possible resolution. The second, a conjecture of Conca for Koszul rings, is motivated by the Taylor resolution for monomial ideals. I'll discuss recent work on these topics and how the algebra structure of Koszul homology has given us hints about to the structure of these algebras.- EDGE: Mara Ungureanu (HU Berlin) - From billiards in polygons to de Jonquières divisors
5th October 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstractAbstract: The aim of this talk is to give a broad overview of the circle of ideas surrounding the de Jonquières divisors on algebraic curves. I will introduce the notion of translation surfaces and their parameter spaces by means of billiards on rational polygons. These parameter spaces lend themselves to a natural algebro-geometric interpretation as strata of holomorphic differentials on the moduli space of curves and as such can be related to certain cohomological field theories. Finally, the de Jonquières divisors will emerge as generalisations of the previous structures and I will discuss related recent progress and open questions.- EDGE: Gwyn Bellamy (Glasgow) - Counting symplectic resolutions of quotient singularities
28th September 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstractAbstract: In this talk I will explain how one can use the representation theory of symplectic reflection algebras to count the number of symplectic resolutions admitted by a symplectic quotient singularity. In particular, I will explain the role played by Calogero-Moser families in this story. This is based partially on joint work with T. Schedler and U. Thiel, and builds on the general theory developed by Namikawa.- EDGE: Lorenzo Foscolo (Heriot-Watt) - Complete non-compact G2 manifolds from asymptotically conical Calabi-Yau 3-folds
21st September 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstractAbstract: G2 manifolds are the Riemannian 7-manifolds with G2 holonomy and in many respects can be regarded as analogues of Calabi-Yau 3-folds. In joint work with Mark Haskins and Johannes Nordström we construct infinitely many families of new complete non-compact G2 manifolds (only four such manifolds are currently known). The underlying smooth 7-manifolds are all circle bundles over asymptotically conical Calabi-Yau 3-folds, the metrics are circle-invariant and have an asymptotic geometry that is the 7-dimensional analogue of the geometry of 4-dimensional ALF hyperkähler metrics. After describing the main features of our construction I will concentrate on some illustrative examples, describing how results in algebraic geometry about isolated singularities and their resolutions can be used to produce examples of complete G2 manifolds and interesting submanifolds inside them.- EDGE: Elizabeth Gasparim (Universidad Católica del Norte) - A Landau--Ginzburg model without projective mirrors
8th August 2017, 2:05pm to 2:55pm JCMB 5323 -- Show/hide abstractAbstract: Using methods of Lie theory we construct a family of Symplectic Lefschetz fibrations. I will explain the construction, and discuss mirror symmetry in a simple case, the adjoint orbit of sl(2). This is joint work with Ballico, Barmeier, Grama, and San Martin.- EDGE: Benjamin Schmidt (UT Austin) - The Genus of Space Curves
29th June 2017, 5:30pm to 6:30pm JCMB 5326 -- Show/hide abstractAbstract: A 19th century problem in algebraic geometry is to understand the relation between the genus and the degree of a curve in complex projective space. This is easy in the case of the projective plane, but becomes quite involved already in the case of three dimensional projective space. In this talk I will give an introduction to the topic, introduce stability conditions in the derived category, and explain how the two can be related. This is based on joint work in progress with Emanuele Macri.- EDGE: Francis Bischoff (Toronto University) - Morita equivalence and the generalized Kahler potential
15th June 2017, 2:10pm to 3:00pm Seminar room JCMB 5323 -- Show/hide abstractAbstract: In this talk I will present a new approach to Generalized Kahler geometry in which a GK structure of symplectic type can be described in terms of a holomorphic symplectic Morita equivalence along with a brane bisection. I will then explain how this new approach can be applied to the problem of describing a GK structure in terms of holomorphic data and a single real-valued function (the generalized Kahler potential). This is joint work with Marco Gualtieri and Maxim Zabzine.- EDGE: De-Qi Zhang (NUSingapore) - Polarized endomorphisms of projective varieties
30th May 2017, 10:10am to 11:00am JCMB 5323 -- Show/hide abstractAbstract: An endomorphism f on a normal projective variety X is polarized if the f-pullback of an ample divisor H on X is linearly equivalent to the multiple qH for some natural number q larger than 1. Examples of such f include self-maps of the projective spaces (or more generally Fano varieties of Picard number 1) and multiplication map of complex tori. We show that we can run the f-equivariant minimal model program (MMP) on smooth or mildly singular X, and conclude that the building blocks of polarized endomorphisms are those on Fano varieties or complex tori and their quotients. This is a joint work with S. Meng.- EDGE: Jason Lo (CSU) - Behaviour of slope stable sheaves under a Fourier-Mukai transform
25th May 2017, 2:10pm to 3:00pm JCMB Seminar room 5323 -- Show/hide abstractAbstract: Over the years, Fourier-Mukai transforms have been used in constructing birational maps or isomorphisms between moduli spaces, explaining symmetries in counting invariants, and constructing Bridgeland stability conditions on Abelian threefolds. All these problems involve the question of how stable objects behave under a Fourier-Mukai transform. In this talk, I will consider this question for slope-stable torsion-free sheaves on a product elliptic threefold. If time permits, I will also discuss this question for stable 1-dimensional sheaves.- EDGE: Xiaolei Zhao (Northeastern) - Canonical points on K3 surfaces and hyper-Kähler varieties.
4th May 2017, 2:10pm to 3:00pm JCMB 5323 -- Show/hide abstractAbstract: The Chow groups of algebraic cycles on algebraic varieties have many mysterious properties. For K3 surfaces, on the one hand, the Chow group of 0-cycles is known to be huge. On the other hand, the 0-cycles arising from intersections of divisors and the second Chern class of the tangent bundle all lie in a one-dimensional subgroup. A conjecture of Beauville and Voisin gives a generalization of this property to hyper-Kähler varieties. In my talk, I will recall these beautiful stories, and explain a conjectural connection between the K3 surface case and the hyper-Kähler case. If time permits, I will also explain how to extend this connection to Fano varieties of lines on some cubic fourfolds. This is based on a joint work with Junliang Shen and Qizheng Yin.- EDGE: Richard Szabo (Heriot-Watt) - Orientifolds and Real bundle gerbes
20th April 2017, 2:10pm to 3:00pm JCMB Seminar room 5323 -- Show/hide abstractAbstract: We describe a generalisation of the notions of bundle gerbe and bundle gerbe modules appropriate to spaces with involution, which realise a twisted version of Atiyah's Real K-theory. This construction is applied to the topological classification of D-branes in orientifolds of string theory with H-flux.- EDGE: Emilio Franco (Porto) - Moduli spaces of Lambda-modules on abelian varieties
6th April 2017, 2:40pm to 3:30pm Seminar room JCMB 5323 -- Show/hide abstractAbstract: Let \Lambda be a D-algebra in the sense of Bernstein and Beilinson. Higgs bundles, vector bundles with flat connections, co-Higgs bundles... are examples of \Lambda-modules for particular choices of \Lambda. Simpson studied the moduli problem for the classification of \Lambda-modules over Kahler varieties, proving the existence of a moduli space Lambda-modules. Using the Polishchuck-Rothstein transform for modules of D-algebras over abelian varieties, we obtain a description of the moduli spaces of \Lambda-modules of rank 1. We also proof that polystable \Lambda module decompose as a direct sum of rank 1 \Lambda-modules. This allow us to describe the module spaces for arbitrary rank and trivial Chern classes.- EDGE/EMPG: Dennis The (University of Tromsø) - Exceptionally simple PDE
5th April 2017, 12:10pm to 1:00pm JCMB 5323 -- Show/hide abstractAbstract: In back-to-back articles in 1893, Cartan and Engel gave the first realisations of $G_2$, the smallest of the exceptional complex simple Lie groups, as the symmetries of a geometric object. I will show how to generalize this story in a remarkably uniform manner to obtain analogous explicit geometric realisations for any complex simple Lie group (…except for $SL_2$).- EDGE: Alexey Bondal (Steklov/Higher School of Economics) - Flobers: Flops and Schobers
4th April 2017, 12:10pm to 1:00pm JCMB 4312 -- Show/hide abstractAbstract: I shall explain how the ideology of schobers, the categorification of the perverse sheaves on stratified topological spaces, should amplify our understanding of some parts of the Minimal Model Program in Birational Geometry.- EDGE: Philip Boalch (Paris Sud) - Nonperturbative hyperkahler manifolds and H3 surfaces
30th March 2017, 2:40pm to 3:30pm JCMB Seminar room 5323 -- Show/hide abstractAbstract: In the 80s and 90s many hyperkahler manifolds were constructed out of additive/Lie algebraic objects such as coadjoint orbits, cotangent bundles and quiver representations. I'll explain how some of these have multiplicative or ``more transcendental'' analogues in wild nonabelian Hodge theory (i.e. moduli of Higgs bundles or connections or solutions of Hitchin's equations, on noncompact curves). In particular 12 deformation classes of complete hyperkahler four-manifolds occur, including multiplicative analogues of the hyperkahler manifolds of Eguchi-Hanson and Atiyah-Hitchin. We call them H3 surfaces in honour of Higgs, Hitchin and Hodge. For the most part we will work algebraically and describe the construction of the underlying holomorphic symplectic manifolds as finite dimensional multiplicative symplectic quotients. This construction was recently completed in joint work with D. Yamakawa, extending the author's construction in the untwisted case 2002-2014.- EDGE: Mihai Fulger (Princeton) - Seshadri constants for curves
23rd March 2017, 2:10pm to 3:00pm JCMB Seminar room 5323 -- Show/hide abstractAbstract: The Seshadri constants of nef divisors are important local measures of positivity of the divisor. We develop a natural dual theory for movable curves and show how it mirrors the case of divisors. For example they give a criterion of "ampleness", their vanishing locus is an analogue of the nonample locus for divisors, and in a sense they measure jet separation at smooth points. We also present examples.- EDGE: Jenia Tevelev - The Craighero-Gattazzo surface is simply-connected
16th March 2017, 4:10pm to 5:00pm JCMB 5323 -- Show/hide abstractAbstract: We show that the Craighero–Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply-connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero–Gattazzo surface is the only explicit example of a smooth simply-connected complex surface of geometric genus zero with ample canonical class. We hope that our method will find other applications: to prove a topological fact about a complex surface, we use an algebraic reduction mod p technique and deformation theory. Joint work with Julie Rana and Giancarlo Urzua.- EDGE: Giovanni Mongardi (Milan) - The last involutions
16th March 2017, 2:10pm to 3:00pm JCMB Seminar Room 5323 -- Show/hide abstractAbstract: In this joint work with C. Camere and G. and M. Kapustka, we give two construction of the last two families of nonsymplectic involutions on deformations of K3^[2] using twisted sheaves on K3 surfaces and special lagrangians, proving that these families are also unirational.- EDGE: Marcos Jardim - Moduli spaces of instanton sheaves on projective space
9th March 2017, 2:10pm to 3:00pm JCMB Seminar room 5323 -- Show/hide abstractAbstract: Instanton bundles were introduced by Atiyah, Drinfeld, Hitchin and Manin in the late 1970s as the holomorphic counterparts, via twistor theory, to anti-self-dual connections (a.k.a. instantons) on the sphere S^4. We will revise some recent results regarding some of the basic geometrical features of their moduli spaces, and on its possible degenerations. We will describe the singular loci of instanton sheaves, and how these lead to new irreducible components of the moduli space of stable sheaves on the projective space.- EDGE: Yanki Lekili (KCL) - Fukaya categories of plumbings and multiplicative preprojective algebras
2nd March 2017, 2:10pm to 3:00pm JCMB Seminar room (5th floor) -- Show/hide abstractAbstract: Given an arbitrary graph, I will construct an open symplectic 4-manifold obtained by plumbing cotangent bundles of 2-spheres. I will then describe a rigorous and explicit computation of the wrapped Fukaya category of this symplectic 4-manifold using techniques of Legendrian surgery. It turns out that the endomorphism algebra of a generating set of objects of this Fukaya category can be identified with the (derived) multiplicative preprojective algebra associated to the original graph. In the case, the underlying graph is Dynkin, we have a (derived) Koszul duality between wrapped and compact Fukaya categories. This leads to computations of Hochschild cohomology of these categories in this case. In the case the graph is of extended Dynkin type, a version of homological mirror symmetry can be confirmed as a result of our computations. This talk is an update on a previous joint work with Tolga Etgu.- EDGE: Boris Kruglikov(Tromsø) - The gap phenomenon and submaximally symmetric structures
16th February 2017, 2:10pm to 3:00pm JCMB 5215 -- Show/hide abstractAbstract: In 2014 together with Dennis The we resolved the gap problem in complex or split-real parabolic geometry, i.e. we computed the amount of submaximal symmetry for every geometry in the class. Results of this type have been known for selected geometries since Ricci, Tresse, Fubini, Cartan, Egorov, Kobayashi, Sinyukov, Yano and some others via specific techniques. However it was in our paper that we first presented a universal solution for a large class of geometries, including conformal structures, systems of second order ODE, almost Grassmanian and Lagrangian structures, generic parabolic distributions, exceptional geometries etc. In later development we covered CR-structures, c-projective structures and some other real (non-split) specification. I will review the results and overview further developments and problems. References: [1] Boris Kruglikov, Dennis The, The gap phenomenon in parabolic geometries, Journal für die reine und angewandte Mathematik (Crelle's Journal) DOI 10.1515/crelle-2014-0072 (2014). [2] Boris Kruglikov, Vladimir Matveev, Dennis The, Submaximally symmetric c-projective structures, International Journal of Mathematics 27, No. 3, 1650022 - 34 pp, (2016). [3] Boris Kruglikov, Submaximally symmetric CR-structures, Journal of Geometric Analysis DOI: 10.1007/s12220-015-9663-x (2015). [4] Boris Kruglikov, Henrik Winther, Lenka Zalabova, Submaximally symmetric quaternionic structures, arXiv: 1607.02025 (2016).- EDGE: Ruadhai Dervan (Cambridge) - K-stability for Kähler manifolds
9th February 2017, 2:10pm to 3:00pm JCMB Seminar room 5323 -- Show/hide abstractAbstract: K-stability is an important algebro-geometric concept introduced by Tian and Donaldson, which is related to the existence of "canonical" Kähler metrics on smooth projective varieties. We formulate a notion of K-stability for Kähler manifolds, and prove that Kähler manifolds admitting canonical Kähler metrics are K-stable. We also show how Kähler techniques can prove stronger results even in the projective case. Some of this is joint work with Julius Ross.- EDGE: George Dimitrov (ICTP) - Norms of non-commutative projective spaces via Bridgeland stability conditions
2nd February 2017, 2:10pm to 3:00pm JCMB Seminar Room 5323 -- Show/hide abstractAbstract: Tom Bridgeland assigned to any triangulated category a complex manifold: the space of stability conditions on it. In a joint work with Ludmil Katzarkov we prove that the Bridgeland stability spaces on wild Kronecker quivers are CxH and these calculations suggest a new notion of a norm. To a triangulated category T which has property of a phase gap, we attach a number ||T||_epsilon \in [0,(1-epsilon)\pi] depending on a parameter epsilon \in (0,1). In this talk, I will tell more about this.- EDGE: Alex Perry (Columbia University)- Derived categories of Gushel-Mukai varieties
26th January 2017, 2:10pm to 3:00pm JCMB 1501 -- Show/hide abstractAbstract: I will discuss the derived categories of Fano varieties of Picard number 1, degree 10, and coindex 3. In particular, I will describe an interesting semiorthogonal component of the derived category of such a variety, and discuss its behavior for some birationally special families of fourfolds. This is joint work with Alexander Kuznetsov.- EDGE: Caucher Birkar(Cambridge)- Birational geometry of Fano varieties
19th January 2017, 2:10pm to 3:00pm JCMB 1501- EDGE: Jinhyung Park(KIAS) - Newton-Okounkov bodies and asymptotic invariants of divisors
8th December 2016, 2:10pm to 3:00pm JCMB 1501 -- Show/hide abstractAbstract: A Newton-Okounkov body is a convex body in Euclidean space associated to a divisor on an algebraic variety with respect to an admissible flag. After briefly recalling basics of Newton-Okounkov bodies of ample or big divisors, I introduce two natural ways to associate Newton-Okounkov bodies to pseudoeffective divisors. We then study various asymptotic invariants of pseudoeffective divisors using these convex bodies. This is joint work with Sung Rak Choi, Yoonsuk Hyun, and Joonyeong Won.- EDGE: Theo Raedschelders (Vrije Universiteit Brussel) - Derived categories of noncommutative quadrics and Hilbert schemes of points
1st December 2016, 2:10pm to 3:00pm JCMB1501 -- Show/hide abstractAbstract: Recent work of Orlov's suggests that for a smooth projective rational surface S, it should be possible to embed Perf-S into Perf-M_S, where M_S is also smooth projective and represents some moduli problem on S. Moreover, noncommutative deformations of S should embed into commutative deformations of M_S. I will discuss Orlov's work and consider the specific example of a smooth quadric surface in more detail. This is joint work with Pieter Belmans.- EDGE: Maxym Fedorchuk (Boston) - Invariant theory of Artinian Gorenstein algebras
24th November 2016, 2:10pm to 3:00pm JCMB1501 -- Show/hide abstractAbstract: I will discuss the interplay between hypersurface singularities, their Milnor algebras, and classical invariant theory of homogeneous forms. In particular, I will prove that a contravariant that associates to a smooth homogeneous form the Macaulay inverse system of its Milnor algebra preserves GIT stability. I will discuss some applications of this result, for example to the direct sum decomposability of polynomials, and many related open problems.- EDGE: Michael Wemyss (Glasgow) - On the Classification of 3-fold Flops
17th November 2016, 2:10pm to 3:10pm JCMB 1501 -- Show/hide abstractAbstract: I will outline a still largely conjectural framework in which smooth flops of irreducible curves in 3-folds are classified by certain elements of the free algebra in two variables. The correspondence is explicit, and by manipulating the noncommutative side, we are able to use this framework to produce many new examples of 3-fold flops, the first since the early 1980s. In particular, we show that there are more type D flops than simply the standard Laufer-type examples, and we also give the first examples in type E. Our flops come equipped with their Gopakumar--Vafa invariants, and I will discuss some of the consequences of our constructions to curve counting invariants. This is all joint work with Gavin Brown.- EDGE: Piotr Pragacz (Polisch Academy of Sciences) - Duality on Grassmann bundles and applications
10th November 2016, 2:10pm to 3:00pm JCMB 1501 -- Show/hide abstractAbstract: The duality of Schubert calculus allows one to present any class on a Grassmannian as an integer combination of Schubert classes. We state and prove a duality theorem on a Grassmann bundle using its Gysin map and skew Schur functions. We give new Gysin formulas for flag bundles and an alternative derivation of the Kempf-Laksov formula. This is a joint work with Lionel Darondeau.- EDGE: Dimitra Kosta (Edinburgh) - Maximum Likelihood Estimation for models corresponding to toric del Pezzo surfaces
3rd November 2016, 2:10pm to 3:10pm JCMB 1501 -- Show/hide abstractAbstract: I will present the correspondence between some statistical models and toric varieties and show how one can obtain a closed-form for the Maximum Likelihood Estimate of algebraic statistical models which correspond to cubic and quartic toric del Pezzo surfaces with Du Val singular points.- EDGE: Seung-Jo Jung (KIAS, Seoul) - G-constellations and quotient singularities
27th October 2016, 2:10pm to 3:00pm JCMB 1501 -- Show/hide abstractAbstract: Let G be a finite group in GL_n(C). A G-equivariant sheaf F on C^n is called a G-constellation if H^0(F) is isomorphic to the regular representation of G as a G-representation. In this talk, we discuss moduli interpretations of many interesting birational models of C^n/G using G-constellations.- EDGE: Diletta Martinelli (Edinburgh) - On the number of minimal models of a smooth threefold of general type
20th October 2016, 2:10pm to 3:10pm JCMB 1501 -- Show/hide abstractAbstract: Finding minimal models is the first step in the birational classification of smooth projective varieties. After it is established that a minimal model exists some natural questions arise such as: is it the minimal model unique? If not, how many are they? After recalling all the necessary notions of the Minimal Model Program, I will explain that varieties of general type admit a finite number of minimal models. I will talk about a recent joint project with Stefan Schreieder and Luca Tasin where we prove that in the case of threefolds this number is bounded by a constant depending only on the Betti numbers. I will also show that in some cases it is possible to compute this constant explicitly.- EDGE: Jesus Martinez-Garcia (Max Planck) - Moduli space of cubic surfaces and their anticanonical divisors
13th October 2016, 2:10pm to 3:00pm JCMB 1501 -- Show/hide abstractAbstract: We study variations of GIT quotients of log pairs (X,D) where X is a hypersurface of some fixed degree and D is a hyperplane section. GIT is known to provide a finite number of possible compactifications of such pairs, depending on one parameter. Any two such compactifications are related by birational transformations. We describe an algorithm to study the stability of the Hilbert scheme of these pairs, and apply our algorithm to the case of cubic surfaces. Finally, we relate this compactifications to the (conjectural) moduli space of log K-semistable pairs showing that any log K-stable pair is an element of our moduli and that there is a canonically defined CM line bundle that polarizes our moduli. This is joint work with Patricio Gallardo (University of Georgia) and Cristian Spotti (Aarhus University).- EDGE: Julian Holstein (Lancaster) - The derived period map
6th October 2016, 2:10pm to 3:00pm JCMB 1501 -- Show/hide abstractAbstract: In this talk I develop the global period map in the context of derived geometry, generalising Griffiths' classical period map as well as the infinitesimal derived period map. (I will not assume previous knowledge of derived algebraic geometry.)- EDGE: Roberto Fringuelli (Edinburgh) - The Picard group of the universal moduli space of vector bundles on stable curves.
29th September 2016, 2:10pm to 3:00pm JCMB 1501 -- Show/hide abstractAbstract: In this talk, we present the moduli stack of properly balanced vector bundles on semistable curves and we determine explicitly its Picard group. As a consequence, we obtain an explicit description of the Picard groups of the universal moduli stack of vector bundles on smooth curves and of the Schmitt's compactification over the stack of stable curves. We show some results about the gerbe structure of the universal moduli stack over its rigidification by the natural action of the multiplicative group. In particular, we give necessary and sufficient conditions for the existence of a universal family of an open substack of the rigidification. In the remaining time, we discuss some consequences for the associated moduli varieties.- EDGE: Igor Pak (UCLA) - What is a formula?
19th September 2016, 2:10pm to 3:10pm JCMB 6201 -- Show/hide abstractAbstract: Integer sequences arise in a large variety of combinatorial problems as a way to count combinatorial objects. Some of them have nice formulas, some have elegant recurrences, and some have nothing interesting about them at all. Can we characterize when? Can we even formalize what is a "formula"? I will give a mini-survey aiming to answer these questions. At the end, I will present some recent results counting certain permutation classes, and finish with open problems.- EDGE: Benjamin Bakker (Humboldt-Universitaet Berlin) - The birational geometry of complex ball quotients
21st June 2016, 1:45pm to 2:45pm JCMB 5327 -- Show/hide abstractAbstract: Quotients of the complex ball by discrete groups of holomorphic isometries naturally arise in many moduli problems---for instance, those of low genus curves, del Pezzo surfaces, certain K3 surfaces, and cubic threefolds. On the other hand, the complex ball is the only bounded symmetric domain to admit nonarithmetic lattices, so birationally classifying such quotients is of particular interest. In joint work with J. Tsimerman, we show that in dimension $n\geq4$ every smooth complex ball quotient is of general type, and further that the canonical bundle $K_{\overline{X}}$ is ample on the toroidal compactification $\overline{X}$ for $n\geq 6$. The proof uses a hybrid technique employing both the hyperbolic geometry of the uniformizing group and the algebraic geometry of the toroidal compactification. We will also discuss applications to bounding the number of cusps and the Green--Griffiths conjecture.- EDGE: Asher Auel (Yale) - Brill-Noether theory for cubic fourfolds
2nd June 2016, 2:00pm to 3:00pm JCMB 4325A -- Show/hide abstractAbstract: Certain cubic fourfolds have K3 surfaces associated to them via Hodge theory. I will discuss how the Brill-Noether properties of special divisors on curves in those associated K3 surfaces can be reflected in the geometry of the cubic fourfolds. For example, cubic fourfolds containing two disjoint planes have an associated K3 surface of degree 14 that is Brill-Noether special in the sense of Lazarsfeld and Mukai. One application is to a description of the boundary of the locus of cubic fourfolds that have pfaffian presentations.- EDGE: Dario Beraldo (Oxford) - On the notion of temperedness in geometric Langlands
5th May 2016, 2:00pm to 4:00pm JCMB 6206 -- Show/hide abstractAbstract: Using the rich algebraic structure enjoyed by Hochschild cochains, I will define a new notion of "sheaf of categories", called $ShvCat^{HC}$, on an arbitrary prestack. This theory resembles the theory of D-modules, in the same way as Gaitsgory's notion of sheaf of categories resembles the theory of quasi-coherent sheaves. For a nice stack \mathcal{Y}, the \infty-category ShvCat^{HC}(\mathcal{Y}) is equivalent to the \infty-category of modules for an explicit monoidal DG category, denoted QCoh^{HC}(\mathcal{Y}). This category is closely related to ind-coherent sheaves on the formal completion of the diagonal of \mathcal{Y}. As an application, I will show that the two DG categories appearing in the geometric Langlands correspondence, Dmod(Bun_G) and IndCoh_N(LocSys_{\check{G}}), are both equipped with a QCoh^{HC}(LocSys_{\check G})-action. The compatibility of geometric Langlands with geometric Satake predicts that these actions be intertwined by the conjectural Langlands equivalence. The action of QCoh^{HC}(\mathcal Y) on a DG category \mathcal{C} allows to define tempered objects of \mathcal{C}. In particular, we obtain a new definition of tempered D-modules on Bun_G, thereby proving a conjecture of Arinkin and Gaitsgory.- EDGE: Alexey Bondal (Steklov/IPMU/Higher School of Economics) - Noncommutative moduli of genus 0 n-punctured curves
21st April 2016, 2:00pm to 3:00pm JCMB 4325B -- Show/hide abstractAbstract: Generalizations of the moduli space of stable n-punctured curves of genus zero will be discussed. One such generalized version is assigned to any finite dimensional algebra. In some cases, they are identified with moduli of quiver representations, thus giving a new geometric point of view on the later.- EDGE: Emile Bouaziz (Edinburgh) - The Twisted Chiral de Rham Complex
31st March 2016, 2:10pm to 3:00pm JCMB 5327 -- Show/hide abstractAbstract: The Chiral de Rham Complex of a smooth complex variety is an infinite dimensional refinement of the usual de Rham complex whose cohomology has modular properties. In joint work in progress with Ian Grojnowski we are studying an analogous object in the presence of a potential on the variety, refining instead the twisted de Rham complex, familiar from vanishing cycles theory.- EDGE: Kai Behrend (UBC) - The spectrum of the inertia operator on the motivic Hall algebra
24th March 2016, 2:10pm to 3:00pm JCMB 5327 -- Show/hide abstractAbstract: Following an idea of Bridgeland, we study the operator on the K-group of algebraic stacks, which takes a stack to its inertia stack. We prove that the inertia operator is diagonalizable when restricted to nice enough stacks, including those with algebra stabilizers. We use these results to prove a structure theorem for the motivic Hall algebra of a projective variety, and give a more conceptual definition of virtually indecomposable stack function. This is joint work with Pooya Ronagh.- EDGE: Bruce Bartlett (Oxford) - Three-dimensional bordism representations
17th March 2016, 1:10pm to 2:00pm JCMB 5327 -- Show/hide abstractAbstract: The collection of all oriented compact 1-, 2- and 3-manifolds can be assembled into a bicategory, which has a fairly simple "generators and relations" presentation obtained using Morse theory. Via the graphical calculus of "internal string diagrams", a representation of this bordism bicategory corresponds to an algebraic structure known as a modular category. I will give an overview of these results and speculate on the relationship to factorization homology.- EDGE: Roland Abuaf (IHES) - Compact hyperkähler categories
10th March 2016, 2:10pm to 3:00pm JCMB 5327 -- Show/hide abstractAbstract: Compact Calabi-Yau categories play a preeminent role in non-commutative geometry and in the mathematical background of string theory. Indeed, many manifestations of Kontsevich's Homological Mirror Symmetry conjectures are best understood when interpreted in the framework of 3-Calabi-Yau categories. In this talk, I want to introduce and discuss the basic properties of a new class of compact Calabi-Yau categories : the hyperkaehler categories. They are categorical analogues of compact hyperkaehler manifolds. The theory of non-commutative resolutions of singularities allows us to construct a large number of deformation classes of such categories in each dimension. For instance, I can construct at least 243 deformation classes of such categories in dimension 4 (compare with the only 2 deformation classes of hyperkaehler spaces of dimension 4 which are known in commutative geometry). If time permits, I would like to discuss a specific modular example (of dimension 4), for which Hochschild co-homology reveals some very intriguing features.- EDGE: Igor Krylov - Classification and birational rigidity of del Pezzo fibrations with an action of the Klein simple group
3rd March 2016, 2:10pm to 3:00pm JCMB 5327 -- Show/hide abstractAbstract: Cremona group of rank b is the group of birational transformations of the projective n-space. One was to study Cremona group is to study its finite subgroups. This problem can be translated to the geometric language: instead of subgroups of Cremona group isomorphic to a group G we can study rational G-Mori fiber spaces. This idea works particularly well for simple subgroups of Cremona group. I prove that any del Pezzo fibration over projective line with an action of the Klein simple group is either a direct product or a certain singular del Pezzo fibration X_n of degree 2. It is known that del Pezzo fibrations of degree 2 satisfying the K^2-condition are birationally superrigid. I extend this result to singular del Pezzo fibrations and prove that X_n are superrigid, in particular not rational, for n>2.- EDGE: Kaie Kubjas (Aalto) - Semialgebraic geometry of nonnegative and psd rank
25th February 2016, 2:10pm to 3:00pm JCMB 5327 -- Show/hide abstractAbstract: One of many definitions gives the rank of an $m \times n$ matrix $M$ as the smallest natural number $r$ such that $M$ can be factorized as $AB$, where $A$ and $B$ are $m \times r$ and $r \times n$ matrices respectively. In many applications, we are interested in factorizations of a particular form. For example, factorizations with nonnegative entries define the nonnegative rank and are closely related to mixture models in statistics. Another rank I will consider in my talk is the positive semidefinite (psd) rank. Both nonnegative and psd rank have geometric characterizations using nested polytopes. I will explain how to use these characterizations to derive a semialgebraic description of the set of matrices of nonnegative/psd rank at most $r$ in some small cases, and to study boundaries of this set. The talk is based on joint work with Rob H. Eggermont, Emil Horobet, Elina Robeva, Richard Z. Robinson, and Bernd Sturmfels- EDGE: Mohammad Akhtar (IHES) - Mutations and the Classification of Fano Varieties
18th February 2016, 2:10pm to 3:00pm JCMB 5327 -- Show/hide abstractAbstract: The classification of Fano varieties is an important long-standing problem in algebraic geometry. Mirror symmetry predicts that this problem should be equivalent to classifying a suitable class of Laurent polynomials up to an appropriate notion of equivalence. Recent work of Coates, Corti, Golyshev et al. suggests that the correct equivalence relation to impose is algebraic mutation of Laurent polynomials. This talk will introduce algebraic mutations and discuss the notion of combinatorial mutations, which are transformations of lattice polytopes induced by algebraic mutations of Laurent polynomials supported on them. Our focus will be on the case of surfaces, where the theory is particularly rich. Particular attention will be given to the role played by combinatorial mutations in the classification of Fano orbifold surfaces. This is joint work with Tom Coates, Alessio Corti and Alexander Kasprzyk.- EDGE: Yuki Hirano (Tokyo Metropolitan University/Edinburgh) - Derived Knoerrer periodicity and Orlov's theorem for gauged Landau-Ginzburgh models
11th February 2016, 2:10pm to 3:00pm JCMB 5327 -- Show/hide abstractAbstract: We prove Kn"orrer periodicity type equivalence between derived factorization categories of gauged LG models, which is an analogy of a theorem proved by Shipman and Isik independently. Combining the Kn"orrer periodicity type equivalence and the theory of variations of GIT quotients, we obtain a LG version of a Orlov's theorem which describes semi-orthogonal decompositions between categories of graded matrix factorizations and derived categories of hypersurfaces in projective spaces.- EDGE: Matilde Marcolli (Caltech) - Motives in Quantum Field Theory
1st February 2016, 1:00pm to 2:00pm JCMB 5327 -- Show/hide abstractAbstract: I will give an overview of the algebro-geometric approach to Feynman integral in perturbative quantum field theory and the occurrence of motives and periods in parametric Feynman integrals in momentum space, focusing on joint work with Paolo Aluffi.- EDGE: Roberto Svaldi (Cambridge) - A geometric characterization of toric varieties.
28th January 2016, 2:10pm to 3:00pm JCMB 5327 -- Show/hide abstractAbstract: Given a pair (X, D), where X is a proper variety and D a divisor with mild singularities, it is natural to ask how to bound the number of components of D. In general such bound does not exist. But when -(K_X+D) is positive, i.e. ample (or nef), then a conjecture of Shokurov says this bound should coincide with the sum of the dimension of X and its Picard number. We prove the conjecture and show that if the bound is achieved, or the number of components is close enough to said sum, then X is a toric variety and D is close to being the toric invariant divisor. This is joint work with M. Brown, J. McKernan, R. Zong.- Big data/EDGE seminar: Marta Casanellas Rius (Barcelona) - The link between pure mathematics and phylogenetics
22nd January 2016, 2:10pm to 3:00pm JCMB 4325A -- Show/hide abstractAbstract: Many of the usual statistical evolutionary models can be viewed as algebraic varieties and a deep understanding of these varieties may solve open problems in phylogenetics. We show how different mathematical areas such as linear and commutative algebra, algebraic geometry, group representation theory, or numerical methods show up when one studies these varieties. Moreover, we prove that an in-depth geometric study leads to improvements on phylogenetic reconstruction methods. We illustrate these improvements by showing results on simulated data and by comparing them to widely used methods in phylogenetics. In order to follow this talk it is not required a previous knowledge on algebraic varieties or phylogenetics.- EDGE: Lotte Hollands (Heriott-Watt) - Spectral networks and the T3 theory
3rd December 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstractAbstract: A spectral network is a collection of trajectories on a (punctured) Riemann surface. Given a spectral network we can define a notion of "abelianization", which relates flat SL(K) connections on the Riemann surface to flat C^* connections on a covering. For any spectral network abelianization gives a construction of a local Darboux coordinate system on the moduli space of flat SL(K) connections. In this seminar we will take a look at a particularly rich example for rank K=3, with interesting applications to WKB analysis and to quantum physics. This is based on work in progress with Andy Neitzke.- EDGE seminar: James Pascaleff (UIUC) - Equivariant structures on Lagrangian submanifolds
19th November 2015, 3:10pm to 4:00pm JCMB 6311 -- Show/hide abstractAbstract: In this talk, based on joint work with Y. Lekili and N. Sheridan, I will describe some equivariant structures in symplectic geometry and their relationship to mirror symmetry. This will be illustrated with some low dimensional examples.- EDGE seminar: Wouter Castryck (Gent) - Geometric invariants encoded in the Newton polygon
19th November 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstractAbstract: Let f be a sufficiently generic bivariate Laurent polynomial with Newton polygon D, and let C be the plane curve defined by f. It is known by work of Khovanskii that the geometric genus of C equals the number of points in the interior of D having integer coordinates. Building on recent work of Kawaguchi, we give similar combinatorial interpretations for various other geometric invariants, such as the gonality, the Clifford index and the scrollar invariants. We will also report on work in progress, where the hope is to find combinatorial interpretations for the graded Betti numbers of the canonical model, a problem which is motivated by Green's canonical syzygy conjecture. The talk will contain joint work with Filip Cools and Alexander Lemmens.- EDGE seminar: Renzo Cavalieri (CSU) - Open invariants and Crepant Transformations
12th November 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstractAbstract: The question that the Crepant Resolution Conjecture (CRC) wants to address is: given an orbifold X that admits a crepant resolution Y, can we systematically compare the Gromov-Witten theories of the two spaces? That this should happen was first observed by physicists and the question was imported into mathematics by Y.Ruan, who posited it as the search for an isomorphism in the quantum cohomologies of the two spaces. In the last fifteen years this question has evolved and found different formulations which various degree of generality and validity. Perhaps the most powerful approach to the CRC is through Givental's formalism. In this case, Coates, Corti, Iritani and Tseng propose that the CRC should consist of the natural comparison of geometric objects constructed from the GW potential fo the space. We explore this approach in the setting of open GW invariants. We formulate an open version of the CRC using this formalism, and make some verifications. Our approach is well tuned with Iritani's approach to the CRC via integral structures, and it seems to suggest that open invariants should play a prominent role in mirror symmetry.- EDGE seminar: Oscar García-Prada (ICMAT) - Higgs bundles and representations of surface groups
12th November 2015, 1:10pm to 2:00pm JCMB 6311 -- Show/hide abstractAbstract: In this talk I will show how Higgs bundles over a compact Riemann surface can be used to study the moduli space of representations of the fundamental group of the surface in a non-compact semisimple Lie group. Special attention will be given to the case in which the symmetric space defined by the Lie group is of Hermitian type.- EDGE seminar: Nikita Kalinin (Geneva) - Tropical geometry in sandpiles, singularity theory and Legendrian geometry
5th November 2015, 3:10pm to 4:00pm JCMB 6311 -- Show/hide abstractAbstract: We encounter tropical curves and their different aspects in three different situations. 1) A planar tropical curve may appear as the scaling limit of the result of the relaxation of a maximal stable state of a sandpile configuration on a large polygon, perturbed in several points. 2) A singular point on a planar curve over a valuation field ``influences’’ a part of the Newton polygon of the curve. This allows us to prove some Nagata’s type estimates for curves in toric varieties, just by area counting. 3) We consider algebraic curves in $\mathbb C P^3$ tangent to the distribution, given by the form $ydx-xdy+wdz-zdw=0$. If the tropical limit of a family of such curves contains a part, locally looking like a tropical line, then we can observe some nice divisibility property. This property is a corollary of a much more general lemma in tropical integration theory (joint work in progress with G. Mikhalkin). This talk will contain no proofs and a lot of pictures.- EDGE seminar: Agnieszka Bodzenta (Edinburgh) - Perverse schobers and flops
5th November 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstractAbstract: I will consider X and Z related by flops f: X \to Y, g: Z \to Y of relative dimensions one and the fiber product W of X and Z over Y. I will show that an appropriate quotient of the derived category of W admits a semi-orthogonal decomposition into the derived category of X and derived category of the null category for g. I will prove that derived categories of null categories of f and g form a spherical pair in the quotient of D(W) and that the associated spherical twist is the flop-flop equivalence of D(X). This is joint work with A. Bondal.- Hodge seminar: Sarah Zerbes (UCL) - Euler systems and the conjecture of Birch and Swinnerton-Dyer
2nd November 2015, 1:10pm to 2:00pm JCMB 5327 -- Show/hide abstractAbstract: The Birch—Swinnerton-Dyer conjecture is one of the most mysterious open problems in number theory, predicting a relation between arithmetic objects, such as the points on an elliptic curve, and certain complex-analytic functions. A powerful approach to the conjecture is via a tool called an ‘Euler system’. I will explain the idea behing this approach, and some recent new results in this direction.- EDGE: Alexander Pukhlikov (Liverpool) - Birationally rigid Fano-Mori fibre spaces
29th October 2015, 3:10pm to 4:00pm JCMB 6206 -- Show/hide abstractAbstract: Starting from the pioneer work of Iskovskikh and Manin on three-dimensional quartics, birational rigidity has been gradually understood as one of the key phenomena in higher-dimensional birational geometry. I will discuss the recent progress in birational rigidity of fibre spaces over a non-trivial base in the general context of higher-dimensional birational geometry of rationally connected varieties.- EDGE: Jørgen Rennemo (Oxford) - Homological projective duality for Sym^2 P^n
22nd October 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstractAbstract: In 2011, Hosono and Takagi constructed an interesting example of two derived equivalent, non-birational Calabi-Yau 3-folds. This example be explained by phrasing it in terms of Kuznetsov's theory of homological projective duality. With this as motivation, we compute the homological projective dual of Sym^2 P^n, and by taking n = 4 we recover Hosono and Takagi's example. I will explain this result and its proof, which is based on the honestly-less-complicated-than-it-sounds technique of "gauged LG models and variation of GIT stability".- EDGE: Joseph Karmazyn (Bath) - Noncommutative Knörrer periodicity
15th October 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstractAbstract: The singularity category of a type $A_n$ Kleinian surface singularity can be understood by using Kn\"{o}rrer periodicity to show it is equivalent to the singularity category of the finite dimensional algebra $k[x]/(x^n)$. However, Knörrer periodicity only applies to hypersurface singularities so cannot be used to study more general cyclic quotient surface singularities. I will discuss joint work with Martin Kalck that extends the Knörrer periodicity phenomenon outside of the Kleinian case by producing finite dimensional algebras with singularity categories equivalent to general cyclic quotient surface singularities. Intriguingly the algebras produced are noncommutative in general. The proof of this result proceeds via studying different subcategories of the derived category of a type $A$ configuration of rational curves in a surface, and it provides a geometric interpretation of these equivalences that is new even in the Kleinian case.- EDGE: Dmitri Orlov (Steklov) - Derived noncommutative schemes, their geometric realizations, and quiver algebras
8th October 2015, 3:10pm to 4:00pm JCMB 6311- EDGE: Victor Przyjalkowski (Steklov) - On Hodge numbers for Landau-Ginzburg models
8th October 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstractAbstract: Abstract: We discuss various definitions of Hodge numbers for Landau-Ginzburg models and their relation to Hodge numbers of Fano varieties.- EDGE: Igor Krylov (Edinburgh) - Rationally connected non Fano type varieties
1st October 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstractAbstract: Abstract: The class of varieties of Fano type is a generalization of Fano varieties which is very well behaved under the MMP. It is known that all varieties of Fano type are rationally connected. The converse is true in a sense in dimension 2. I will give counterexamples in dimension 3 and higher using the technique of singularities of linear systems which is typically used for proving birational rigidity.- EDGE: Matthew Woolf (Edinburgh) - Rational curves on general hypersurfaces in positive characteristic
(Open in Google Calendar)
24th September 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstractAbstract: Abstract: In characteristic 0, it is a classical fact that smooth hypersurfaces of large degree cannot be rationally connected, i.e., have a rational curve passing through a general pair of points. In positive characteristic, though, there are examples of smooth hypersurfaces of a fixed dimension and characteristic and arbitrarily large degree which are rationally connected. In this talk, we will see that nevertheless, the general hypersurface outside of the Fano range is not rationally connected. We will also discuss some applications of this result to other questions about rational curves on hypersurfaces in positive characteristic
Past MAXIMALS events at University of Edinburgh-
Recycling Centre Booking Ref:576771
23rd July 2021, 12:00pm to 12:30pm Sighthill HWRC, Bankhead Avenue, EH11 4EA -- Show/hide abstractAbstract: Your booking is for Sighthill HWRC, Bankhead Avenue, EH11 4EA. If you can no longer attend your booking then please cancel using the link provided in your confirmation email. -
Recycling Centre Booking Ref:576771
23rd July 2021, 12:00pm to 12:30pm Sighthill HWRC, Bankhead Avenue, EH11 4EA -
Web MAXIMALS: Maria Chlouveraki (Versailles-St Quentin) - Are complex reflection groups real?
9th June 2020, 10:30am to 11:30am Microsoft Teams -- Show/hide abstractAbstract:
Real reflection groups are finite groups of real matrices generated by reflections, and they include several known families of groups, such as symmetric groups and dihedral groups. Their Hecke algebras appear as endomorphism algebras in the study of finite reductive groups. Complex reflection groups generalize real reflection groups in a natural way and their Hecke algebras were defined by Broué, Malle and Rouquier twenty years ago. However, many basic properties of Hecke algebras associated with real reflection groups were simply conjectured in the complex case. In this talk we will discuss some of the most fundamental conjectures, their state of art and our contribution towards their proof.
Instructions for joining the seminar:
This seminar will be hosted on Microsoft Teams. The Web MAXIMALS team can be accessed at https://teams.microsoft.com/l/channel/19%3ab3e8e7d9dfbd4fbd96e6e087666744e5%40thread.tacv2/General?groupId=00457dab-4343-4bb3-b941-077c303454fb&tenantId=2e9f06b0-1669-4589-8789-10a06934dc61. External attendees will need to be added by hand by the organiser: please email dougal.davis@ed.ac.uk to be added. Please note that this seminar will be recorded unless objections are raised with the organisers beforehand. -
Web MAXIMALS: Neil Saunders (University of Greenwich) - The Exotic Nilpotent Cone and Type C Combinatorics
5th May 2020, 11:00am to 12:00pm Microsoft Teams -- Show/hide abstractAbstract: The exotic nilpotent cone as defined by Kato gives a 'TypeA-like' Springer correspondence for Type C. In particular, there is a bijection between the symplectic group orbits on the exotic nilpotent cone and the irreducible representations of the Weyl group of Type C. In this talk, I will outline the various geometric and combinatorial results that follow from this. These results are joint work with Vinoth Nanadakumar and Daniele Rosso, and Arik Wilbert.
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Instructions for joining the seminar:
This seminar will be hosted on Microsoft Teams. All attendees will need to join the University of Edinburgh team "Web MAXIMALS" before the seminar. University of Edinburgh students and staff should be able to add themselves by opening Microsoft Teams in Office365, clicking on "Join or create team" under the "Teams" tab, and searching for "Web MAXIMALS". External attendees will need to be added by hand by the organiser: please email dougal.davis@ed.ac.uk to be added. -
MAXIMALS: Tim Magee (University of Birmingham) - Convexity in tropical spaces and compactifications of cluster varieties
10th March 2020, 2:00pm to 3:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Cluster varieties are a relatively new, broadly interesting class of geometric objects that generalize toric varieties. Convexity is a key notion in toric geometry. For instance, projective toric varieties are defined by convexlattice polytopes. In this talk, I'll explain how convexity generalizes to the cluster world, where "polytopes" live in a tropical space rather than a vector space and "convex polytopes" define projective compactifications of cluster varieties. Time permitting,I'll conclude with two exciting applications of this more general notion of convexity: 1) an intrinsic version of Newton-Okounkov bodies and 2) a possible cluster version of a classic toric mirror symmetry construction due to Batyrev. Based on joint work withMan-Wai Cheung and Alfredo Nájera Chávez. -
MAXIMALS: Konstanze Rietsch (King's College London) - Mirror symmetry and the positive critical point
3rd March 2020, 3:30pm to 4:30pm Bayes Centre, Room 5.10 -- Show/hide abstractAbstract:
In mirror symmetry for non-Calabi-Yau varieties, a big role is played by Laurent polynomials. For example a kind of mirror object that first appeared in works of Batyrev and Givental on smooth projective toric varieties, is given by an explicitLaurent polynomial living on the dual torus. Namely this Laurent polynomial is read off from the rays of the fan defining the toric variety. In this talk I will explain a joint work with Jamie Judd about Laurent polynomials over a field of generalised Puiseauxseries (Novikov field). Namely our result characterises the positive Laurent polynomials which have a unique positive critical point, as those whose Newton polytope has 0 in their interior. One application of this result using mirror symmetry and work of Fukaya,Oh, Ohta, Ono, and Woodward, is to the construction of non-dispaceable Lagrangians in smoth toric manifolds (resp. orbifolds). -
MAXIMALS: No seminar
25th February 2020, 2:00pm to 3:00pm -
MAXIMALS: Martin Gonzalez (Université de Marseille) - The ellipsitomic KZB associator: universal aspects and realisations
4th February 2020, 2:00pm to 3:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: The universal elliptic KZB connection has a twisted (or cyclotomic) counterpart. This is a flat connection defined on a G-principal bundle over the moduli space of elliptic curves with n marked points and a (M; N)-level structure. Here the Lie algebra associated to G is constructed from a twisted elliptic Kohno-Drinfeld Lie algebra, the Lie algebra sl2, and a twisted derivation algebra controlling the algebraic information of some modular forms. After presenting this connection I will retrieve an ellipsitomic (or twisted elliptic) KZB associator from its monodromy and I will relate it to a connection associated to classical dynamical r-matrices with spectral parameter, following the work of Etingof and Schiffmann. Some parts of the results come from a joint work with Damien Calaque. -
MAXIMALS: Kevin McGerty (University of Oxford) - The pure cohomology of multiplicative quiver varieties
28th January 2020, 3:30pm to 4:30pm Bayes Centre 5.02 -- Show/hide abstractAbstract: Multiplicative quiver varieties are a variant of Nakajima's "additive" quiver varieties which were introduced by Crawley-Boevey and Shaw. They arise naturally in the study of various moduli spaces, in particular in Boalch's work on irregular connections. Inthis talk we will discuss joint work with Tom Nevins which shows that the tautological classes for these varieties generate the largest possible subalgebra of the cohomology ring, namely the pure part. Time permitting we may discuss the relation of this resultto a conjecture of Hausel. -
MAXIMALS: Mauro Porta (IRMA Strasbourg) - Categorification of 2-dimensional K-theoretical Hall algebras
28th January 2020, 2:00pm to 3:00pm Bayes Centre 5.02 -- Show/hide abstractAbstract: In this talk I will survey the results obtained in my joint paper with Francesco Sala, arXiv 1903.07253. Using techniques from derived geometry, I will explain how to find a natural categorification of K-theoretical Hall algebras associated to 2-dimensional objects. Among the examples, I will discuss the Hall algebra attached to a surface and the ones attached to Higgs bundles and flat vector bundles on a curve. If time permits, I will sketch how to obtain the categorification of certain natural representations. -
MAXIMALS: Eleonore Faber (University of Leeds) - Infinite constructions: from Grassmannian cluster categories to singularities of countable Cohen-Macaulay type
21st January 2020, 2:00pm to 3:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: The homogeneous coordinate ring $\mathbb{C}[Gr(k,n)]$ of the Grassmannian of $k$-dimensional subspaces in $n$-space carries a natural structure of a cluster algebra. There is an additive categorification of this coordinate ring into a so-called Grassmanniancluster category $C(k,n)$, as shown by Jensen-King-Su in 2016. In particular, the cluster category $C(2,n)$ models triangulations of a regular $n$-gon. A natural question is, if there is some kind of limit construction, i.e., the category ``$C(2,\infty)$''and how to model triangulations of a regular ``$\infty$-gon''. In this talk we show how the category $CM(R)$ of maximal Cohen-Macaulay modules over the coordinate ring $R$ for the $A_{\infty}$-curve allows us to construct triangulations of the $\infty$-gon, making use of the language of Grassmannian cluster categories. This is joint work with J. August, M. Cheung, S. Gratz, and S. Schroll. -
MAXIMALS: André Henriques (University of Oxford) - Representations of based loop groups
14th January 2020, 3:30pm to 4:30pm Bayes Centre 5.10 -- Show/hide abstractAbstract: The free loop group of a compact Lie group has a very interesting representation theory, related to modular forms, integrable systems, quantum groups, vertex algebras, string theory... Of particular interest is the fusion product of representations. The representation theory of the *based* loop groups have never been considered before (even though individual examples of representations have been considered in disguise). We will explain how to extend the fusion product to representations of the based loop groups, and how to recover the category of representations of the free loop group from the category of representations of the based loop group. At last, I will explain in what sense I expect the representation theory of based loop groups to be wild (unlike that of the based loop groups), and which representations one might have a hope to classify. -
Special MAXIMALS: Bernard Leclerc (Université de Caen) - Cluster structure on strata of flag varieties
13th December 2019, 2:00pm to 3:30pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Flag varieties have well known stratifications by Schubert cells. Intersecting two such stratifications corresponding to opposite Borel subgroups, one obtains finer stratifications which play a role in describing the totally nonnegative part of the flag varieties (in the sense of Lusztig). In joint work with Geiss and Schröer, we studied Frobenius categories attached to Schubert cells of flag varieties of type A,D,E, yielding cluster algebra structures on their coordinate rings. After recalling this, I will explain how one can extend this work to the above finer stratifications of the flag varieties. -
MAXIMALS: Tomasz Przezdziecki (University of Edinburgh) - Quiver Schur algebras and cohomological Hall algebras
5th December 2019, 2:00pm to 3:00pm JCMB 6201 -- Show/hide abstractAbstract: Quiver Schur algebras are a generalization of Khovanov-Lauda-Rouquier algebras, well known for their role in the categorification of quantum groups. In this talk I will discuss their basic structural properties, as well as their connection to the cohomological Hall algebras defined by Kontsevich and Soibelman. -
ARTIN 56
29th November 2019, 10:00am to 6:00pm Bayes Center 5.10/5.02 -- Show/hide abstractAbstract: https://www.maths.ed.ac.uk/~aappel/artin56.html -
ARTIN 56
28th November 2019, 10:00am to 6:00pm Bayes Center 5.10 -- Show/hide abstractAbstract: https://www.maths.ed.ac.uk/~aappel/artin56.html -
MAXIMALS: Noah Arbesfeld (Imperial College) - K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface
12th November 2019, 2:00pm to 3:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Tautological bundles on Hilbert schemes of points often enter into enumerative and physical computations. I will explain how to use the Donaldson-Thomas theory of toric threefolds to produce combinatorial identities that are expressed geometrically using tautological bundles on the Hilbert scheme of points on a surface. I'll also explain how these identities can be used to study Euler characteristics of tautological bundles over Hilbert schemes of points on general surfaces. -
MAXIMALS: Michael Wemyss (University of Glasgow) - Tits Cone Intersections and 3-fold flops
29th October 2019, 2:00pm to 3:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: The first half of the talk, which will be very algebraic, will describe intersection hyperplane arrangements inside Tits cones of (both finite and affine) Coxeter groups. I will demonstrate the local wall crossing rules, compute some examples, and state the classification of such hyperplane arrangements in small dimension. For example, these give precisely 16 tilings of the plane, with only 3 being the "traditional" Coxeter tilings. The new hyperplane arrangements turn out to be quite fundamental: they (1) classify all noncommutative resolutions for cDV singularities, (2) classify tilting theory for (contracted) projective algebras, and (3) also have many algebraic-geometric consequences. One is that they describe the stability manifold for an arbitrary 3-fold flop X --> Spec R, where X can have terminal singularities. Another is that they describe a large part of the autoequivalence group. Another is that they give lower bounds for certain curve-counting GV invariants. Parts of the talk are joint with Iyama, parts with Hirano, and parts with Donovan. -
MAXIMALS: Maxime Fairon (University of Glasgow) - From double brackets to integrable systems
8th October 2019, 2:00pm to 3:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Double brackets were introduced by M. Van den Bergh in his successful attempt to understand the Poisson geometry of (multiplicative) quiver varieties directly at the level of the path algebra of quivers. I will begin with a review of the basics of this theory and its relation to usual geometric structures. I will then move on to the properties of double brackets that can be used to study integrable systems. As a first application, I will explain how the double Poisson bracket on the path algebra of an extended Jordan (or one-loop) quiver can be used to easily derive integrable systems of Calogero-Moser type. As a second application, I will explain the corresponding relation between double quasi-Poisson brackets and Ruijsenaars-Schneider systems based on recent works with O. Chalykh (Leeds). -
Mathematics, grammars... and babies?!
2nd October 2019, 5:30pm to 6:30pm Heriot-Watt University, James Watt Centre 2, Heriot-Watt University, Edinburgh, EH14 4AS -- Show/hide abstractAbstract: A public lecture by Dr. Laura Ciobanu on the mathematics behind the languages we speak, and their connections to computer science. https://www.eventbrite.co.uk/e/mathematics-grammars-and-babies-tickets-72088389313?aff=eand -
MAXIMALS: Joshua Wen (University of Illinois Urbana-Champaign) - Wreath Macdonald polynomials as eigenstates
1st October 2019, 2:00pm to 3:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Wreath Macdonald polynomials were defined by Haiman as generalizations of transformed Macdonald polynomials from the symmetric groups to their wreath products with cyclic groups of order m. In a sense, their definition was given in the hope that they would correspond to K-theoretic fixed point classes of cyclic quiver varieties, much like how Haiman's proof of Macdonald positivity assigns Macdonald polynomials to fixed points of Hilbert schemes of points on the plane. This hope was realized by Bezrukavnikov and Finkelberg, and the subject has been relatively untouched until now. I will present a first result exploring possible ties to integrable systems. Using work of Frenkel, Jing, and Wang, we can situate the wreath Macdonald polynomials in the vertex representation of the quantum toroidal algebra of sl_m. I will present the result that, in this setting, the wreath Macdonald polynomials diagonalize the horizontal Heisenberg subalgebra of the quantum toroidal algebra---a first step towards developing a notion of 'wreath Macdonald operators'. -
Structure and Symmetry Theme Day
27th September 2019, 10:00am to 5:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Morning Schedule: @10:00 Speaker: Minhyong Kim (University of Oxford) Title: Principal Bundles in Diophantine Geometry. Abstract: Principal bundles and their moduli have been important in various aspects of physics and geometry for many decades. It is perhaps not so well-known that a substantial portion of the original motivation for studying them came from number theory, namely the study of Diophantine equations. I will describe a bit of this history and some recent developments. * @11:30 Speaker: James Lucietti (MPI Bonn) Title: Black holes in higher dimensions. Abstract: Black holes are one of the most remarkable predictions of Einstein's theory of General Relativity. I will give an overview of general results which constrain the topology and geometry of black holes in four and higher-dimensional spacetimes. While in four dimensions this leads to the celebrated black hole uniqueness theorem, in higher-dimensions this is not the case and their classification remains an open problem. I will describe recent progress in the classification of higher-dimensional black holes. * -
MAXIMALS: Dougal Davis (University of Edinburgh) - Families of principal bundles and the elliptic Grothendieck-Springer resolution
24th September 2019, 2:00pm to 3:00pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Given a family of principal bundles on an elliptic curve, a natural problem is to describe the locus of unstable bundles, and the fibres of the map from its complement to the coarse moduli space of semistable bundles. While the coarse moduli space map can be somewhat difficult to understand directly, it has a simultaneous resolution of singularities at the level of stacks, called the elliptic Grothendieck-Springer resolution, which is often easier to analyse in practice. After explaining this general machinery and its extension from semistable to unstable bundles, I will give some examples of families for which the unstable part of the resolution can be computed explicitly, from which a complete description of the entire picture miraculously follows. -
Special MAXIMALS: Hoel Queffelec (Université de Montpellier) - Surface skein algebras and categorification
24th July 2019, 11:00am to 12:00pm JCMB 5323 -- Show/hide abstractAbstract: Skein modules are a natural extension of the Jones polynomial to 3-manifolds. In the case where the manifold is a thickened surface, they naturally come with a (usually) non-commutative multiplicative structure. I'll review basic definitions, formulas and conjectures about these skein modules, before discussing their categorification by foams. We will study simplicity properties, and explain the difference between the torus, where I will give an almost-proven categorified Frohman-Gelca formula, and other surfaces, where we will discuss reformulations of the Fock-Goncharov-Thurston positivity conjecture. This is joint with P. Wedrich. -
Special MAXIMALS: Leandro Vendramin (Universidad de Buenos Aires) - On the classification of Nichols algebras
23rd July 2019, 2:00pm to 3:00pm JCMB 5323 -- Show/hide abstractAbstract: Nichols algebras appear in several branches of mathematics going from Hopf algebras and quantum groups, to Schubert calculus and conformal field theories. In this talk, we review the main problems related to Nichols algebras and we discuss some classification theorems. The talk is mainly based on joints works with I. Heckenberger. -
Special EDGE/MAXIMALS: Andrei Caldararu (University of Wisconsin, Madison) - Categorical Gromov-Witten invariants: a computable definition
9th July 2019, 4:00pm to 5:00pm JCMB 5323 -- Show/hide abstractAbstract: In his 2005 paper "The Gromov-Witten potential associated to a TCFT" Kevin Costello described a procedure for recovering an analogue of the Gromov-Witten potential directly out of a Calabi-Yau category. The main difficulty to be overcome in that paper was dealing with the fact that TCFT's constructed form Calabi-Yau categories are always required to have at least one input. This problem was originally solved in a non-constructive fashion using dg-Weyl algebras and associated Fock spaces. In my talk I shall describe recent work on giving a new definition of Costello's invariants. We bypass the dg-Weyl algebra approach completely. Instead we use a Koszul resolution of the space of Sigma_n-invariant chains on M_{g,n}. This approach involves no choices, and makes the new invariants amenable to explicit computer calculations. I willl list some of the higher genus invariants that we computed; they agree with predictions from mirror symmetry. This talk is based on joint works with Junwu Tu and with Kevin Costello. -
Special MAXIMALS - Geetha Venkataraman - Enumeration of groups in varieties of A-groups
21st June 2019, 2:00pm to 3:00pm 5323 JCMB -- Show/hide abstractAbstract: Let $f(n)$ denote the number of isomorphism classes of groups of order $n$. Let $\cal S$ be a class of groups and let $f_{\cal S}(n)$ be the number of isomorphism classes of groups in $\cal S$, of order $n$. Some interesting classes that have been studied are the class of soluble groups, varieties of groups, $p$-groups and $A$-groups. Finite $A$-groups are those with abelian Sylow subgroups. In 1993, L Pyber proved a result, which bettered a conjecture, when he showed that $ f(n) \leq n^{\frac{2}{27}{\mu(n)}^2^2 + O({\mu(n)}^^{\frac{5}{3}})}$ where $\mu(n)$ denotes the maximum $\alpha$, such that $p^{\alpha}$ divides $n$ for any prime $p$. While Pyber's upper bound has the correct leading term, it is certainly not the case for the error term. The key to this puzzle may lie in deeper investigation of $A$-groups and varieties of $A$-groups. We present results concerned with asymptotic bounds for $f_{\cal S}(n)$ when ${\cal S}$ is a variety of $A$-groups, namely, ${\mathfrak U} = {\mathfrak A}_p{\mathfrak A}_q$, and ${\mathfrak V} = {\mathfrak A}_p{\mathfrak A}_q \vee {\mathfrak A}_q{\mathfrak A}_p$ and discuss open questions in this area. The talk will be self-contained and should be accessible to anyone with a basic knowledge of group theory. -
Special MAXIMALS: Martin Gonzalez (UPMC & Max Planck Bonn) - (CANCELLED)
17th June 2019, 2:00pm to 3:00pm TBC -
MAXIMALS: Kobi Kremnitzer (Oxford University) - Global analytic geometry and Hodge theory
21st May 2019, 3:30pm to 4:30pm JCMB 5323 -- Show/hide abstractAbstract: In this talk I will describe how to make sense of the function (1+t)^x over the integers. I will explain how different rings of analytic functions can be defined over the integers, and how this leads to global analytic geometry and global Hodge theory. If time permits I will also describe an analytic version of lambda-rings and how this can be used to define a cohomology theory for schemes over Z. This is joint work with Federico Bambozzi and Adam Topaz. -
MAXIMALS: Leonid Rybnikov (HSE) - Gaudin model and crystals
20th May 2019, 11:45am to 12:45pm JCMB 5323 -- Show/hide abstractAbstract: Drinfeld-Kohno theorem relates the monodromy of KZ equation to the braid group action on a tensor product of $U_q(\mathfrak{g})$-modules by R-matrices. The KZ equation depends on the parameter $\kappa$ such that $q=\exp(\frac{\pi i}{\kappa})$. We study the limit of the Drinfeld-Kohno correspondence when $\kappa\to 0$ along the imaginary line. Namely, on the KZ side this limit is the Gaudin integrable magnet chain, while on the quantum group side the limit is the tensor product of $\mathfrak{g}$-crystals. The limit of the braid group action by the monodromy of KZ equation is the action of the fundamental group of the Deligne-Mumford space of real stable rational curves with marked points (called cactus group) on the set of eigenlines for Gaudin Hamiltonians (given by algebraic Bethe ansatz). On the quantum group side, the cactus group acts by crystal commutors on the tensor product of $\mathfrak{g}$-crystals. We construct a bijection between the set of solutions of the algebraic Bethe ansatz for the Gaudin model and the corresponding tensor product of $\mathfrak{g}$-crystals, which preserves the natural cactus group action on these sets. If time allows I will also dicuss some conjectural generalizations of this result relating it to works of Losev and Bonnafe on cacti and Kazhdan-Lusztig cells. This is a joint work with Iva Halacheva, Joel Kamnitzer, and Alex Weekes (https://arxiv.org/abs/1708.05105) -
Junior Hodge Day
3rd July 2022, 5:34am to 5:34am Bayes Centre 5.10 -
MAXIMALS: Bernhard Keller (Université Paris Diderot - Paris 7) - Tate-Hochschild cohomology from the singularity category
13th May 2019, 3:30pm to 4:30pm Bayes Centre 5.10 -- Show/hide abstractAbstract: The singularity category (or stable derived category) was introduced by Buchweitz in 1986 and rediscovered in a geometric context by Orlov in 2003. It measures the failure of regularity of an algebra or scheme. Following Buchweitz, one defines the Tate-Hochschild cohomology of an algebra as the Yoneda algebra of the identity bimodule in the singularity category of bimodules. In recent work, Zhengfang Wang has shown that Tate-Hochschild cohomology is endowed with the same rich structure as classical Hochschild cohomology: a Gerstenhaber bracket in cohomology and a B-infinity structure at the cochain level. This suggests that Tate-Hochschild cohomology might be isomorphic to the classical Hochschild cohomology of a (differential graded) category, in analogy with a theorem of Lowen-Van den Bergh in the classical case. We show that indeed, at least as a graded algebra, Tate-Hochschild cohomology is the classical Hochschild cohomology of the singularity category with its canonical dg enhancement. In joint work with Zheng Hua, we have applied this to prove a weakened version of a conjecture by Donovan-Wemyss on the reconstruction of a (complete, isolated) compound Du Val singularity from its contraction algebra, i.e. the algebra representing the non commutative deformations of the exceptional fiber of a resolution. -
MAXIMALS: Fan Qin (Shanghai Jiao Tong University) - Bases for upper cluster algebras and tropical points
6th May 2019, 2:00pm to 3:00pm JCMB 5323 -- Show/hide abstractAbstract: It is known that many (upper) cluster algebras possess very different good bases which are parametrized by the tropical points of Langlands dual cluster varieties. For any given injective reachable upper cluster algebra, we describe all of its bases parametrized by the tropical points. In addition, we obtain the existence of the generic bases for such upper cluster algebras. Our results apply to many cluster algebras arising from representation theory, including quantized enveloping algebras, quantum affine algebras, double Bruthat cells, etc. -
MAXIMALS: Yvain Bruned (University of Edinburgh) - Algebraic structures for singular SPDEs.
30th April 2019, 2:00pm to 3:00pm JCMB 5323 -- Show/hide abstractAbstract: We will review the main structures used for renormalising singular SPDEs. Indeed, the solutions of these equations are described using stochastic processes indexed by decorated trees. Hopf algebras techniques have been successful for recentering these objects and proving their convergence. -
MAXIMALS: Ivan Ip (Hong Kong University of Science and Technology) - Positive Peter-Weyl Theorem
23rd April 2019, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: For a compact Lie group G, the classical Peter-Weyl Theorem states that the regular representation of G on L^2(G) decomposes as the direct sum of its irreducible unitary representations. These results were generalized for real reductive groups by Harish-Chandra, and for compact quantum groups by Woronowicz, at the same time, the case of non-compact quantum groups remained open. In this talk I will explain the Peter-Weyl Theorem for split real quantum groups of type A_n. I will discuss ingredients necessary to formulate and prove the theorem, including the GNS representations of C*-algebras, quantum parallel transports, and cluster realization of positive representations. This is a joint work with Gus Schrader and Alexander Shapiro. -
MAXIMALS: Lang Mou (UC Davis) - Scattering diagrams of quivers with potentials
16th April 2019, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: In the study of cluster algebras, a combinatorial tool named scattering diagram plays an important role. In this talk, we will investigate the relation between the cluster scattering diagram of Gross-Hacking-Keel-Kontsevich and the stability scattering diagram of Bridgeland. We will also discuss how scattering diagrams can be used to study the Donaldson-Thomas invariants of a quiver with potential. -
MAXIMALS: Michael Shapiro (Michigan State University) - Darboux coordinates on the Poisson space of triangular bilinear forms
26th March 2019, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: In this project we give a construction of a system of Darboux-type coordinates on the space of unipotent upper triangular bilinear forms equipped with the Poisson structure discovered by A. Bondal in 1995. Some special cases of low dimensional symplectic leaves were identified earlier by L. Chekhov and M. Mazzocco with the Poisson algebras of hyperbolic length functions where Darboux-type coordinates are obtained by hyperbolic lengths of special system of loops. Utilizing the construction of Fock-Goncharov coordinates for flat SL_N connections on the disc with 3 marked points on the boundary, we compute Darboux-type coordinates for the maximal symplectic leaves. This is a joint work with L. Chekhov. -
MAXIMALS: Alexander Veselov (Loughborough University) - Conway's topograph, PGL(2,Z)-dynamics and two-valued groups
19th March 2019, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: In 1990s John H. Conway proposed "topographic" approach to describe the values of the binary quadratic forms, which can be applied also to the description of the celebrated Markov triples, featured in many areas of mathematics, including algebraic and hyperbolic geometry, theory of Frobenius manifolds and quiver mutations. In the talk I will review Conway’s approach from the point of view of the theory of two-valued groups. The first important examples of such groups were discovered by Buchstaber and Novikov in algebraic topology, which was developed further by Buchstaber and Rees. I will explain some classification results in the theory of two-valued groups, which emphasize again the role of the group PGL(2,Z) and present a novel view on the results of Conway, Markov and Mordell. The talk is based on a joint work with V.M. Buchstaber. -
MAXIMALS: Oleg Chalykh (University of Leeds) - Cherednik operators, Lax pairs, and twisted Ruijsenaars models.
12th March 2019, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.02 -- Show/hide abstractAbstract: Trigonometric Cherednik operators are a remarkable commutative family of commuting difference-reflection operators, arising in the basic representation of the double affine Hecke algebra of a root system R. There exists also their elliptic version, due to Komori and Hikami. I will show how to use these operators to construct Lax pairs (previously unknown beyond type A) for the Ruijsenaars models for all root systems. I will then explain how one can modify the elliptic R-matrices of Komori—Hikami and construct new generalisations of the Ruijsenaars models. Partly based on arXiv:1804.01766. -
Special MAXIMALS: Lukas Woike (Hamburg) - Derived modular functors
7th March 2019, 3:30pm to 4:30pm Bayes Centre 5.10 -- Show/hide abstractAbstract: For a semisimple modular tensor category the Reshetikhin-Turaev construction yields an extended three-dimensional topological field theory and hence by restriction a modular functor. By work of Lyubachenko-Majid the construction of a modular functor from a modular tensor category remains possible in the non-semisimple case. We explain that the latter construction is the shadow of a derived modular functor featuring homotopy coherent mapping class group actions on chain complex valued conformal blocks and a version of factorization and self-sewing via homotopy coends. On the torus we find a derived version of the Verlinde algebra, an algebra over the little disk operad (or more generally a little bundles algebra in the case of equivariant field theories). The concepts will be illustrated for modules over the Drinfeld double of a finite group in finite characteristic. This is joint work with Christoph Schweigert (Hamburg). -
Joint HW-UoE MAXIMALS: Ben Davison (UoE) - Coloured 3d partitions
5th March 2019, 4:40pm to 5:30pm Bayes Centre 5.10 -- Show/hide abstractAbstract: I will discuss the theory of 3d partitions, the counting of the number of ways to arrange boxes in the corner of a room (as opposed to boxes in a Young diagram). In applications to algebra and geometry, one introduces a colour scheme to these boxes, and refines the partition function to take account of the number of boxes of each colour. I will report on some recent work with Ongaro and Szendroi on the resulting partition functions, as well as q-refinements and positivity conjectures. -
Joint HW-UoE MAXIMALS: Alexandre Martin (HW) - The Tits alternative for two-dimensional Artin groups
5th March 2019, 3:45pm to 4:35pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Many groups of geometric interest present an interesting dichotomy at the level of their subgroups: their subgroups are either very large (they contain a free subgroup) or very small (they are virtually abelian). In this talk, I will explain how one can use ideas from group actions in negative curvature to prove such a dichotomy. In particular, I will show how one can prove such a strengthening of the Tits Alternative for a large class of Artin groups. This is joint work with Piotr Przytycki. -
Joint HW-UoE MAXIMALS: Francesca Carocci (UoE) - Endomorphisms of the Koszul complex and deformations of lci ideal sheaves
5th March 2019, 3:00pm to 3:20pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Given a regular sequence (f1,..,fn) in a commutative K algebra R, we will study the homotopy abelianity (over K and over R) of the differential graded Lie algebra of endomorphisms of the Koszul resolution of the regular sequence. At the end, we will briefly discuss how the result give an annihilation theorem for obstructions to deformations of lci ideal sheaves. -
Joint HW-UoE MAXIMALS: Alan Logan (HW) - On the isomorphism problem for one-relator groups
5th March 2019, 2:30pm to 2:50pm Bayes Centre 5.10 -- Show/hide abstractAbstract: In the 1960s Magnus conjectured that two one-relator groups -
Joint HW-UoE MAXIMALS: Anna Mkrtchyan (UoE) - Gradings on the Brauer algebra
5th March 2019, 2:00pm to 2:20pm Bayes Centre 5.10 -- Show/hide abstractAbstract: Brauer algebras B_n(\delta) are finite dimensional algebras introduced by Richard Brauer in order to study the n-th tensor power of the defining representations of the orthogonal and symplectic groups. They play the same role that the group algebras of the symmetric groups do for the representation theory of the general linear groups in the classical Schur-Weyl duality. We will discuss two different constructions which show that the Brauer algebras are graded cellular algebras and then show that they define the same gradings on B_n(\delta). -
MAXIMALS: Catharina Stroppel(HCM Bonn) - Categorified coideal subalgebras and Deligne categories
26th February 2019, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: In this talk I will consider the Deligne category Rep((δ)) and its module category. The main point is to show that it categorifies a certain Fock space for a coideal subalgebra inside a quantum group. As an application we obtain a connection between decomposition numbers in Brauer centralizer algebras and Kazdhan-Lusztig polynomials of type D. If time allows we will also mention the relevance of this construction to the representation theory of Lie superalgebras. -
MAXIMALS: Robert Weston (Heriot-Watt University) - A Q-operator for Open Quantum Systems
26th February 2019, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: R-matrices arising from quantum affine algebras may be used to define the transfer matrix of a closed quantum spin chain. The 'prime directive' of the field of quantum integrable systems is to find the eigenvectors and eigenvalues of this transfer matrix. A powerful approach involves first constructing a new `Q-operator' defined as a trace over an infinite-dimensional module of a Borel subalgebra of the quantum affine algebra. A short exact sequence for infinite-dimensional modules then leads to a functional relation for simultaneous eigenvalues of the transfer matrix and Q-operator. These functional relations may then be solved exactly. In this talk I will summarise this approach and then extend it to 'open' quantum systems. The transfer matrix of an open system involves both the quantum affine algebra R-matrix and a solution of Cherednik's reflection equation associated with a coideal subalgebra. I will give a new construction of the Q-operator for such open systems and derive functional relations. This is joint work with Bart Vlaar. -
MAXIMALS: Camille Laurent-Gengoux (Université de Lorraine) - The Lie algebra up to homotopy hidden in a singular foliation
12th February 2019, 2:00pm to 3:00pm JCMB 5323 -- Show/hide abstractAbstract: We show that behind any Lie-Rinehart algebra, and in particular behind any singular foliation or behind any affine variety, there is a canonical (homotopy class) of Lie-infinity algebroid (also called "dg-manifolds" or "Q-manifolds"). We are able to give an explicit construction. Also, we shall try to explain the algebraic and geometrical meanings of this higher structure. Joint works with Sylvain Lavau and Thomas Strobl. -
MAXIMALS: Alexander Samokhin (Institute for Information Transmission Problems) - On the Dubrovin conjecture for horospherical varieties of Picard rank one
5th February 2019, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: Given a Fano variety X, Dubrovin's conjecture relates semisimplicity of the big quantum cohomology ring of X to the existence of a full exceptional collection in the derived category of coherent sheaves on X. One of the goals of our joint work with C.Pech, R.Gonzales, and N. Perrin is to establish Dubrovin's conjecture for the varieties in the title. In this talk, I will focus on the derived category side of the conjecture, and try to explain a connection to categorical joins that have recently been introduced by A.Kuznetsov and A.Perry. This is based on https://arxiv.org/abs/1803.05063. -
MAXIMALS: Shengyong Pan (Beijing Jiaotong University) - Stable functors of derived equivalences
5th February 2019, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: From certain triangle functors, called non-negative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the stable categories of the abelian categories. The construction generalizes a previous work by Hu and Xi. We show that the stable functors of non-negative functors have nice exactness property and are compatible with composition of functors. This allows us to compare conveniently the homological properties of objects linked by the stable functors. Particularly, we prove that the stable functor of a derived equivalence between two arbitrary rings provides an explicit triangle equivalence between the stable categories of Gorenstein projective modules. This generalizes a result of Y. Kato. This is joint work with Wei Hu. -
MAXIMALS: Ruth Reynolds (University of Edinburgh) - Idealisers in Skew Group Rings
15th January 2019, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.02 -- Show/hide abstractAbstract: Let C be a commutative noetherian domain, G be a finitely generated abelian group which acts on C and B = C#G be the skew group ring. For a prime ideal I in C, we study the largest subring of B in which the right ideal IB becomes a two-sided ideal - the idealiser subring. In this talk we will introduce the idealizer and describe some interesting results about how the noetherianity of these subrings is closely linked to the orbit of I under the G-action. We will also give examples to show how this works in practice. -
MAXIMALS: David Ayala (University of Montana) - Orthogonal group and higher categorical adjoints
10th December 2018, 3:05pm to 3:55pm JCMB 5323 -- Show/hide abstractAbstract: In this talk I will articulate and contextualize the following sequence of results. 1) The Bruhat decomposition of the general linear group defines a stratification of the orthogonal group. 2) Matrix multiplication defines an algebra structure on its exit-path category in a certain Morita category of categories. 3) In this Morita category, this algebra acts on the category of n-categories -- this action is given by adjoining adjoints to n-categories. This result is extracted from a larger program -- entirely joint with John Francis, some parts joint with Nick Rozenblyum -- which proves the cobordism hypothesis. -
Special EDGE: Harold Williams (UC Davis): Kasteleyn operators from mirror symmetry
6th December 2018, 2:00pm to 3:30pm JCMB 5323 -- Show/hide abstractAbstract: Given a consistent bipartite graph $\Gamma$ in $T^2$ with a complex-valued edge weighting $\mathcal{E}$ we show the following two constructions are the same. The first is to form the Kasteleyn operator of $(\Gamma, \mathcal{E})$ and pass to its spectral transform, a coherent sheaf supported on a spectral curve in $(\mathcal{C}^\times)^2$. The second is to form the conjugate Lagrangian $L \subset T^* T^2$ of $\Gamma$, equip it with a brane structure prescribed by $\mathcal{E}$, and pass to its mirror coherent sheaf. This lives on a stacky toric compactification of $(\mathcal{C}^\times)^2$ determined by the Legendrian link which lifts the zig-zag paths of $\Gamma$ (and to which the noncompact Lagrangian $L$ is asymptotic). We work in the setting of the coherent-constructible correspondence, a sheaf-theoretic model of toric mirror symmetry. This is joint work with David Treumann and Eric Zaslow. -
MAXIMALS: Ian Le (Perimeter Institute) - Cluster Structures on Configurations of Flags via Tensor Invariants
4th December 2018, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: The theory of cluster algebras gives a concrete and explicit way to quantize character varieties for a group G and a surface S. In order to do this, one must show that character varieties admit natural cluster structures. It turns out to be enough to carry this out in the case of a disc with marked points. I will briefly explain the theory of cluster algebras and then describe a general procedure for constructing the cluster structure on Conf_n, the space of configurations of n flags for the group G. The key step will be to show that cluster variables can be realized in terms of tensor invariants of G. -
Special MAXIMALS: Adam Chapman (Tel-Hai College) - Alternative Clifford Algebras
3rd December 2018, 3:05pm to 3:55pm JCMB 5323 -- Show/hide abstractAbstract: Associative Clifford algebras have long been in use in the algebaic theory of quadratic forms, as well as in geometry and physics. Attempts to formualate the Rost invariant for quadratic forms led to the definition of the alternative Clifford algebra by Musgrave. We describe the structure of the alternative Clifford algebra of a ternary quadratic form, and present some other preliminary results and open problems. This talk is based on join work with Uzi Vishne. -
MAXIMALS: Misha Feigin (University of Glasgow) - Laplace-Runge-Lenz-Dunkl vector
27th November 2018, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: Laplace-Runge-Lenz vector represents hidden symmetry of Coulomb problem (equivalently, hydrogen atom), which is so(4). I am going to discuss its generalisation for the Dunkl settings in which a Coxeter group is present. The corresponding model is related to Calogero-Moser system, and the arising symmetry algebra is related to the Dunkl angular moment subalgebra of the rational Cherednik algebra. The talk is based on joint works with T. Hakobyan and A. Nersessian. -
MAXIMALS: Alexey Sevastyanov (University of Aberdeen) - q-W algebras, Mickelsson algebras, and Zhelobenko operators
22nd November 2018, 3:00pm to 4:00pm JCMB 5323 -- Show/hide abstractAbstract: In this talk I will show that q-W algebras, which are quantum group analogues of W-algebras, belong to the class of the so-called Mickelsson algebras. They can be described in terms of certain analogues of Zhelobenko operators. -
MAXIMALS: Gus Schrader (Columbia University) - Cluster theory of the quantum Toda chain
20th November 2018, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.02 -- Show/hide abstractAbstract: The classical open relativistic Toda chain is a well-known integrable Hamiltonian system which appears in various different contexts in Lie theory and mathematical physics. As was observed by Gekhtman-Shapiro-Vainshtein, the phase space of the relativistic Toda chain admits the additional structure of a cluster variety. I will explain how this cluster structure can also be used to analyze the quantization of the relativistic Toda chain. In particular, we will see that the Baxter Q-operator for the quantum system can be realized as a sequence of quantum cluster mutations, which allows us to obtain a Givental-type integral representation of the Toda eigenvectors, the q-Whittaker functions. Joint work with Alexander Shapiro. -
MAXIMALS: Clément Dupont (Université de Montpellier) - Single-valued integration and superstring amplitudes
30th October 2018, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: The classical theory of integration concern integrals of differential forms over domains of integration. In geometric terms, this corresponds to a canonical pairing between de Rham cohomology and singular homology. For varieties defined over the reals, one can make use of complex conjugation to define a real-valued pairing between de Rham cohomology and its dual, de Rham homology. The corresponding theory of integration, that we call single-valued integration, pairs a differential form with a `dual differential form’. We will explain how single-valued periods are computed and give an application to superstring amplitudes in genus zero. This is joint work with Francis Brown. -
MAXIMALS: Bart Vlaar (Heriot-Watt) - Quasitriangular coideal subalgebras of U_q(g) in terms of generalized Satake diagrams
23rd October 2018, 2:00pm to 3:00pm JCMB 5323 -- Show/hide abstractAbstract: Let g be a finite dimenional simple complex Lie algebra. Following M. Noumi et al. (1995) and G. Letzter (1999) a fixed-point subalgebra of g with respect to an involutive automorphism can be quantized, which yields a coideal subalgebra B of U_q(g). This was re-engineered and generalized to arbitrary quantized Kac-Moody algebras by S. Kolb (2014). More recently, M. Balagović and S. Kolb showed that, for g of finite type, B is quasitriangular with respect to the category of finite-dimensional representations of U_q(g). As a consequence, associated to B there is a "universal solution" K of the reflection equation (4-braid relation). This entire story works in a more general setting, where the relevant subalgebra of g is no longer the fixed-point subalgebra of a semisimple automorphism. The underlying combinatorial data, dubbed "generalized Satake diagrams", arose previously in work by A. Heck on the classification of involutive automorphisms of root systems. Conjecturally, for g of finite type, this classifies all quasitriangular coideal subalgebras. Joint work with Vidas Regelskis (arXiv:1807.02388). -
MAXIMALS: Vladimir Fock (IRMA Strasbourg) - Flag configurations and integrable systems.
9th October 2018, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: A configuration space of $k$ framed flags in an $N$-dimensional space is a pre-symplectic manifold admitting distinguished coordinates constructed as wedge products. If we take infinite $k$ but impose a symmetry with respect to a certain action of the fundamental group of a surface we obtain the coordinates on the (framed) $SL(N)$ character variety. If we take infinite $N$, but impose a symmetry with respect to a group $\mathbb{Z}^2$ we obtain coordinates on the phase space of a Goncharov-Kenyon (GK) integrable system - generalization of many known integrable systems arising from Poisson-Lie groups. This phase space can be identified with the space of planar curves provided with a line bundle. The GK integrable system admits continuous and discrete commuting flows. (The simplest example of the latter is the Poncelet porism). In coordinates this discrete flow corresponds to birational transformations composed of elementary ones called cluster mutations. We will give explicit formula of cluster coordinates, discuss stable points of these discrete flows and if time permits explain what the $\tau$ functions of Sato is and show that our cluster coordinates special values of this function. -
MAXIMALS: Sira Gratz (University of Glasgow) - Homotopy invariants of singularity categories
9th October 2018, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: The existence of a grading on a ring often makes computations a lot easier. In particular this is true for the computation of homotopy invariants. For example one can readily compute such invariants for the stable categories of graded modules over connected graded self-injective algebras. Using work of Tabuada, we will show how to deduce from this knowledge the homotopy invariants of the ungraded stable categories for such algebras. This is based on joint work with Greg Stevenson. -
MAXIMALS: Daniel Chan (University of New South Wales) - Degenerations of weighted projective lines
2nd October 2018, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstractAbstract: In the 1960s, Deligne and Mumford compactified the moduli space of smooth curves by adding nodal curves at the boundary. We will look at the analogous question of compactifying the moduli space of weighted projective lines. Our interest is mainly in the various commutative and noncommutative degenerations that can naturally arise. This talk is about preliminary work done in this direction with Abdelgadir, Okawa and Ueda. -
MAXIMALS: Vassily Gorbunov (University of Aberdeen): Quantum integrable systems and quantum Schubert calculus
25th September 2018, 2:00pm to 3:00pm JCMB 5323 -- Show/hide abstractAbstract: In the talk we describe a natural solution to the quantum Yang-Baxter equation associated to the equivariant cohomology of the Grassmanian manifolds and study the appropriate quantum integrable system. We discuss the connection of the features of this quantum integrable system to Schubert calculus. -
MAXIMALS: Ben Davison (University of Edinburgh) - A new, positive(?) partition function controlling the enumerative invariants in the 3d McKay correspondence
18th September 2018, 2:05pm to 2:55pm JCMB 5323 -- Show/hide abstractAbstract: Given G a finite Abelian subgroup of SL_3(C) there is an associated coloured planar partition counting problem, which may be approached via studying Euler characteristics of moduli of sheaves on a crepant resolution of C^3/G. If G is trivial, this gives the McMahon partition function counting uncoloured planar partitions. A prediction from DT theory is that this partition function should be determined by a much sparser one, obtained by dividing by the McMahon partition function e times, where e is the Euler characteristic of any crepant resolution. I will discuss a recent conjecture with Szendroi and Ongaro, regarding this "reduced" partition function, namely, that it has only positive coefficients. The approach I will discuss for proving this conjecture is somewhat algebraic, and is an upgrade of the famous PT/DT correspondence to cohomology. Namely, the route to proving the conjecture consists of showing that the unreduced partition function is a module over a certain quantum group (the degree zero cohomological Hall algebra of sheaves on the resolution), where this quantum group has characteristic function given by the required power of the McMahon partition function. Freeness of this module then provides a categorification of the reduced partition function - it is the characteristic function of the set of generators for this module. -
MAXIMALS: Uzi Vishne (Bar-Ilan University) - Locally central simple algebras
17th July 2018, 2:05pm to 3:55pm JCMB 5323 -- Show/hide abstractAbstract: The building blocks of ring theory are finite dimensional central simple algebras, governed by the Brauer group of the base field. Stretching the theory to infinity, we consider the class of algebras which are locally central simple, such as "supernatural" matrix algebras, and some infinite dimensional algebraic division algebras. We develop a theory for this class, generalizing the basic notions from the Brauer group, such as a supernatural degree and supernatural matrix degree, and define a Brauer monoid for countably generated locally central simple algebras. Examples such as infinite dimensional Clifford algebras and infinite dimensional crossed products serve to demonstrate the limitations of the new theory. This is based on joint work with Eli Matzri and our PhD students Shira Gilat and Tamar Bar-On. -
MAXIMALS: Valerio Toledano Laredo (Northeastern University)- Yangians, quantum loop algebras and elliptic quantum groups
19th June 2018, 2:05pm to 3:55pm JCMB 5323 -- Show/hide abstractAbstract: I will talk about an ongoing project with Sachin Gautam aimed at computing the monodromy of differential and difference equations associated to Yangians. As corollaries, one obtains a meromorphic braided tensor equivalence between finite-dimensional representations of Yangians and quantum loop algebras, as well as a classification of finite-dimensional representations of elliptic quantum groups. -
MAXIMALS: Joao Faria Martins (University of Leeds)- Categorifications of the KZ-connection
5th June 2018, 2:05pm to 3:55pm JCMB 5323 -- Show/hide abstractAbstract: I will review the work done over the last few years on categorifications of the Knizhnik–Zamolodchikov connection via a differential crossed module of 2-chord diagrams. Possible applications to higher category theory and to the topology of knotted surfaces in the 4-sphere will be explored. All algebraic and differential-geometrical background will be carefully explained. -
MAXIMALS: Adrien Brochier (Hamburg University)- Towards a quantum Riemann-Hilbert correspondence
29th May 2018, 3:05pm to 3:55pm JCMB 5323 -- Show/hide abstractAbstract: An important theorem by Kohno, Drinfeld and Kazhdan-Lusztig states that the explicit representations of braid groups obtained from the representation theory of quantum groups compute the monodromy of the so-called KZ equation in conformal field theory. Remarkably this connection can also be interpreted as a quantization of the action by isomonodromy of the mapping class group on the moduli space of flat connections on a Riemann spheres with several punctures. In this talk I will sketch some recent progresses towards a higher genus version of this result, computing the monodromy of an analog of the KZ connection on the moduli spaces of Riemann surfaces in terms of a certain canonical quantization of the character varieties of those surfaces. This can be interpreted as a quantization of the symplectic nature and the mapping class group equivariance of the Riemann-Hilbert correspondence. This is partly based on joint work with D. Ben-Zvi, D. Jordan and N. Snyder. -
MAXIMALS: Noah Snyder (Indiana University)- Exceptional Fusion Categories
29th May 2018, 2:05pm to 2:55pm JCMB 5323 -- Show/hide abstractAbstract: Fusion categories are tensor categories that look much like the category of complex representations of a finite group: they have duals, are semisimple, and have finitely many simple objects. In addition to finite groups, the main source of examples are the semisimplified quantum groups at roots of unity. Moore and Seiberg asked whether quantum group categories might explain all fusion categories. The goal of this talk is to survey the current state of knowledge about ''exceptional'' fusion categories which don't seem to come from groups or quantum groups. In a sense this talk will be more like an experimental physics talk, in that one is searching for ''new particles'' in various regimes (e.g. ''low index subfactors'') and seeing what you can find. The punchline is that we know one new large family of fusion categories (the Izumi quadratic categories) and four isolated examples (the Extended Haagerup Subfactors). This will include some of my own work joint with Bigelow, Grossman, Izumi, Morrison, Penneys, Peters, and others, but also summarize the work of many other people (especially Asaeda, Bisch, Ocneanu, Haagerup, Izumi, Jones, and Popa working in Subfactor theory). -
MAXIMALS: Olivier Schiffmann (Université Paris Sud) - Cohomological hall algebra of coherent sheaves on a smooth projective curve
22nd May 2018, 2:05pm to 3:55pm JCMB 5323 -- Show/hide abstractAbstract: We define and study Hall algebra structures on the (co)homology of the moduli stacks of coherent sheaves and Higgs sheaves on compact Riemann surfaces. We provide some generation and torsion-freeness results in both cases, and an algebraic presentation in the case of coherent sheaves. This is joint work with F. Sala, and E. Vasserot respectively. -
MAXIMALS: Kayvan Nejabati Zenouz (University of Edinburgh) - Hopf-Galois Theory and the Yang-Baxter Equation
15th May 2018, 2:05pm to 3:55pm JCMB 5323 -- Show/hide abstractAbstract: For L/K a finite Galois extension of fields with Galois group G the existence of a normal basis implies that L is a free K[G]-module of rank one. In general there can be other Hopf algebras acting on L with similar properties, namely those which endow L/K with a Hopf-Galois structure. Hopf-Galois theory was initially introduced in 1969 by S. Chase and M. Sweedler and has applications in Galois module theory. On the other hand, the Yang-Baxter equation is a matrix equation for the linear automorphisms of the tensor product of a vector space with itself. The Yang-Baxter equation is one of the important equations in quantum group theory, which has applications in mathematical physics. In 1992 V. Drinfeld suggested studying the set-theoretic version of this equation as a simpler way of solving some instances of it. Currently, the classification of Hopf-Galois structures and the set-theoretic solutions of the Yang-Baxter equation are among important topics of research. In this talk we will explain how the study of Hopf-Galois theory and the Yang-Baxter equation came to be connected via algebraic objects called skew braces. Then we will explain how one can classify and study the Hopf-Galois structures and skew braces of order p^3 for a prime number p. -
MAXIMALS: Stefan Kolb (University of Newcastle) - Braided module categories via quantum symmetric pairs
8th May 2018, 2:05pm to 3:55pm JCMB 5323 -- Show/hide abstractAbstract: The theory of quantum symmetric pairs provides coideal subalgebras of quantized enveloping algebras which are quantum group analogs of Lie subalgebras fixed under an involution. The finite dimensional representations of a quantized enveloping algebra form a braided monoidal category C, and the finite dimensional representations of any coideal subalgebra form a module category over C. In this talk I will discuss the notion of a braided module category over C as introduced by A. Brochier in 2013, and I will explain how quantum symmetric pairs provide examples. Time permitting, I will also indicate how the underlying universal K-matrix can be employed to describe a basis of the centre of quantum symmetric pair coideal subalgebras. This simplifies joint work with G. Letzter from 2006. -
MAXIMALS: Michael Groechenig (Freie Universität Berlin) - p-adic integration for the Hitchin system
1st May 2018, 3:05pm to 3:55pm JCMB 5323 -- Show/hide abstractAbstract: I will report on joint work with D. Wyss and P. Ziegler. We prove a conjecture by Hausel-Thaddeus which predicts an agreement of appropriately defined Hodge numbers for certain moduli spaces of Higgs bundles over the complex numbers. Despite the complex-analytic nature of the statement our proof is entirely arithmetic. If time permits I will also discuss the connection to the fundamental lemma. -
MAXIMALS: Thomas Nevins (University of Illinois Urbana-Champaign) - Cohomology of quiver varieties and other moduli spaces
1st May 2018, 2:05pm to 2:55pm JCMB 5323 -- Show/hide abstractAbstract: Nakajima's quiver varieties form an important class of algebraic symplectic varieties. A quiver variety comes naturally equipped with certain “tautological vector bundles”; I will explain joint work with McGerty that shows that the cohomology ring of the quiver variety is generated by the Chern classes of the tautological bundles. Analogous results (work in preparation with McGerty) also hold for the Crawley-Boevey—Shaw “multiplicative quiver varieties,’’ in particular for twisted character varieties; and the cohomology results in both cases generalize to other cohomology theories, derived categories, etc. I hope to explain the main ideas behind the proofs of such theorems and how they form part of a general pattern in noncommutative geometry. -
MAXIMALS: Omar Leon Sanchez (University of Manchester) - The Dixmier-Moeglin equivalence: a differential and a model-theoretic version.
24th April 2018, 2:05pm to 3:55pm JCMB 1501 -- Show/hide abstractAbstract: The classical Dixmier-Moeglin equivalence for noetherian algebras studies when three seemingly distinct notions for prime ideals are in fact equivalent (more precisely, the notions of locally-closed, primitive, and rational). This equivalence is known to hold for a large class of algebras (including commutative algebras, and quantised coordinate rings). In the last four years, there has been applications of the model theory of differential fields that yield new examples where the equivalence does not hold, and, more recently, also establish the equivalence for certain families of Hopf-Ore extensions. In this talk, I will give an overview of how this connection between "model theory" and "the Dixmier-Moeglin equivalence" arises. This will cover several results obtained jointly with Jason Bell, Stephane Launois, and Rahim Moosa. -
MAXIMALS: Martina Balagovic (Newcastle University) - The affine VW supercategory
17th April 2018, 2:05pm to 3:55pm JCMB 5323 -- Show/hide abstractAbstract: We define the affine VW supercategory sW, which arises from studying the action of the periplectic Lie superalgebra p(n) on the tensor product of an arbitrary representation M with several copies of the vector representation V of p(n). It plays a role analogous to that of the degenerate affine Hecke algebras in the context of representations of the general linear group. The main obstacle was the lack of a quadratic Casimir element for p(n). When M is the trivial representation, the action factors through the action of the previously known Brauer supercategory sBr. Our main result is an explicit basis theorem for the morphism sW, and as a consequence we recover the basis theorem for sBr. The proof utilises the close connection with the representation theory of p(n). As an application we explicitly describe the centre of all endomorphism algebras, and show that it behaves well under the passage to the associated graded and under deformation. This is joint work with Zajj Daugherty, Inna Entova-Aizenbud, Iva Halacheva, Johanna Hennig, Mee Seong Im, Gail Letzter, Emily Norton, Vera Serganova and Catharina Stroppel, arising from the WINART workshop. -
MAXIMALS: Jan Grabowski (Lancaster University) - Recovering automorphisms of quantum spaces
27th March 2018, 2:05pm to 3:55pm JCMB 5323 -- Show/hide abstractAbstract: It has long been expected, and is now proved in many important cases, that quantum algebras are more rigid than their classical limits. That is, they have much smaller automorphism groups. This begs the question of whether this broken symmetry can be recovered. I will outline an approach to this question using the ideas of noncommutative projective geometry, from which we see that the correct object to study is a groupoid, rather than a group, and maps in this groupoid are the replacement for automorphisms. I will illustrate this with the example of quantum projective space. This is joint work with Nicholas Cooney (Clermont-Ferrand). -
MAXIMALS: Jamie Judd (King's College London) - Tropical critical points of the superpotential for the flag variety in type A.
21st March 2018, 2:05pm to 2:55pm JCMB 5323 -- Show/hide abstractAbstract: In this talk I will look at the notion of tropical critical points of the superpotential of the flag variety (in type A). A superpotential for any flag variety of general type was defined in the work of K. Rietsch, and there it was also shown how the critical points of this superpotential describe the quantum cohomology of the original flag variety. By tropicalising this superpotential, one can associate to any finite dimension representation of the group SLn, a family of polytopes indexed by the reduced expressions for the longest element of the Weyl group. I will then show how looking at the tropical critical points of the superpotential gives a distinguished point in each of these polytopes, and will also interpret this point via a construction coming just from the original flag variety. -
MAXIMALS: Daniele Valeri (Tsinghua University) - Algebraic structures arising from physics
13th February 2018, 2:05pm to 3:55pm JCMB 5323 -- Show/hide abstractAbstract: In 1985 Zamolodchikov constructed a "non-linear" extension of the Virasoro algebra known as W(3)-algebra. This is the one of the first appearance of a rich class of algebraic structures, known as W-algebras, which are intimately related to physical theories with symmetry and revealed many applications in mathematics . In the first part of the talk I will review some facts about the general theory of W-algebras. Then, I will explain how to describe quantum finite and classical affine W-algebras using Lax operators. In the quantum finite case this operator satisfies a generalized Yangian identity, while in the classical affine case it is used to construct an integrable Hamiltonian hierarchy of Lax type equations. This is a joint work with A. De Sole and V.G. Kac. -
MAXIMALS: Clark Barwick (University of Edinburgh) - Exodromy
6th February 2018, 2:05pm to 3:55pm JCMB 5323 -- Show/hide abstractAbstract: It is a truth universally acknowledged, that a local system on a connected topological manifold is completely determined by its attached monodromy representation of the fundamental group. Similarly, lisse ℓ-adic sheaves on a connected variety determine and are determined by representations of the profinite étale fundamental group. Now if one wants to classify constructible sheaves by representations in a similar manner, new invariants arise. In the topological category, this is the exit path category of Robert MacPherson (and its elaborations by David Treumann and Jacob Lurie), and since these paths do not ‘run around once’ but ‘run out’, we coined the term exodromy representation. In the algebraic category, we define a profinite ∞-category – the étale fundamental ∞-category – whose representations determine and are determined by constructible (étale) sheaves. We describe the étale fundamental ∞-category and its connection to ramification theory, and we summarise joint work with Saul Glasman and Peter Haine. -
MAXIMALS: Karin Baur (University of Graz) - Dimers with boundary, associated algebras and module categories
30th January 2018, 3:05pm to 3:55pm JCMB 5323 -- Show/hide abstractAbstract: Dimer models with boundary were introduced in joint work with King and Marsh as a natural generalisation of dimers. We use these to derive certain infinite dimensional algebras and consider idempotent subalgebras w.r.t. the boundary. The dimer models can be embedded in a surface with boundary. In the disk case, the maximal CM modules over the boundary algebra are a Frobenius category which categorifies the cluster structure of the Grassmannian. -
Joint UoE-HW Maximals seminar: Collin Bleak (St. Andrews) - Generalised Ping-Pong Lemmas, and the group of PL homeomorphisms of the unit interval.
23rd January 2018, 3:05pm to 3:55pm James Clerk Maxwell Building, Edinburgh EH9, UK, room 5323 -- Show/hide abstractAbstract: We define a notion of fast generating sets, for groups of self-homeomorphisms of a space (under composition), and apply it to the group Pl_o(I) of piecewise-linear homeomorphisms of the unit interval. As a consequence, we build some general forms of Ping-Pong Lemmas for this group, which lemmas guarantee isomorphism types for certain fg subgroups of Plo(I), based on simple combinatorial data. We also find a lemma which guarantees that some particular (unexpectedly large) set of subgroups of Pl_o(I) also embed in R. Thompson’s group F. Joint with Matt Brin and Justin Moore. -
Joint UoE-HW Maximals seminar: Yuri Bazlov (Manchester) - H-cross products
23rd January 2018, 2:05pm to 2:55pm James Clerk Maxwell Building, Edinburgh EH9, UK, room 5323 -- Show/hide abstractAbstract: This talk is based on joint work with Arkady Berenstein. It frequently happens that an algebra C factors as C=AB, meaning a vector space isomorphism between C and the tensor product of its subalgebras A and B. The classical PBW theorem and its more recent incarnations -- think quantum groups and affine Hecke algebras -- are statements about algebra factorizations. Conversely, an algebra structure on the tensor product of can be established in many cases: semidirect product, braided tensor product, etc, which all fit the situations when A is an H-module algebra and B is an H-comodule algebra for some bialgebra H. We show that, quite surprisingly, any algebra factorization C=AB can be realised in this way for a suitable H: an ordinary bialgebra if the factorization is tame (which is typically the case), or a topological bialgebra in general. In particular, when C is a rational Cherednik algebra or a Kostant-Kumar nilHecke algebra, reconstructing H leads us to a Nichols algebra. DAHA and its generalizations correspond to the Hecke-Hopf algebras H, recently found by Berenstein and Kazhdan. Even in more straightforward examples of algebra factorisations, H can be a new and interesting Hopf algebra the representation theory of which begs to be explored. -
Joint EDGE-MAXIMALS: Peter Samuelson (University of Edinburgh) - The Hall algebra of the Fukaya category of a surface
16th January 2018, 3:05pm to 3:55pm JCMB 5323 -- Show/hide abstractAbstract: The construction of the Fukaya category of a symplectic manifold is inspired by string theory: roughly, objects are Lagrangians, morphisms are intersection points, and composition of morphisms comes from "holomorphic disks." For surfaces, a combinatorial construction of the (partially wrapped) Fukaya category was recently given by Haiden, Katzarkov, and Kontsevich. We will discuss this category and some recent results involving its (derived) Hall algebra (joint with B. Cooper). -
Joint EDGE-MAXIMALS: Peter Samuelson (University of Edinburgh) - The Hall algebra of an elliptic curve
16th January 2018, 2:05pm to 2:55pm JCMB 5323 -- Show/hide abstractAbstract: The Hall algebra of an abelian category has structure constants coming from "counting extensions" in the category. In this talk we give a survey of some recent results involving the Hall algebra of the category of coherent sheaves on an elliptic curve. Some topics involve an explicit description of the algebra by Burban and Schiffmann and a construction of Schiffmann and Vasserot of an action on the space of symmetric functions using Hilbert schemes and double affine Hecke algebras. -
MAXIMALS: Richard Timoney, Associative universal enveloping triples for Jordan triples
12th December 2017, 3:05pm to 3:55pm James Clerk Maxwell Building, Edinburgh EH9, UK, seminar room 5323 -- Show/hide abstractAbstract: In recent work with Les Bunce we have investigated the different ways to embed a JC*-triple as JC*-subtriple of all operators. This relates to the associative algebraic structure generated by the triple and is also reflected in the geometry of matrices over the triple. The concept of universal reversibility plays a significant role. We will describe some of this work when restricted to the finite dimensional case. -
MAXIMALS: Richard Timoney, A brief survey of Jordan triples
12th December 2017, 2:05pm to 2:55pm James Clerk Maxwell Building, Edinburgh EH9, UK, seminar room 5323 -- Show/hide abstractAbstract: We summarize part of the basic theories of Jordan algebras and (positive hermitian) Jordan triple systems in finite dimensions. This will include connections with other topics including several complex variables and homogeneous cones. -
MAXIMALS: Yuri Bahturin, Growth functions and embeddings of algebras
5th December 2017, 3:05pm to 3:55pm JCMB, Edinburgh, United Kingdom, seminar room 5323 -- Show/hide abstractAbstract: The aim of the talk is to present some joint results with Alexander Olshansky. When an algebra B is embedded in an algebra A then the growth functions of A produce some growth-like functions on B. Comparing these functions with the growth functions of B, one can speak about embeddings with or without distortion. We study these and related phenomena for general algebras, but the main results are in the case of associative and Lie algebras. -
MAXIMALS: Yuri Bahturin, Associative and Lie algebras defined by generators and relations
5th December 2017, 2:05pm to 2:55pm JCMB, Edinburgh, United Kingdom, seminar room 5323 -- Show/hide abstractAbstract: The aim of this lecture is to describe techniques that enable one to provide vector space bases for associative and Lie algebras which are given in terms of generators and defining relations. -
MAXIMLS: Jessica Sidman, Bar-and-joint frameworks: Stresses and Motions
4th December 2017, 3:05pm to 3:55pm James Clerk Maxwell Building, Edinburgh EH9, UK, room 6201 -- Show/hide abstractAbstract: Suppose that we have a framework consisting of finitely many fixed-length bars connected at universal joints. Such frameworks (and variants) arise in many guises, with applications to the study of sensor networks, the matrix completion problem in statistics, robotics and protein folding. The fundamental question in rigidity theory is to determine if a framework is rigid or flexible. The standard approach in combinatorial rigidity theory is to differentiate the quadratic equations constraining the distances between joints, and work with these linear equations to determine if the framework is infinitesimally rigid or flexible. In this talk I will discuss recent progress using algebraic matroids that gives further insight into the infinitesimal theory and also provides methods for identifying special bar lengths for which a generically rigid framework is flexible. We use circuit polynomials to identify stresses, or dependence relations among the linearized distance equations and to find bar lengths which give rise to motions. This is joint work with Zvi Rosen, Louis Theran, and Cynthia Vinzant. -
MAXIMALS: Lewis Topley, The orbit method for Poisson orders
28th November 2017, 3:05pm to 3:55pm JCMB, Edinburgh, United Kingdom, seminar room 5323 -- Show/hide abstractAbstract: In the first talk I will give an introduction to the theory of complex affine Poisson varieties. I will explain how they arise in deformation theory, how they can be stratified into symplectic leaves and into symplectic cores. Finally I will recall the Poisson Dixmier-Moeglin equivalence (PDME) for affine Poisson algebras and explain some consequences. The second talk will focus on Poisson orders, which can be seen as coherent sheaves of non-commutative algebras carrying a Poisson module structure, over some Poisson variety. I will explain how to stratify the primitive spectrum of a Poisson order into symplectic cores, and introduce the category of Poisson modules over a Poisson order. The main result of this talk states that when the Poisson variety is smooth with locally closed symplectic leaves, the spectrum of annihilators of simple Poisson modules over a Poisson order is homeomorphic to the space of symplectic cores of the Poisson order, once both spaces have been endowed with suitable topologies. We view this as an expression of the orbit method from Lie theory. I will explain that the theorem follows from an upgraded version of the PDME for Poisson orders. Our main new tool is the enveloping algebra of a Poisson order, an associative algebra which captures the Poisson representation theory of the Poisson order. This is joint work with Stephane Launois (arXiv:1711.05542). -
MAXIMALS: Lewis Topley - The orbit method for Poisson orders
28th November 2017, 2:05pm to 2:55pm Seminar room, JCMB 5323 -- Show/hide abstractAbstract: In the first talk I will give an introduction to the theory of complex affine Poisson varieties. I will explain how they arise in deformation theory, how they can be stratified into symplectic leaves and into symplectic cores. Finally I will recall the Poisson Dixmier-Moeglin equivalence (PDME) for affine Poisson algebras and explain some consequences. The second talk will focus on Poisson orders, which can be seen as coherent sheaves of non-commutative algebras carrying a Poisson module structure, over some Poisson variety. I will explain how to stratify the primitive spectrum of a Poisson order into symplectic cores, and introduce the category of Poisson modules over a Poisson order. The main result of this talk states that when the Poisson variety is smooth with locally closed symplectic leaves, the spectrum of annihilators of simple Poisson modules over a Poisson order is homeomorphic to the space of symplectic cores of the Poisson order, once both spaces have been endowed with suitable topologies. We view this as an expression of the orbit method from Lie theory. I will explain that the theorem follows from an upgraded version of the PDME for Poisson orders. Our main new tool is the enveloping algebra of a Poisson order, an associative algebra which captures the Poisson representation theory of the Poisson order. This is joint work with Stephane Launois (arXiv:1711.05542). -
MAXIMALS: Andrea Santi, talk title - On a class of ternary algebras
21st November 2017, 3:05pm to 3:55pm JCMB, Edinburgh, United Kingdom, Seminar room 5323 -- Show/hide abstractAbstract: In the first part of the seminar, I will review the traditional realizations by Chevalley, Jacobson and Schafer of the exceptional simple Lie groups as automorphisms of algebraic structures. In particular, I will recall the role played by a certain class of ternary algebras introduced by Freudenthal in the process of constructing the 56-dimensional representation of E7 from the 27-dimensional exceptional Jordan algebra. Kantor ternary algebras are a natural generalization of both Jordan and Freudenthal ternary algebras. In the second part of the seminar, I will describe the classification problem of simple linearly compact Kantor ternary algebras (over the complex field) and propose a solution to this problem. I will show that every such ternary algebra is finite-dimensional and provide a classification in terms of Satake diagrams. The Kantor ternary algebras of exceptional type can be divided into three main classes, a concrete example for each class will be given. This is a joint work with N. Cantarini and A. Ricciardo. -
MAXIMALS: Andrea Santi, On a class of ternary algebras
21st November 2017, 2:05pm to 2:55pm JCMB, Edinburgh, United Kingdom, Seminar room 5323 -- Show/hide abstractAbstract: In the first part of the seminar, I will review the traditional realizations by Chevalley, Jacobson and Schafer of the exceptional simple Lie groups as automorphisms of algebraic structures. In particular, I will recall the role played by a certain class of ternary algebras introduced by Freudenthal in the process of constructing the 56-dimensional representation of E7 from the 27-dimensional exceptional Jordan algebra. Kantor ternary algebras are a natural generalization of both Jordan and Freudenthal ternary algebras. In the second part of the seminar, I will describe the classification problem of simple linearly compact Kantor ternary algebras (over the complex field) and propose a solution to this problem. I will show that every such ternary algebra is finite-dimensional and provide a classification in terms of Satake diagrams. The Kantor ternary algebras of exceptional type can be divided into three main classes, a concrete example for each class will be given. This is a joint work with N. Cantarini and A. Ricciardo. -
MAXIMALS: Angus Macintyre, Exponential algebra and its relevance to analysis
14th November 2017, 3:05pm to 3:55pm JCMB, Edinburgh, United Kingdom, seminar room 5323 -- Show/hide abstractAbstract: I will explain how the ideas from Part 1 connect to difficult parts of analysis and/or topology(Nevanlinna Theory, Morse Theory and Shapiro’s Conjecture from 1950 about common zeros of exponential polynomials). Serious work from Diophantine geometry is involved, due to Bombieri, Masser and Zannier. -
MAXIMALS: Angus Macintyre, Exponential algebra and its relevance to analysis
14th November 2017, 2:05pm to 2:55pm JCMB, Edinburgh, United Kingdom, Seminar room 5323 -- Show/hide abstractAbstract: Exponential algebra extends commutative algebra by taking account of rings (particularly fields) with an exponential function. Classically there are few examples, but those, the real and the complex exponentials, are of great importance for all of mathematics. It is not even clear what should be the definition of an exponential ring, and it is certainly not at all clear what exponentially algebraic should mean. Historically Hardy (around 1911) used some tricks of the trade to get good information about zeros of one variable exponential polynomials, and Ritt, in the late 1920’s, established a quite subtle factorization theorem for one variable exponential polynomials. These in turn linked to questions about the distribution of zeros of systems of exponential polynomials. Some of these problems have remained open, and turn out to be connected both to transcendental number theory and to mathematical logic (decidability and definability issues). In the first part I will explain some basic definitions and constructions (e.g of free exponential rings),and sketch the connection to Schanuel’s Conjecture from transcendental number theory. I will also explain how logicians came to these problems, and what difference their ideas made in establishing a quite elaborate subject of exponential algebra. -
MAXIMALS: Gabor Elek, Almost commuting matrices
7th November 2017, 3:05pm to 3:55pm Seminar Room, JCMB 5323 -- Show/hide abstractAbstract: I will give some details about the proof of our results on almost commuting matrices that includes effective algebraic geometry and commutative algebra as well as the algebraic Ornstein Weiss principle. -
MAXIMALS: Gabor Elek, Almost commuting matrices
7th November 2017, 2:05pm to 2:55pm Seminar Room, JCMB 5323 -- Show/hide abstractAbstract: I will talk about a classical problem of Halmost on almost commuting matrices and our recent result with Lukasz Grabowski. -
MAXIMALS: Stephane Launois - Total positivity and quantum algebras
31st October 2017, 3:05pm to 3:55pm JCMB, Edinburgh, United Kingdom, Seminar Room 5323 -- Show/hide abstractAbstract: I will discuss links between total positivity and the ideal structure of quantum algebras. In the first talk, I will focus on the matrix case and show how tools developed in the quantum setting are relevant for the study of totally nonnegative matrices. In the second part of the talk, I will focus on the grassmannian case. -
MAXIMALS: Stephane Launois, Total positivity and quantum algebras
31st October 2017, 2:05pm to 2:55pm JCMB, Edinburgh, United Kingdom, Seminar room 5323 -- Show/hide abstractAbstract: I will discuss links between total positivity and the ideal structure of quantum algebras. In the first talk, I will focus on the matrix case and show how tools developed in the quantum setting are relevant for the study of totally nonnegative matrices. In the second part of the talk, I will focus on the grassmannian case. -
MAXIMALS: Dom Hipwood - Blowing up a noncommutative surface
17th October 2017, 3:05pm to 3:55pm Seminar Room, JCMB 5323 -- Show/hide abstractAbstract: A major current goal for noncommutative projective geometers is the classification of so-called “noncommutative surfaces”. Let S denote the 3-dimensional Sklyanin algebra, then S can be thought of as the generic noncommutative surface. In recent work Rogalski, Sierra and Stafford have begun a project to classify all algebras birational to S. They successfully classify the maximal orders of the 3-Veronese subring T of S. These maximal orders can be considered as blowups at (possibly non-effective) divisors on the elliptic curve E associated to S. We are able to obtain similar results in the whole of S. -
MAXIMALS: Dom Hipwood - Introduction to noncommutative projective geometry
17th October 2017, 2:05pm to 2:55pm Seminar Room, JCMB 5323 -- Show/hide abstractAbstract: I will introduce a few key concepts in the theory of noncommutative projective geometry. In particular, I aim to give an idea of how one should think of a noncommutative curve/ surface. I will also describe a key example called a twisted homogeneous coordinate ring: a ring built out of geometry which plays a vital role in the theory. -
MAXIMALS: Stanislav Shkarin, Automaton algebras versus finite Groebner basis
10th October 2017, 3:05pm to 3:55pm JCMB, Edinburgh, United Kingdom, seminar room 5323 -- Show/hide abstractAbstract: We answer a question of Ufnarovskii whether an automaton algebra must possess a generating set and an order on monomials with respect to which the reduced Groebner basis in the ideal of relations is finite. Namely, we present an example of a quadratic algebra given by three generators and three relations, which is automaton (the set of normal words forms a regular language) and such that its ideal of relations does not possess a finite Groebner basis with respect to any choice of generators and any choice of a well-ordering of monomials compatible with multiplication. Note that extending the ground field does not help. The proof is partially based on sensitivity of the growth of an algebra to characteristic of the ground field, which is restricted in case of finite Groebner basis. -
MAXIMALS: Stanislav Shkarin, pre-talk: Intermediate growth via Groebner basis
10th October 2017, 2:05pm to 2:55pm JCMB, Edinburgh, United Kingdom, seminar room 5323 -- Show/hide abstractAbstract: We introduce the concepts of a Groebner basis, Hilbert series and growth of an algebra. These notions will be demonstrated on the following result. We present a simple example (4 generators and 7 relations) of a quadratic semigroup algebra of intermediate growth. The proof is obtained by computing the (infinite) reduced Groebner basis in the ideal of relations. Although the basis follows a clear and simple pattern, the corresponding set of normal words fails to form a regular language. The latter is noteworthy in its own right. The only previously known example of a quadratic algebra of intermediate growth due to Kocak is non-semigroup and is given by 14 generators and 96 quadratic relations. -
MAXIMALS: Kevin Tucker, talk - Bertini Theorems for F-signature
3rd October 2017, 3:05pm to 3:55pm JCMB, Edinburgh, United Kingdom, seminar room 5323 -- Show/hide abstractAbstract: Bertini Theorems for F-signature Abstract: In characteristic zero, it is well known that multiplier ideals and log terminal singularities satisfy Bertini-type theorems for hyperplane sections. The analogous situation in characteristic p > 0 is more complicated. While F-regular singularities satisfy Bertini, the test ideal does not. In this talk, I will describe joint work with Karl Schwede and Javier Carvajal-Rojas showing that the F-signature -- a numerical invariant of singularities that detects F-regularity -- satisfies the relevant Bertini statements for hyperplane sections. In particular, one can view this as a generalization of the corresponding results for F-regularity. -
MAXIMALS: Kevin Tucker, pre-talk 'An introduction to F-invariants'
3rd October 2017, 2:05pm to 2:55pm JCMB, Edinburgh, United Kingdom, seminar room 5323 -- Show/hide abstractAbstract: We will give a gentle introduction to some of the basic invariants of singularities of rings in positive characteristic defined via the Frobenius endomorphism. In particular, we will pay close attention to F-signature and Hilbert-Kunz multiplicity, highlighting the known examples for each. -
MAXIMALS: Andrea Appel - The isomorphism between classical and quantum sl(n)
26th September 2017, 3:05pm to 3:55pm Seminar room JCMB 5323 -- Show/hide abstractAbstract: It is well known that the universal enveloping algebra of a finite dimensional Lie algebra admits no non-trivial deformations as an algebra. In particular, there exists a (non-explicit) isomorphism between the trivial deformation of the enveloping algebra and the corresponding quantum group. An explicit description of such isomorphism was known only for sl(2). In this talk, we introduce a new realisation of the evaluation homomorphism of the Yangian of sl(n) and we use it to obtain an explicit isomorphism between classical and quantum sl(n). This is a work in progress with S. Gautam. -
MAXIMALS: Adrea Appel - The Yangian and the Capelli identities for gl(n) and sl(n)
26th September 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstractAbstract: In this talk, we recall some basic facts about the Yangian of gl(n) and, in particular, the role played by the Capelli identity in the definition of the Yangian and in the construction of the evaluation homomorphism. We then describe a similar result for sl(n), which leads to an apparently new presentation of the evaluation homomorphism in this case. -
MAXIMALS: Greg Stevenson - "Non-crossing partitions as lattices of localizations"
19th September 2017, 3:05pm to 3:55pm video conference room 5323, JCMB -- Show/hide abstractAbstract: By a result of Ingalls and Thomas, one can think of the bounded derived category of finite dimensional representations of an ADE quiver as a categorification of non-crossing partitions of the corresponding type. The non-crossing partitions are precisely the lattice of exact localizations of the bounded derived category. I'll discuss various directions in which one can generalise this, such as the extension to doubly infinite Dynkin type A, representations over more general rings, and (time permitting) the situation for tame quivers. This is based on joint work with Gratz, with Antieau, and with Krause. -
MAXIMALS: Greg Stevenson - "Quivers, their derived categories, and lattices of subcategories"
19th September 2017, 2:05pm to 2:55pm video conference room 5323, JCMB -- Show/hide abstractAbstract: Abstract TBA -
Theo Johnson-Freyd (Perimeter Institute) - The Moonshine Anomaly
19th July 2017, 4:00pm to 5:00pm -- Show/hide abstractAbstract: Conformal field theories, and the fusion categories derived from them, provide classes in group cohomology that generalize characteristic classes. These classes are called "anomalies", and obstruct the existence of constructing orbifold models. I will discuss two of the most charismatic groups --- the Conway group Co_0 and the Fischer--Griess Monster group M --- and explain my calculation that in both cases the anomaly has order exactly 24. The Monster calculation relies on a version of "T-duality" for finite groups which in turn relies on fundamental results about fusion categories. I will try to explain everything from the beginning, and assume no knowledge of the Monster or its cousins. -
MAXIMALS Seminar - Travis Schedler: Poisson (co)homology via D-modules
23rd May 2017, 2:00pm to 3:00pm Seminar room, JCMB 5323 -- Show/hide abstractAbstract: Abstract: I will explain how to study the Poisson (co)homology of a Poisson variety locally via D-modules. When there are finitely many symplectic leaves, the zeroth Poisson homology is finite-dimensional, and as an application, one has finitely many irreducible representations of every quantization. In the case that the variety is conical and admits a symplectic resolution, this conjecturally recovers the cohomology of the resolution and equips it with filtrations recording the order of vanishing of fiberwise closed differential forms on smoothings. In the case of nilpotent cones, this recovers a conjectural formula of Lusztig in terms of Kostka polynomials. In the case of smooth Poisson varieties, work in progress with Brent Pym shows that the entire Poisson cohomology is finite-dimensional when the modular filtration is finite and defines a perverse sheaf. This has applications to Feigin-Odesski Poisson structures on even-dimensional projective spaces. -
GEARS Seminar - Travis Schedler: Poisson traces, D-modules, and symplectic resolutions II
22nd May 2017, 4:00pm to 5:00pm Seminar room, JCMB 5323 -- Show/hide abstractAbstract: Abstract: Given a Poisson algebra A, the space of Poisson traces are those functionals annihilating {A,A}, i.e., invariant under Hamiltonian flow. I explain how to study this subtle invariant via D-modules (the algebraic study of differential equations), conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. -
GEARS Seminar - Travis Schedler: Poisson traces, D-modules, and symplectic resolutions I
22nd May 2017, 2:30pm to 3:30pm Seminar room JCMB 5323 -- Show/hide abstractAbstract: Abstract: Given a Poisson algebra A, the space of Poisson traces are those functionals annihilating {A,A}, i.e., invariant under Hamiltonian flow. I explain how to study this subtle invariant via D-modules (the algebraic study of differential equations), conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. -
David Ssevviiri (University of Makerere): Completely prime modules and 2-primal modules
8th May 2017, 3:00pm to 4:00pm 4325B -- Show/hide abstractAbstract: A notion of prime ideal in a commutative ring can be expressed in many equivalent statements that become distinct when the ring is assumed to be noncommutative. This leads to: completely prime ideals, strictly prime ideals, strongly prime ideals, s-prime ideals, l-prime ideals, etc. Each of the aforementioned “prime” has a module analogue; and these analogues collapse to just one notion when a module is defined over a commutative ring. In this talk, I discuss the completely prime (sub)modules, their properties and torsion theories induced by the completely prime radical. Secondly, by comparing completely prime (sub)modules with prime (sub)modules, I talk about a class of 2-primal modules which is a module analogue of 2-primal rings. -
MAXIMALS: Brent Pym - "Multiple zeta values in deformation quantization"
18th April 2017, 3:10pm to 4:00pm JCMB 5323 (video conference room) -- Show/hide abstractAbstract: The integrals appearing in Kontsevich's deformation quantization formula are notoriously difficult to compute. As a result, direct calculations with the formula have so far been intractable, even in very simple examples. In forthcoming work with Peter Banks and Erik Panzer, we give an algorithm for the exact evaluation of the integrals in terms of special transcendental constants: the multiple zeta values. It allows us to calculate the formula on a computer for the first time. I will give an overview of our approach, which recasts the integration problem in purely algebraic terms, using Francis Brown's theory of single-valued multiple polylogarithms. -
MAXIMALS: Brent Pym - "Introduction to Kontsevich's quantization formula"
18th April 2017, 2:10pm to 3:00pm JCMB 5323 (video conference room) -- Show/hide abstractAbstract: Deformation quantization is a process that assigns to any classical mechanical system its quantum mechanical analogue. The problem can be phrased in purely algebraic terms: we would like to start with a commutative ring equipped with a Poisson bracket, and produce a noncommutative deformation of its product. A priori, the Poisson bracket only specifies the deformation to first order in the deformation parameter, but a deep theorem of Maxim Kontsevich extends the deformation to all orders in a canonical way. While the problem is algebraic, his solution is transcendental: it involves integrals over high-dimensional configuration spaces. I will give an elementary introduction to his formula, and talk about the (very few) examples in which it can actually be computed by hand. -
MAXIMALS: Elena Gal -- "A geometric approach to Hall algebra"
4th April 2017, 3:10pm to 4:10pm video conference room (5323) JCMB -- Show/hide abstractAbstract: The Hall algebra associated to a category is related to the Waldhausen S-construction in the work of Kapranov and Dyckerhoff. We explain how the higher associativity data can be extracted from this construction in a natural way, thus allowing for various higher categorical versions of Hall algebra. We then discuss a natural and systematic extension of this construction providing a bi-algebraic structure. We show how it can be used to provide a more transparent proof for the Green's theorem for the Hall algebras of hereditary categories and discuss possible extension to the higher categorical setting. -
MAXIMALS: Adam Gal -- "Base change and categorification"
4th April 2017, 2:10pm to 3:00pm video conference room 5323, JCMB -- Show/hide abstractAbstract: We discuss the notion of a “mate” of a square in a 2-category. We will explain how it is related to base change in algebraic geometry, and that it can be understood as a homotopic condition. We then explain how this can be used to categorify the notion of Hopf algebra, and the Heisenberg double construction. -
MAXIMALS: Leandro Vendramin - Set-theoretical solutions of the Yang-Baxter equation
28th March 2017, 2:10pm to 3:00pm -- Show/hide abstractAbstract: The Yang-Baxter equation is an important tool in theoretical physics and pure mathematics, with many applications in different domains going from condensed matter to topology. The importance of this equation led Drinfeld to ask for studying the simplest family of solutions: combinatorial or set-theoretical solutions. In this talk we review the basic theory of set-theoretical solutions, we discuss some problems and solutions and we give some application. -
MAXIMALS: Pieter Belmans - Construction of noncommutative surfaces of rank 4
21st March 2017, 3:10pm to 4:00pm video conference room (5323) JCMB -- Show/hide abstractAbstract: The numerical classification of "noncommutative surfaces" of rank 4 suggests the existence of a family previously not considered in the literature. By generalising Orlov's blowup formula to blowups of sheaves of maximal orders outside the ramification locus, we construct these starting from Artin--Schelter regular algebras which are finite over their center for all cases in the classification. Previously the first non-trivial case in the classification was constructed by de Thanhoffer--Presotto using noncommutative P^1-bundles. We can compare this to the blowup construction using some very classical geometry of linear systems. This comparison can be seen as a noncommutative instance of the classical isomorphism between the first Hirzebruch surface and the blowup of P^2 in a point. This is joint work with Dennis Presotto and Michel Van den Bergh. -
MAXIMALS: Pieter Belmans Exceptional collections, mutations and Serre functors
21st March 2017, 2:10pm to 3:00pm video conference room (5323) JCMB -- Show/hide abstractAbstract: I will explain how one can describe derived categories of smooth projective varieties using full and strong exceptional collections, i.e. via an explicit finite-dimensional algebra. I will also explain the mutation of exceptional collections, and the role of the braid group. Another important ingredient in the description of a derived category is the Serre functor. It is given by Serre duality in algebraic geometry, and the Auslander-Reiten translation in the representation theory of algebras. The Serre functor induces an automorphism of the Grothendieck group, and in the case of a smooth projective surface this automorphism satisfies additional strong properties. I will review these results, and explain how it leads to the numerical classification of "noncommutative surfaces" of rank 4 due to de Thanhoffer--Van den Bergh. -
MAXIMALS: Bart Van Steirteghem - A characterization of the weight monoids of smooth affine spherical varieties
14th March 2017, 3:10pm to 4:10pm video conference room (5323) JCMB -- Show/hide abstractAbstract: Spherical varieties form a remarkable class of complex algebraic varieties equipped with an action of a reductive group G, which includes toric, flag and symmetric varieties. Smooth affine spherical varieties are the local models of multiplicity free (real) Hamiltonian and quasi-Hamiltonian manifolds. A natural invariant of an affine spherical variety X is its weight monoid, which is the set of irreducible representations (or dominant weights) of G that occur in the coordinate ring of X. In the 1990s F. Knop conjectured that the weight monoid is a complete invariant for smooth affine spherical varieties, and in 2006 I. Loseu proved this conjecture. I will present joint work with G. Pezzini in which we use the combinatorial theory of spherical varieties and a smoothness criterion of R. Camus to characterize the weight monoids of smooth affine spherical varieties. I will also discuss some applications obtained with Pezzini and K. Paulus. -
MAXIMALS: Bart Van Steirteghem - Representation theory and combinatorial invariants of algebraic varieties
14th March 2017, 2:10pm to 3:00pm video conference room (5323) JCMB -- Show/hide abstractAbstract: I will illustrate with many examples how one can use the representation theory of complex reductive groups to obtain combinatorial invariants of algebraic varieties equipped with an action of such a group. The main focus will be on invariants of (affine) spherical varieties. -
MAXIMALS: Iva Halacheva (Lancaster) - The odd Brauer category and the periplectic Lie superalgebra
14th February 2017, 3:10pm to 4:00pm video conference room (5323) JCMB -- Show/hide abstractAbstract: The representation theory of Lie algebras of type D is studied by Ehrig and Stroppel in an approach generalizing Arakawa and Suzuki’s work in type A. More specifically, they use affine Nazarov-Wenzl algebras, and their cyclotomic quotients, instead of the degenerate affine Hecke algebra to describe the endomorphism ring of certain representations in type D. We will discuss a signed version, or odd VW algebras living in an extension of the odd Brauer category, in a similar approach towards describing endomorphism rings of the periplectic Lie superalgebra p(n). The key ingredient will be a certain sneaky quadratic Casimir element and the corresponding Jucys-Murphy components. -
MAXIMALS (Preseminar):Iva Halacheva (Lancaster) - Schur-Weyl duality for gl(n) and beyond
14th February 2017, 2:10pm to 3:00pm video conference room (5323) JCMB -- Show/hide abstractAbstract: As motivation for the work to be described in the second hour, I will discuss the classical Schur-Weyl duality for gl(n), describing the endomorphism ring of tensor powers of the vector representation in terms of the symmetric group, as well as higher Schur-Weyl duality involving more general representations and the degenerate affine Hecke algebra in place of the symmetric group. This construction has been further extended in using diagrammatic algebras to describe the representation theory of other Lie algebras and Lie superalgebras. We will also discuss some Lie supertheory towards looking at the periplectic Lie superalgebra. -
MAXIMALS - Ben Davison: Hidden properness and Kac-Moody algebras
7th February 2017, 3:10pm to 4:10pm -- Show/hide abstractAbstract: I'll explain the sense in which the map from the stack of finite-dimensional representations of an algebra to the coarse moduli space behaves as though it were a proper map. This turns out to be a vital piece in proving the following statement: the cohomological Hall algebra associated to the preprojective algebra of a quiver is a quantum enveloping algebra, for the strictly positive part of a new type of Kac-Moody algebra, which carries a cohomological grading. This gives a mathematical formulation for the physicists' Lie algebra of BPS states. The cohomological degree zero piece of this algebra is the positive part of the usual Kac-Moody algebra of the largest subquiver without imaginary simple roots, but even if this is the entire quiver, there's much more to this algebra than the usual Kac-Moody algebra of the quiver. -
MAXIMALS: Ben Davison - Hunting for BPS algebras
7th February 2017, 2:10pm to 3:10pm -- Show/hide abstractAbstract: The main talk concerns the search for the "Lie algebra of BPS states" associated to a preprojective algebra. In the pretalk I'll explain why, as a mathematician, one would go looking for such a thing. The evidence pointing to the existence of this algebra involves many nice results from combinatorics of representations of quivers, Donaldson-Thomas theory, and Nakajima quiver varieties. I'll try to give a geographical sketch of these other results in order to motivate the main talk. -
Joint EMPG-MAXIMALS: Bart Vlaar (York) - TBA
1st February 2017, 2:30pm to 3:30pm ICMS New Seminar Room -
MAXIMALS - Simon Crawford. Singularity categories of deformations of Kleinian singularities
31st January 2017, 3:00pm to 4:00pm 5323 JCMB (next to common room) -- Show/hide abstractAbstract: The Kleinian singularities make up a family of well-understood (commutative) surface singularities. In 1998, Crawley-Boevey and Holland introduced a family of algebras which may be viewed as noncommutative deformations of Kleinian singularities. Using singularity categories, I will make comparisons between the types of singularity arising in the commutative and noncommutative settings. I will also show that the "most singular" of these noncommutative deformations has a noncommutative resolution for which an analogue of the geometric McKay correspondence holds. -
Joint UoE&HW MAXIMALS: Ben Martin (Aberdeen) - Complete reducibility and geometric invariant theory
25th January 2017, 3:10pm to 4:00pm ICMS Lecture Theatre -- Show/hide abstractAbstract: Let G be a reductive algebraic group over a field k. The notion of a $G$-complete reducible subgroup of G was introduced by Serre; in the special case G= GLn(k), a subgroup H of G is G-completely reducible if and only if the inclusion of H in G is completely reducible in the sense of representation theory. G-complete reducibility has turned out to be an important tool for investigating the subgroup structure of simple algebraic groups. In this talk I will discuss the interplay between geometric invariant theory and the theory of G-complete reducibility. -
Joint UoE&HW MAXIMALS: Armando Martino (Southampton) - The Lipschitz Metric on Culler Vogtmann Space and Automorphisms of Free Groups
25th January 2017, 2:10pm to 3:10pm ICMS Lecture Theatre -- Show/hide abstractAbstract: One of the main tools for understanding automorphisms of free groups is via the action on Culler Vogtmann Space. More recently, the geometry of this space has been the subject of intense study. We will provide an introduction to these objects as well as presenting a report on some recent joint work with Stefano Francaviglia showing that the set of "minimally displaced points" for a given automorphism is connected, and that this is enough to solve the conjugacy problem in some limited cases. -
MAXIMALS: José Figueroa-O'Farrill (Edinburgh)- Filtered deformations in Algebra, Geometry and Physics
6th December 2016, 3:10pm to 4:00pm JCMB 6206 -- Show/hide abstractAbstract: In these talks I attempt to intercontextualise recent results obtained in collaboration with Andrea Santi on what could be termed an “Erlangen Programme for Supergravity”. (You don’t need to know what supergravity is to understand this talk.) We recently realised that an object I introduced many years ago — a Lie superalgebra with a geometric origin — has a precise algebraic structure that suggests a means of classifying certain geometries of interest. This is reminiscent of Klein’s Erlangen Programme: to study a geometry via its group of automorphisms. The algebraic structure in question is that of a filtered deformation of a Z-graded Lie superalgebra. In the main seminar I would like to explain these results, but in the pre-seminar I would like to explore other contexts where filtered deformations arise, such as quantisation, automorphisms of geometric structures,… -
MAXIMALS(preseminar): José Figueroa-O'Farrill (Edinburgh)- Filtered deformations in Algebra, Geometry and Physics
6th December 2016, 2:10pm to 3:00pm JCMB 6206 -- Show/hide abstractAbstract: In these talks I attempt to intercontextualise recent results obtained in collaboration with Andrea Santi on what could be termed an “Erlangen Programme for Supergravity”. (You don’t need to know what supergravity is to understand this talk.) We recently realised that an object I introduced many years ago — a Lie superalgebra with a geometric origin — has a precise algebraic structure that suggests a means of classifying certain geometries of interest. This is reminiscent of Klein’s Erlangen Programme: to study a geometry via its group of automorphisms. The algebraic structure in question is that of a filtered deformation of a Z-graded Lie superalgebra. In the main seminar I would like to explain these results, but in the pre-seminar I would like to explore other contexts where filtered deformations arise, such as quantisation, automorphisms of geometric structures,… -
MAXIMALS: Mehdi Aaghabali (Edinburgh) - "Left algebraic division rings".
29th November 2016, 4:10pm to 5:00pm JCMB 1501 -- Show/hide abstractAbstract: In recent years there has been renewed interest in the construction of finitely generated algebraic division algebras that are not finite-dimensional. This is division ring version of Kurosh Problem. There are results as local versions of Kurosh problem, for example a theorem due to Jacobson asserts: every division ring whose elements are algebraic of bounded degree over its center, is centrally finite. Recently, this result has been generalized by Jason Bell et al to left algebraic division rings over not necessarily central subfields. Using combinatorics of words, in this seminar we show the statement holds for division rings whose commutators are left algebraic over not necessarily central subfields. -
MAXIMALS (preseminar): Mehdi Aaghabali (Edinburgh) - "Commutators contain important information about division rings".
29th November 2016, 3:10pm to 4:00pm JCMB 1501 -- Show/hide abstractAbstract: In this talk I am going to give a brief historic overview about the origin of commutators in group theory. Then I will pass to division rings and will show how one can obtain essential information about the structure of a division ring in terms of commutators and structures generated by commutators. Also, you will find generalization of some classic results due to Jacobson, Kaplansky and Noether about division rings. -
MAXIMALS: Kevin De Laet (Antwerp)- Quantum algebras with an action of a finite group
15th November 2016, 4:10pm to 5:00pm JCMB 1601 -- Show/hide abstractAbstract: Consider a positively graded, connected algebra A, finitely generated in degree 1, for example the polynomial ring of global dimension n. Assume that there exists some reductive group G acting on A as gradation preserving algebra automorphisms, then each degree k part decomposes as a finite sum of simple G-modules. Then a natural question is: do there exist other graded algebras B such that 1) G acts on B, with the action preserving the gradation and 2) the degree k parts of A and B are isomorphic as G-modules for each natural number k ? As one may suspect, this depends greatly on G and A itself. Some constructions and the motivating example of the 3-dimensional Sklyanin algebras will be discussed. For this example, if time permits, I will show that some additional information about these algebras can be gained by this construction. -
MAXIMALS (preseminar): Kevin De Laet (Antwerp)- A short introduction to noncommutative projective geometry
15th November 2016, 3:10pm to 4:00pm JCMB 1501 -- Show/hide abstractAbstract: A quick introduction to noncommutative projective geometry in the style of Artin-Tate-Van den Bergh. In this field, one studies noncommutative graded algebras with 'similar' properties to the commutative polynomial ring. Such properties can be either of homological or algebraic nature. I will talk about the classification of AS-regular algebras and define the problem I've been working on in the context of this field. -
MAXIMALS: Dave Benson (Aberdeen) - Module categories for finite groups, finite group schemes, and finite supergroup schemes.
8th November 2016, 4:10pm to 5:00pm JCMB 1501 -- Show/hide abstractAbstract: The starting point for this talk is Chouinard’s theorem, which states that a module for a finite group is projective if and only if its restriction to every elementary abelian p-subgroup is projective; and Dade’s lemma, which gives an easy test for whether a module for an elementary abelian group is projective. I shall talk about analogous results for finite group schemes and finite supergroup schemes, and their consequences for the structure of the stable module category. Parts of this talk represent joint work with Iyengar, Krause and Pevtsova. -
MAXIMALS(preseminar): Dave Benson (Aberdeen) - Module categories for finite groups, finite group schemes, and finite supergroup schemes.
8th November 2016, 3:10pm to 4:00pm JCMB 1501 -
Joint EMPG-MAXIMALS Seminar: Sanjaye Ramgoolam (Queen Mary) - Algebras, Invariants and Gauge-String Duality.
25th October 2016, 4:10pm to 5:00pm JCMB 1501 -- Show/hide abstractAbstract: Permutations subject to equivalences can be used to classify invariants of unitary group actions on polynomial functions of matrices and tensors. These equivalence classes are related to permutation centralizer algebras. One sequence of these algebras is closely related to Littlewood-Richardson coefficients. Structural properties of these algebras as well as Fourier transforms on them have applications in dualities between gauge theories and string theories. They yield results on the counting and correlators of multi-matrix invariants, relevant to the physics of super-symmetric states in the AdS/CFT correspondence. Combinatoric questions on the structure of these algebras are related to the complexity of spaces of super-symmetric states. -
Joint EMPG-MAXIMALS Seminar(preseminar): Sanjaye Ramgoolam (Queen Mary) - Algebras, Invariants and Gauge-String Duality.
25th October 2016, 3:10pm to 4:00pm JCMB 1501 -- Show/hide abstractAbstract: In the pre-seminar, I will give an overview of some aspects of conformal field theories and representation theory which play an important role in understanding the duality (the AdS/CFT correspondence) between strings in 10 dimensions and conformal field theories in 4 dimensions. This will include BPS states, matrix correlators, large N expansions and Schur-Weyl duality. -
MAXIMALS: David Jordan (University of Edinburgh) - A topological field theory in dimension four
18th October 2016, 4:10pm to 5:00pm JCMB 1501 -- Show/hide abstractAbstract: A character variety of a manifold X is a moduli space of representations of pi_1(X). It was shown by Atiyah-Bott and Goldman that character varieties of surfaces are naturally symplectic, and that character varieties of 3-manifolds define Lagrangian subvarieties in the character varieties of their boundary surfaces. In this talk, I'll explain that all this structure can be ``quantized", giving rise to a TFT which we call the quantum character TFT. We obtain in this way manifestly topological constructions of many gadgets traditionally thought of as living in the world of representation theory: quantum coordinate algebras, q-difference operator algebras, double affine Hecke algebras, etc. Quantizing the Lagrangians of different 3-manifolds gives a new approach to studying the representation theory of these objects: the quantization of a Lagrangian should be (roughly) a simple module for the quantization of the symplectic variety it lives on. The main technical ingredient in the construction and any computations with it is the Morita theory of tensor categories, as developed in the preseminar. This is joint work with Ben-Zvi, Brochier and Snyder. -
MAXIMALS (preseminar): David Jordan (University of Edinburgh) - Elementary representation theory, categorified
18th October 2016, 3:10pm to 4:00pm JCMB 1501 -- Show/hide abstractAbstract: "Categorification" means replacing vector spaces with categories, in an artful way. When we categorify the notion of a ring, and a module over it, we get the notions of a tensor category, and a module category over it. Examples of these are ubiquitous throughout representation theory and algebraic geometry. If you hand a representation theorist a ring, she will ask "what are its modules?". In this talk, I'll develop some tools which you can use in case someone ever hands you a tensor category, and asks what are its module categories? It turns out that the resulting "Morita theory" for tensor categories plays a crucial role in the next talk. -
MAXIMALS: Gwyn Bellamy (Glasgow)- Symplectic resolutions of quiver varieties.
11th October 2016, 4:10pm to 5:00pm JCMB 1501 -- Show/hide abstractAbstract: Quiver varieties, as introduced by Nakaijma, play a key role in representation theory. They give a very large class of symplectic singularities and, in many cases, their symplectic resolutions too. However, there seems to be no general criterion in the literature for when a quiver variety admits a symplectic resolution. In this talk I will give necessary and sufficient conditions for a quiver variety to admit a symplectic resolution. This result builds upon work of Crawley-Boevey and of Kaledin, Lehn and Sorger. The talk is based on joint work with T. Schedler. -
MAXIMALS(preseminar): Gwyn Bellamy (Glasgow)- Introduction to quiver varieties.
11th October 2016, 3:10pm to 4:00pm JCMB 1501 -- Show/hide abstractAbstract: This will be a short reminder on the basic properties of quiver varieties. As well as giving the basic definitions, I’ll explain how one computes their dimensions, when they are smooth etc. No prior knowledge will be assumed. -
MAXIMALS: Iordan Ganev (Vienna) - The wonderful compactification for quantum groups
4th October 2016, 4:10pm to 5:00pm JCMB 1501 -- Show/hide abstractAbstract: The wonderful compactification of a group encodes the asymptotics of matrix coefficients for the group and captures the rational degenerations of the group. In this talk, we will explain a construction of the wonderful compactification via the Vinberg semigroup which makes these properties explicit. We will then introduce quantum group versions of the Vinberg semigroup, the wonderful compactification, and the latter's stratification by G x G orbits. Our approach relies on a theory of noncommutative projective schemes, which we will review briefly. -
MAXIMALS(preseminar): Iordan Ganev (Vienna) - An introduction to the wonderful compactification
4th October 2016, 3:10pm to 4:00pm JCMB 1501 -- Show/hide abstractAbstract: The wonderful compactification of a group plays a crucial role in several areas of geometric representation theory and related fields. The aim of this talk is to give a construction of the wonderful compactification and explain how its rich structure links the geometry of the group to the geometry of its partial flag varieties. We will describe several examples in detail. The first part of the talk will be an overview of necessary background from the representation theory of complex reductive groups. -
MAXIMALS: Natalia Iyudu (University of Edinburgh)- Sklyanin algebras via Groebner bases
27th September 2016, 4:10pm to 5:00pm JCMB 1501 -- Show/hide abstractAbstract: I will discuss how questions on Sklyanin algebras can be solved using combinatorial techniques, namely, Groebner bases theory. Elements of homological algebra also feature in our proofs. We calculate Hilbert series, prove Koszulity, PBW, Calabi-Yau etc., depending on parameters of Sklyanin algebras. Similar methods are used for generalized Sklyanin algebras, and for other potential algebras. -
MAXIMALS (preseminar): Natalia Iyudu (University of Edinburgh)- Sklyanin algebras via Groebner bases
27th September 2016, 3:10pm to 4:00pm JCMB 1501 -- Show/hide abstractAbstract: TBA -
MAXIMALS: Alexey Petukhov (Manchester): Primitive ideals of U(sl(infinity))
7th September 2016, 3:00pm to 4:00pm 6206 JCMB -- Show/hide abstractAbstract: Infinite-dimensional representation theory of finite dimensional Lie algebras is a rich topic with many interesting results. One of the most beautiful pieces of this subject is a description of primitive and prime ideals of universal enveloping algebras of finite-dimensional Lie algebras, and this involves quite advanced algebraic, geometric, and combinatorial techniques. It is natural to generalize the classification of primitive and prime ideals to the setting of infinite-dimensional Lie algebras, and in my talk I will provide a description of primitive ideals of the universal enveloping algebra of sl(infinity). I hope that I will be able to explain in an understandable way algebraic and combinatorial aspects of this result. -
Sasha Shapiro (UC Berkeley/Toronto) - Cluster structure on quantum groups
19th July 2016, 3:00pm to 5:00pm JCMB 5327 -- Show/hide abstractAbstract: A quantum cluster (or quantum torus) is an algebra over C(q) with q-commuting generators. Various embeddings of quantum groups into quantum tori have been studied over the past twenty years in relation with modular doubles, quantum Gelfand-Kirillov conjecture, and construction of braided monoidal categories. In a recent paper by K. Hikami and R. Inoue, such an embedding of the quantum group U_q(sl_2) was used to relate the corresponding R-matrix with quantum cluster mutations and half-Dehn twists. I will discuss how to generalize the results of Hikami and Inoue to U_q(sl_n). The quantum group is embedded into the tensor square of the quantized categorification space of 3 flags and 3 lines in C^n, which were studied in detail in the works of V. Fock and A. Goncharov. I also plan to show how the conjugation by the R-matrix can be expressed via a sequence of cluster mutations. If time permits, I will outline a way to generalize the above construction to quantum groups of arbitrary finite type and discuss its applications to the representation theory. -
Noah Snyder (IU Bloomington) - The exceptional knot polynomial
12th July 2016, 3:00pm to 4:00pm JCMB 6311 -- Show/hide abstractAbstract: Many Lie algebras fit into discrete families like GL_n, O_n, Sp_n. By work of Brauer, Deligne and others, the corresponding planar algebras fit into continuous familes GL_t and OSp_t. A similar story holds for quantum groups, so we can speak of two parameter families (GL_t)_q and (OSp_t)_q. These planar algebras are the ones attached to the HOMFLY and Kauffman polynomials. There are a few remaining Lie algebras which don't fit into any of the classical families: G_2, F_4, E_6, E_7, and E_8. By work of Deligne, Vogel, and Cvitanovic, there is a conjectural 1-parameter continuous family of planar algebras which interpolates between these exceptional Lie algebras. Similarly to the classical families, there ought to be a 2-paramter family of planar algebras which introduces a variable q, and yields a new exceptional knotpolynomial. In joint work with Scott Morrison and Dylan Thurston, we give a skein theoretic description of what this knot polynomial would have to look like. In particular, we show that any braided tensor category whose box spaces have the appropriate dimension and which satisfies some mild assumptions must satisfy these exceptional skein relations. -
Sam Gunningham (UT Austin) - Induction and Restriction patterns in geometric representation theory
7th July 2016, 3:00pm to 5:00pm JCMB 5327 -- Show/hide abstractAbstract: In geometric representation theory, we typically study certain categories associated to a reductive group G (e.g. certain representations of G or g=Lie(G), D-modules on Bun_G(Curve), Character sheaves on G, ...). Often there are functors of parabolic induction and restriction going between the categories associated to G and to Levi subgroups L of G. I will explain how these functors allow us to break up our category into pieces, indexed by classes of cuspidal objects on Levi subgroup (an object is called cuspidal if it is not seen by induction from a proper Levi). We will see how things like Hecke algebras and (relative) Weyl groups naturally appear. Later, I will focus on the case of adjoint equivariant D-modules on the Lie algebra g, and indicate how this case may be generalized to other settings - mirabolic, quantum, elliptic... -
Andrea Appel (USC) - Quantum Groups, Monodromy, and Generalised Braided Categories
5th July 2016, 3:00pm to 5:00pm JCMB 5327 -- Show/hide abstractAbstract: Quantum groups play a prominent role in many branches of mathematics, from gauge theory and enumerative geometry, to knot theory and quantum computing. In many cases, this is due to their tight relation with the braid groups. More specifically, to the fact that a quantum group is in the first place a Hopf algebra whose representations carry a natural action of the braid group. In the first part of the talk, I will explain how the quantum groups are in fact analytic objects, describing the monodromy of certain systems of difeerential equations arising in Lie theory. I will first review the renowned Drinfeld-Kohno theorem, describing the monodromy of the Knizhnik-Zamolodchikov equations associated to a simple Lie algebra in terms of the universal R-matrix of the corresponding quantum group. I will then present an extension of this result, providing a descrip- tion of the monodromy of the Casimir equations associated to a simple Lie algebra (in fact, to any symmetrisable Kac-Moody algebra) in terms of the quantum Weyl group operators of the corresponding quantum group. The proof relies on the notion of generalised braided category (or quasi-Coxeter category), which is to a generalised braid group what a braided monoidal category is to the standard braid group on n strands. In the second part of the talk, I will explain how Tannaka{Krein dual- ity, quantization of Lie bialgebras, dynamical KZ equations, and Hochschild cohomology in the framework of appropriate PROP categories play a funda- mental role in the proof of the monodromy theorem. -
Maximals: Tom Braden (University of Massachusetts, Amherst) - Modular representation theory and hypertoric varieties.
28th June 2016, 3:10pm to 5:00pm JCMB 5327 -- Show/hide abstractAbstract: One of the earliest successes in geometric representation theory was Springer's construction of the irreducible representations of the symmetric group (or any Weyl group) in the top cohomology of fibers of a resolution of singularities of the nilpotent cone of GL(n) (or a reductive group). More recently there has been considerable progress extending these ideas to representations and sheaves with positive characteristic coefficients. Life is more complicated in positive characteristic: the category of representations is no longer semisimple, and on the geometric side this is reflected in the failure of some important geometric tools from characteristic 0 such as the decomposition theorem. In this talk I will describe work with Carl Mautner giving a picture similar to Springer theory where the role of the nilpotent cone is played by hypertoric varieties. We obtain representations of a new class of algebras which we call "Matroidal Schur algebras", which share many features with their classical cousins. In particular the categories are highest weight, and the categories for Gale dual hypertoric varieties are related by Ringel duality. In the second part of the talk I will explain some of the ideas in the proof and a conjectural geometric approach to proving that certain categories of perverse sheaves are highest weight. -
Noah Arbesfeld (Columbia University) - A geometric R-matrix for the Hilbert scheme of points on a surface
21st June 2016, 3:00pm to 5:00pm -- Show/hide abstractAbstract: We explain two ways in which geometry can be used to produce solutions of the Yang-Baxter equation. First, we introduce Maulik and Okounkov's "stable envelope" construction of R-matrices acting in the cohomology of a symplectic variety, and describe some of the geometric properties these R-matrices enjoy. Second, we produce an R-matrix from the Hilbert scheme of points on a general surface from an intertwiner of certain highest weight Virasoro modules; for the surface C^2, this construction is due to Maulik and Okounkov. We also explain how to modify this construction to produce formulas for multiplication by Chern classes of tautological bundles on the Hilbert scheme. -
Johanna Hennig (Alberta) -- Path algebras of quivers and representations of locally finite Lie algebras
15th June 2016, 3:00pm to 4:00pm 4325B -- Show/hide abstractAbstract: This is joint work with S. Sierra. We explore the (noncommutative) geometry of representations of locally finite Lie algebras. Let L be one of these Lie algebras, and let I ⊆ U(L) be the annihilator of a locally simple L-module. We show that for each such I, there is a quiver Q so that locally simple L-modules with annihilator I are parameterized by “points” in the “noncommutative space” corresponding to the path algebra of Q. We classify the quivers that occur and along the way discover a beautiful connection to characters of the symmetric groups S_n. -
Gus Schrader (UC Berkeley) - Geometric R-matrices, Yangians, and reflection equation algebras
15th June 2016, 11:00am to 1:00pm JCMB 5327 -- Show/hide abstractAbstract: I will describe the geometric R-matrix formalism, developed in the work of Maulik and Okounkov, that leads to the construction of a Hopf algebra $Y_Q$ acting on the equivariant cohomology of the Nakajima varieties associated to a quiver $Q$. In the first hour, I will give an introductory overview of the basic machinery of Nakajima varieties and their equivariant cohomology, which underpins the Maulik-Okounkov construction. In particular, we’ll illustrate the general definitions by focusing on the concrete example of cotangent bundles to Grassmannians. In the second hour, I’ll describe the stable basis construction in the equivariant cohomology of a symplectic variety, which is the key technical tool used to construct the geometric R-matrices. In our example of cotangent bundles to Grassmannians, we’ll see that this yields a geometric construction of the Yangian of $gl_2$. Finally, we will conclude by discussing some work in progress extending the Maulik-Okounkov construction to encompass Yangian coideal subalgebras, such as reflection equation algebras. -
Peter Samuelson (Iowa State --> University of Edinburgh) Kauffman bracket skein modules and double affine Hecke algebras
7th June 2016, 3:00pm to 5:00pm JCMB 4325B -- Show/hide abstractAbstract: First hour: The Witten-Reshetikhin-Turaev knot invariants are polynomials associated to a knot in S^3 constructed using representation theory of quantum groups. A concrete combinatorial construction of these invariants is given by the Kauffman bracket skein relations. In this talk we first discuss some of the background for these constructions. We then discuss how the skein relations are related to representation varieties, and to the Poisson structure on representation varieties of topological surfaces. Second hour: The double affine Hecke algebra is a noncommutative algebra depending on parameters q and t which is associated to a Lie algebra. The DAHA has been connected to various parts of math, including symmetric polynomials, integrable systems, Hilbert schemes, and more. Frohman and Gelca showed that the skein algebra of the torus is the t=q specialization of the sl_2 DAHA. We discuss this result (and some background), and then discuss a conjecture involving the DAHA and modules coming from knot complements. -
Maximals: Ulrich Thiel (Stuttgart)- Finite-dimensional graded algebras with triangular decomposition
24th May 2016, 3:10pm to 5:00pm JCMB 5327 -- Show/hide abstractAbstract: We study the representation theory of finite-dimensional graded algebras which admit a triangular decomposition similar to universal enveloping algebras of Lie algebras. Such a decomposition implies a rich combinatorial structure in the representation theory (this was discovered in this generality by Holmes and Nakano) and there are many examples like restricted quantized enveloping algebras at roots of unity, restricted rational Cherednik algebras, etc. We show that even though our algebras have in general infinite global dimension, the graded module category is in fact a highest weight category (with infinitely many simple objects, however). Under certain conditions we are able to establish a proper tilting theory in this category and use this to show that the degree zero part of the algebra is a cellular algebra. This is joint work with Gwyn Bellamy (Glasgow). -
MAXIMALS: Gwendolyn E. Barnes (Heriot-Watt) - Nonassociative geometry in representation categories of quasi-Hopf algebras
17th May 2016, 2:00pm to 3:00pm JCMB 4312 -- Show/hide abstractAbstract: The noncommutative and nonassociative algebra which arises in the description of the target space of non-geometric string theory fits naturally as a commutative and associative algebra object in a certain closed braided monoidal category, the representation category of a triangular quasi-Hopf algebra. In this talk I will show how exploring the syntax of category theory enables one to express notions of geometry on an algebra object in terms of universal constructions internal to the representation category of any triangular quasi-Hopf algebra. -
Maximals: Dmitry Gurevich - Quantum matrix algebras and braided Yangians
26th April 2016, 3:10pm to 5:00pm JCMB 6206 -- Show/hide abstractAbstract: By quantum matrix algebras I mean algebras related to quantum groups and close in a sense to that Mat(m). These algebras have numerous applications. In particular, by using them (more precisely, the so-called reflection equation algebras) we succeeded in defining partial derivatives on the enveloping algebras U(gl(m)). This enabled us to develop a new approach to Noncommutative Geometry: all objects of this type geometry are deformations of their classical counterparts. Also, with the help of the reflection equation algebras we introduced the notion of braided Yangians, which are natural generalizations of the usual ones and have many beautiful properties. -
Maximals: Stephane Gaussent (Saint-Etienne) - Hovels and Hecke algebras
22nd March 2016, 3:10pm to 5:00pm JCMB 5327 -- Show/hide abstractAbstract: A hovel is degraded building. It can be used to associate to a Kac-Moody group over a local field a bunch of Hecke algebras. First, I will explain the definition of the hovel which generalises the construction of the Bruhat-Tits building associated to a reductive group. Then I will present three kind of algebras that one can associate to the hovel: the spherical Hecke algebra, the Iwahori-Hecke algebra and the Bernstein-Lusztig-Hecke algebra. -
Maximals: Tatiana Gateva-Ivanova (American University in Bulgaria, Sofia)- Set-theoretic solutions of the Yang-Baxter equation and related algebraic objects
15th March 2016, 3:10pm to 5:00pm JCMB 5327 -- Show/hide abstractAbstract: Set-theoretic solutions of the Yang--Baxter equation form a meeting ground of mathematical physics, algebra and combinatorics. Such a solution consists of a set $X$ and a bijective map $r:X\times X\to X\times X$ which satisfies the braid relations. Associated to each set-theoretic solution are several algebraic constructions: the monoid $S(X, r)$, the group $G(X, r)$, the semigroup algebra $kS(X, r)$ over a field k, generated by X and with quadratic relations $xy = .r(x, y)$, a special permutation group $\mathcal{G}$ and a left brace $(G, +,.)$. In this talk I shall discuss some of the remarkable algebraic properties of these object. -
Maximals: Sergey Malev (Edinburgh) - The images of non-commutative polynomials evaluated on matrix algebras.
8th March 2016, 3:10pm to 5:00pm JCMB 5327 -- Show/hide abstractAbstract: Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set $sl_n(K)$ of matrices of trace $0$, or all of $M_n(K)$. We prove the conjecture for $K=\mathbb{R}$ or for quadratically closed field and $n=2$, and give a partial solution for an arbitrary field $K$. We also consider homogeneous and Lie polynomials and provide the classifications for the image sets in these cases. -
Maximals: Robert Marsh (Leeds) - Twists of Pluecker coordinates as dimer partition functions
1st March 2016, 3:30pm to 4:30pm JCMB 5327 -- Show/hide abstractAbstract: By a result of J. Scott, the homogeneous coordinate ring of the Grassmannian Gr(k,n) can be realised as a cluster algebra. The Pluecker coordinates of the Grassmannian are all cluster variables. I will talk about joint work with J. Scott. We introduce a twist map on the Grassmannian and show that it is related to a twist of Berenstein-Fomin-Zelevinsky and can be implemented by a maximal green sequence, up to frozen variables. We give Laurent expansions for twists of Pluecker coordinates as scaled dimer partition functions (matching polynomials) on weighted versions the plabic (planar bicoloured) graphs arising in the cluster structure. -
Maximals: Markus Reineke (Wuppertal) - "Geometry of Donaldson-Thomas invariants of quivers"
23rd February 2016, 3:10pm to 5:00pm JCMB 5327 -- Show/hide abstractAbstract: Motivic Donaldson-Thomas invariants of quivers were defined by M. Kontsevich and Y. Soibelman as a mathematical definition of string-theoretic BPS state counts. We discuss several results relating these invariants to the geometry of moduli spaces of quiver representations. -
MAXIMALS: Sian Fryer (Leeds) - There And Back Again: A Localization's Tale
17th February 2016, 1:10pm to 2:00pm JCMB 6206 -- Show/hide abstractAbstract: The prime spectrum of a quantum algebra has a finite stratification in terms of a set of distinguished primes called H-primes, and we can study these strata by passing to certain nice localizations of the algebra. H-primes are now starting to show up in some surprising new areas, including combinatorics (totally nonnegative matrices) and physics, and we can borrow techniques from these areas to answer questions about quantum algebras and their localizations. In particular, we can use Grassmann necklaces -- a purely combinatorial construction -- to study the topological structure of the prime spectrum of quantum matrices. -
Maximals: Emily Cliff (Oxford) - Factorisation spaces and examples from Hilbert schemes
16th February 2016, 3:10pm to 5:00pm JCMB 5327 -- Show/hide abstractAbstract: Factorisation algebras and equivalently chiral algebras are geometric versions of vertex algebras, introduced by Beilinson and Drinfeld. There is also a non-linear analogue of a factorisation algebra, called a factorisation space. I will define these objects, and furthermore show how we can use the Hilbert scheme of points of a smooth d dimensional variety X to construct examples. -
Maximals: Alberto Facchini (Padova) - Krull-Schmidt-Remak theorem, direct-sum decompositions of modules, direct-product decompositions of groups, G-groups
9th February 2016, 3:10pm to 5:00pm JCMB 5327 -- Show/hide abstractAbstract: I will begin presenting (part of) the history of the Krull-Schmidt-Remak Theorem. Then I will show a number of results concerning uniqueness of direct-sum decompositions of right modules over a ring R and uniqueness of direct-product decompositions of a group G. I will conclude giving some results about the category of G-groups, which is a category rather similar to the category Mod-R of right R-modules. Here a G-group is a group H on which G acts as a group of automorphisms. -
Maximals: CANCELLED
2nd February 2016, 3:10pm to 5:00pm JCMB 5327 -
Maximals: Brendan Nolan (Kent) - A generalised Dixmier-Moeglin equivalence for quantum Schubert cells.
26th January 2016, 3:10pm to 5:00pm JCMB 5327 -- Show/hide abstractAbstract: In the late 1970s and early 1980s, Dixmier and Moeglin gave algebraic and topological conditions for recognising the primitive ideals (namely the kernels of the irreducible representations) of the enveloping algebra of a finite-dimensional complex Lie algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, such algebras are said to satisfy the "Dixmier-Moeglin equivalence". Many interesting families of algebras, including many families of quantum algebras, have since been shown to satisfy this equivalence. I will outline work of Jason Bell, Stephane Launois, and myself, showing that in several families of quantum algebras, an arbitrary prime ideal is equally close (in a manner which I will make precise) to being primitive, rational, and locally closed. The family on which I shall focus is that of the quantum Schubert cells U_q [w]. For a simple complex Lie algebra g, a scalar q which is not a root of unity, and an element w of the Weyl group of g, U_q [w] is a subalgebra of U_q^+(g) constructed by De Concini, Kac, and Procesi; familiar examples include the algebras of quantum matrices. -
Maximals: Jacopo Gandini (Pisa) - On the set of orbits of a spherical subgroup on the flag variety
19th January 2016, 3:10pm to 5:00pm JCMB 5327 -- Show/hide abstractAbstract: Let G be a complex reductive group and B a Borel subgroup of G, a subgroup H of G is called spherical if it acts with finitely many orbits on the flag variety G/B. For example, if H coincides with B, then the orbits are parametrized by the Weyl group of G and the orbits are the Schubert cells. Even though spherical subgroups are classified combinatorially, the corresponding orbit decompositions of G/B are not yet understood in general. In this seminar I will consider two special cases, namely that of a solvable spherical subgroup and that of a symmetric subgroup of G corresponding to an involution of Hermitian type. In these cases, I will explain how to attach a root system to every H-orbit in G/B, and how these root systems allow to parametrize the H-orbits in G/B. These parametrizations are compatible with a general action of the Weyl group of G that Knop defined on the set of H-orbits in G/B, and I will explain how to recover the Weyl group action from the parametrization of the orbits. The talk is based on two joint works, respectively with Andrea Maffei and with Guido Pezzini. -
Maximals: Rishi Vyas (Ben-Gurion University) - A noncommutative Matlis-Greenlees-May equivalence
7th December 2015, 4:10pm to 5:00pm JCMB 4312 -- Show/hide abstractAbstract: Roughly speaking, an element s in a commutative ring A is said to be weakly proregular if every module over A can be reconstructed from its localisation at s considered along with its local cohomology at the ideal generated by s. This notion extends naturally to finite sequences of elements: a precise definition will be given during the talk. An ideal in a commutative ring is called weakly proregular if it has a weakly proregular generating set. In particular, every ideal in a commutative noetherian ring is weakly proregular. It turns out that weak proregularity is the appropriate context for the Matlis-Greenlees-May (MGM) equivalence: given a weakly proregular ideal I in a commutative ring A, there is an equivalence of triangulated categories (given in one direction by derived local cohomology and in the other by derived completion at I) between cohomologically I-torsion (i.e. complexes with I-torsion cohomology) and cohomologically I-complete complexes in the derived category of A. In this talk, we will give a categorical characterization of weak proregularity: this characterization then serves as the foundation for a noncommutative generalisation of this notion. As a consequence, we will arrive at a noncommutative variant of the MGM equivalence. This work is joint with Amnon Yekutieli. -
Maximals: Guido Pezzini (Erlangen) - Spherical subgroups of Kac-Moody groups
1st December 2015, 3:10pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: Spherical subgroups of finite type of a Kac-Moody group have been recently introduced, in the framework of a research project aimed at bringing the classical theory of spherical varieties to an infinite-dimensional setting. In the talk we discuss their definition and some of their properties. We introduce a combinatorial object associated with such a subgroup, its homogeneous spherical datum, which satisfies the same axioms as in the finite-dimensional case. -
Maximals: Antonio Sartori (Freiburg) - Link invariants of type A and categorification.
24th November 2015, 3:10pm to 5:00pm JCBM 6311 -- Show/hide abstractAbstract: We describe the finite dimensional representation category of gl(m|n) and of its quantized enveloping algebra using variations of Howe duality, and we review the Reshetikhin-Turaev construction of the corresponding link invariants of type A. We discuss then some results (and some open questions) on their categorification, in particular using the BGG category O. -
Maximals: Daniel Tubbenhauer (Bonn) - (Singular) TQFT’S, link homologies and Lie theory
17th November 2015, 3:10pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: In pioneering work, Khovanov introduced the so-called arc algebra. His arc algebra is a certain algebra built from cobordisms turning up in TQFT’s and is known to be related to Khovanov homologies, categorification of tensor products of sl(2) and to a certain 2-block parabolic of category O for gl(n) - as follows from work of many researchers in the last 15 years. Sadly, in Khovanov’s original construction, the functoriality of Khovanov homologies cannot be encoded directly nor is it clear how to generalize his construction to obtain relations to e.g. tensor products of sl(N) or N-block parabolics of category O for gl(n). For this purpose, one needs to modify his arc algebra by using singular TQFT’s instead of “usual” TQFT’s. In this talk I will explain Khovanov’s topologically and elementary, yet powerful, construction in details as well as its relations to categorification and category O. I will then sketch how to use singular TQFT’s to generalize the construction. -
Maximals: Michel Van den Bergh (Hasselt) - Resolutions of determinantal varieties.
16th November 2015, 1:10pm to 2:00pm JCMB 5327 -- Show/hide abstractAbstract: If X is a determinantal variety then there are a number of objects that may be regarded as "resolutions of singularities" of X: (1) the classical Springer resolution by a vector bundle over a Grassmannian, (2) a suitable quotient stack, (3) various non-commutative resolutions. In the talk we will discuss how these different resolutions are related. For ordinary determinal varieties this is joint work with Buchweitz and Leuschke. For determinantal varieties of symmetric and skew-symmetric matrices new phenomena occur due to the fact that the Springer resolution is no longer crepant. This is joint work with Špela Špenko. -
Maximals: Xin Fang (Cologne) - Linear degenerations of flag varieties
10th November 2015, 3:10pm to 5:00pm JCBM 6311 -- Show/hide abstractAbstract: Abstract: Flag varieties are fertile soil where it grows geometry, algebra and combinatorics. Motivated by the PBW filtration of Lie algebras, E. Feigin defined the degenerate flag varieties, which are flat degenerations of the corresponding flag varieties. The purpose of this talk is to introduce a new family of (flat) degenerations of flag varieties of type A, called linear degenerate flag varieties, by classifying flat degenerations of a particular quiver Grassmannian. The geometry of these degenerations will also be presented. This talk is based on a joint work (in progress) with G. Cerulli Irelli (Rome), E. Feigin (Moscow), G. Fourier (Glasgow) and M. Reineke (Wuppertal). -
Maximals: Michael Ehrig (Bonn) - How to think of the good old Brauer algebra?
3rd November 2015, 3:10pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: In this talk I will discuss how categorification methods can be used to obtain a graded version of the Brauer algebra, especially in non semi-simple cases. This will involve the category O for certain orthogonal Lie algebras and their parabolic and graded analogues. The categorifications involved in this process are a generalised version of the ones introduced by Rouquier and Khovanov-Lauda in the sense that the categorified object is not a Kac-Moody algebra or quantum group, but a so-called quantum symmetric pair. This is joint work with Catharina Stroppel. -
Hodge seminar: Sarah Zerbes (UCL) - Euler systems and the conjecture of Birch and Swinnerton-Dyer
2nd November 2015, 1:10pm to 2:00pm JCMB 5327 -- Show/hide abstractAbstract: The Birch—Swinnerton-Dyer conjecture is one of the most mysterious open problems in number theory, predicting a relation between arithmetic objects, such as the points on an elliptic curve, and certain complex-analytic functions. A powerful approach to the conjecture is via a tool called an ‘Euler system’. I will explain the idea behing this approach, and some recent new results in this direction. -
Maximals: Martina Lanini (Edinburgh) - Multiplicity formulae and moment graph modules.
27th October 2015, 3:10pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: The natural problem of determining characters of simple objects in suitable representation categories can be rephrased in terms of multiplicities of irreducible modules in standard objects. In this talk, I will focus on the case in which these multiplicities are governed (or expected to be governed) by certain combinatorial families of polynomials, and explain how moment graph techniques can be used to approach such a problem. -
Maximals: Tara Brendle (Glasgow) - Combinatorial models for mapping class groups
20th October 2015, 3:10pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: The mapping class group Mod(S) of a surface S appears in a variety of contexts, for example, as a natural analogue both of arithmetic groups and of automorphism groups of free groups, and as the (orbifold) fundamental group of the moduli space of Riemann surfaces. However, its subgroup structure is not at all well understood. In this talk we will discuss a certain rigidity displayed by a wide class of subgroups of Mod(S): any normal subgroup of Mod(S) that contains a "small" element has Mod(S) as its group of automorphisms. This result is proved using a resolution of a metaconjecture posed by N. Ivanov stating that every stating that every "sufficiently rich" complex associated to S has Mod(S) as its group of automorphisms. (This is joint work with Dan Margalit.) -
Maximals: Deke Zhao (Beijing Normal University at Zhuhai) - Hattori-Stallings traces
13th October 2015, 3:10pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: We review the definition of Hattori-Stallings traces of projective modules and their relation to Morita equivalence. As an application, we will discuss Berest-Etingof-Ginzburg's work on Morita equivalence of rational Cherednik algebras of type A. -
Maximals: Milen Yakimov (LSU) - Cluster structures on open Richardson varieties and their quantizations
6th October 2015, 3:10pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: Open Richardson varieties are the intersections of opposite Schubert cells in full flag varieties. They play a key role in Schubert calculus, total positivity and cluster algebras. We will show how to realize the quantized coordinate ring of each open Richardson variety as a normal localization of a prime factor of a quantum Schubert cell algebra. Using a combination of ring theoretic and representation theoretic methods, we will produce large families of toric frames for all quantum Richardson varieties. This has applications to cluster algebras and to the construction of a Dixmier type map from the symplectic foliation of each Schubert cell to the primitive spectrum of the corresponding quantum Schubert cell algebra. This is a joint work with Tom Lenagan (University of Edinburgh). -
Maximals: Wolfgang Soergel (Freiburg) - Graded Representation Theory and Motives
29th September 2015, 3:10pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: I will discuss what a graded version of a category is and why graded versions of categories of representations are interesting to study. I want to discuss how recent advances in the theory of motives help to better understand these graded versions. -
Maximals: Claire Amiot (Grenoble) - Derived invariants for surface algebras
17th September 2015, 2:00pm to 3:00pm JCMB 6206 -- Show/hide abstractAbstract: In this talk I will explain a joint work with Y. Grimeland. Surface algebras are algebras of global dimension 2 associated to an unpunctured surface $S$ with an admissible cut. It is possible to associate to each such algebra an invariant in an affine space of $H^1(S,\mathbb Z)$ up to an action of the mapping class group which determines the derived equivalence class of the algebra. The proof uses strongly the graded mutation introduced in a joint work with S. Oppermann. This invariant is closely linked with the Avella-Alaminos-Geiss invariant for gentle algebras, and gives some information on the AR structure of the corresponding derived category. -
GEARS: Claire Amiot (Institut Fourier, Grenoble): Cluster categories for algebras of global dimension 2 and cluster-tilting theory
15th September 2015, 3:00pm to 5:30pm ICMS -- Show/hide abstractAbstract: Abstract: In this talk I will present basic results of cluster-tilting theory developed in [Iyama Yoshino 2008: Mutation in triangulated categories and Rigid CM modules] and [Buan-Iyama-Reiten-Scott 2009: Cluster structures for 2-Calabi-Yau categories]. I will explain how these results were a motivation for generalising the construction of cluster categories. I will first recall the motivation and definition of the acyclic cluster category due to Buan Marsh Reineke Reiten Todorov in 2006, and then focus to the construction of the generalised cluster category associated with algebras of global dimension 2 [Amiot 09]. Then I will explain how cluster-tilting theory can apply in classical tilting theory via graded mutation in a joint work with Oppermann. -
Maximals: Monica Vazirani (UC Davis) - Representations of the affine BMW algebra
14th September 2015, 4:00pm to 5:00pm JCMB 4325B -- Show/hide abstractAbstract: The BMW algebra is a deformation of the Brauer algebra, and has the Hecke algebra of type A as a quotient. Its specializations play a role in types B, C, D akin to that of the symmetric group in Schur-Weyl duality. One can enlarge these algebras by a commutative subalgebra $X$ to anaffine, or annular, version. Unlike the affine Hecke algebra, the affine BMW algebra is not of finite rank as a right $X$-module, so induction functors are ill-behaved, and many of the classical Hecke-theoretic constructions of simple modules fail. However, the affine BMW algebra still has a nice class of $X$-semisimple, or calibrated, representations, tha t don't necessarily factor through the affine Hecke algebra. I will discuss Walker's TQFT-motivated 2-handle construction of the $X$-semisimple, or calibrated, representations of the affine BMW algebra. While the construction is topological, the resulting representation has a straightforward combinatorial description. This is joint work with Kevin Walker. -
ARTIN 45
11th September 2015, 1:00pm to 1:00pm JCMB 6206 -- Show/hide abstractAbstract: The 45th ARTIN meeting will take place at the University of Edinburgh on the 11th and 12th of September 2015. All talks will be in the James Clerk Maxwell Building, room 6206. The theme of the meeting is noncommutative ring theory, with an emphasis on noncommutative algebraic geometry. See http://hodge.maths.ed.ac.uk/tiki/ARTIN-45 for more details -
Maximals: Theodore Voronov (Manchester) - Microformal geometry
2nd June 2015, 2:00pm to 3:00pm JCMB 4325B -- Show/hide abstractAbstract: Abstract: In search of L-infinity mappings between homotopy Poisson algebras of functions, we discovered a construction of a certain nonlinear analog of pullbacks. Underlying such "nonlinear pullbacks" there are formal categories that are "formal thickening" of the usual category of smooth maps of manifolds (or supermanifolds). Morphisms in these categories are called ''thick morphisms'' (or microformal morphisms). They are defined by formal canonical relations between the (anti)cotangent bundles of the manifolds. Thick morphisms have beautiful properties. For example, we can define the adjoint of a nonlinear morphism of vector bundles, as a thick morphism of the dual bundles, which reduces to the ordinary adjoint in the linear case. In the talk, I will explain the construction of microformal morphisms and nonlinear pullbacks, and their applications to homotopy Poisson structures, vector bundles and L-infinity algebroids. (See preprints: http://arxiv.org/abs/1409.6475 and http://arxiv.org/abs/1411.6720.) -
Maximals: Charlie Beil (Bristol) - Smooth noncommutative blowups of dimer algebras and isolated nonnoetherian singularities
26th May 2015, 2:00pm to 3:00pm 6206 JCMB -- Show/hide abstractAbstract: Abstract: A nonnoetherian singularity may be viewed geometrically as an algebraic variety with positive dimensional `smeared-out' points. I will describe how isolated nonnoetherian singularities admit noncommutative blowups which are smooth, in a suitable geometric sense. Furthermore, I will describe how a class of isolated nonnoetherian noncommutative singularities, namely non-cancellative dimer algebras, also admit noncommutative blowups which are smooth. -
Alissa Crans (Loyola Marymount University) - Crossed modules of racks
14th May 2015, 2:00pm to 3:00pm JCMB 6311 -- Show/hide abstractAbstract: Abstract: A rack is a set equipped with two binary operations satisfying axioms that capture the essential properties of group conjugation and algebraically encode two of the three Reidemeister moves. We will begin by generalizing Whitehead's notion of a crossed module of groups to that of a crossed module of racks. Motivated by the relationship between crossed modules of groups and strict 2-groups, we then will investigate connections between our rack crossed modules and categorified structures including strict 2-racks and trunk-like objects in the category of racks. We will conclude by considering topological applications, such as fundamental racks. This is joint work with Friedrich Wagemann. -
Arkady Vaintrob (University of Oregon) - Cohomological field theories related to singularities and matrix factorizations
2nd April 2015, 2:00pm to 3:00pm -- Show/hide abstractAbstract: Abstract: I will discuss a cohomological field theory associated to a quasihomogeneous isolated singularity W with a group G of its diagonal symmetries (a Landau-Ginzburg A-model, in physical parlance). The state space of this theory is the equivariant Milnor ring of W and the corresponding invariants can be viewed as analogs of the Gromov-Witten invariants for the non-commutative space associated with the pair (W,G). In the case of simple singularities of type A they control the intersection theory on the moduli space of higher spin curves. The construction is based on derived categories of (equivariant) matrix factorizations of singularities with the role of the virtual fundamental class from the Gromov-Witten theory played by a "fundamental matrix factorization" over a certain moduli space. -
Maximals: Alexey Petukhov (University of Manchester) - Two-sided ideals of U(sl(oo))
31st March 2015, 3:00pm to 4:00pm JCMB 4325B -- Show/hide abstractAbstract: Abstract: The idea behind this talk is to describe two-sided ideals of U(g) for an infinite dimensional Lie algebra g using known classification of prime two-sided ideals of U(g') for finite dimensional Lie subalgebras g' of g. In particular, if g is semisimple, by analogy with a finite dimensional case, one may expect that all primitive two-sided ideals are annihilators of highest weight modules. To start with we focus on infinite-dimensional Lie algebra sl(\infty). We will see that the annihilators in U(sl(\infty)) of most highest weight sl(\infty)-modules equal (0) and explicitly describe all highest weights for which this annihilator is not (0). We also prove that in the latter case the annihilator is an integrable ideal and provide a classification of such ideals. The proof will use the classification of two-sided ideals of U(sl(n)) (and thus a little bit of Young diagrams and Robinson–Schensted algorithm). Title/Abstract for the second part: Title: On ideals in the enveloping algebra of a locally simple Lie algebra Abstract: Let g be a Lie algebra with universal enveloping algebra U(g). To a two-sided ideal I of U(g) one can canonically assign a Poisson ideal gr I in S(g). It turns out that very frequently S(g) has no non-trivial Poisson ideals (and I hope I give some idea why it is so). As a consequence very frequently U(g) has no non-trivial two-sided ideals. As a final result I will provide some quite explicit description of countable dimensional locally simple Lie algebras g such that U(g) affords a non-trivial two-sided ideal. -
Maximals: Michael McBreen (IPFL Lausanne) - Mirror Symmetry for Hypertoric Varieties
17th March 2015, 3:00pm to 5:00pm JCMB 4325B -- Show/hide abstractAbstract: I will discuss work (in progress) with Ben Webster on homological mirror symmetry for hypertoric varieties. Hypertoric varieties are a family of noncompact algebraic symplectic spaces associated to hyperplane arrangements; we show how the quantization of such spaces in finite characteristic has a natural description on the mirror side. -
Misha Feigin (Glasgow) - On PBW subalgebras of Cherednik algerbras
10th March 2015, 3:00pm to 5:00pm JCMB 4325B -- Show/hide abstractAbstract: Abstract: I am going to discuss two subalgebras in the rational Cherednik algebra associated with a Coxeter group. These subalgebras satisfy Poincare-Birkhoff-Witt property and they are given by quadratic relations. They deform semidirect product of quotients of universal enveloping algebras of so(n) and gl(n) with the Coxeter group algebra, and they are related to quantisation of functions on the Grassmanian of two-planes and on the space of matrices of rank at most 1 respectively. The centres of these subalgebras give quantum Hamiltonians related to Calogero-Moser integrable systems which I plan to discuss as well. This is based on joint work with T. Hakobyan. -
Maximals: Raf Bocklandt (Amsterdam)
3rd March 2015, 3:00pm to 5:00pm -
Maximals: Christian Korff (Glasgow) - Yang-Baxter algebras in quantum cohomology
24th February 2015, 3:00pm to 5:00pm JCMB 4325B -- Show/hide abstractAbstract: Starting from solutions of the Yang-Baxter equation we construct a noncommutative bi-algebra which can be described in purely combinatorial terms using non-intersecting lattice paths. Inside this noncommutative algebra we identify a commutative subalgebra, called the Bethe algebra, which we identify with the direct sum of the equivariant quantum cohomology rings of the Grassmannian. We relate our construction to results of Peterson which describe the quantum cohomology rings in terms of Kostant and Kumar's nil Hecke ring and the homology of the affine Grassmannian. This is joint work with Vassily Gorbounov, Aberdeen. -
Pavel Safronov (Oxford) - Quantization of Hamiltonian reduction
19th February 2015, 3:00pm to 5:00pm JCMB 4325A -- Show/hide abstractAbstract: I will explain how (quasi-)Hamiltonian reduction fits into the framework of derived symplectic geometry. (Quasi-) Hamiltonian spaces are interpreted as Lagrangians in shifted symplectic stacks and the reduction corresponds to Lagrangian intersection. This gives a new perspective on deformation quantization of Hamiltonian spaces. This could also be used to make sense of deformation quantization of quasi-Hamiltonian spaces. -
Maximals: Jack Jeffries (University of Utah) - How many invariants are needed to separate orbits?
3rd February 2015, 3:00pm to 5:00pm JCMB 4325B -- Show/hide abstractAbstract: Abstract: The study of separating invariants is a relatively recent trend in invariant theory. For a finite group acting linearly ona vector space, a separating set is a set of invariants whose elements separate the orbits of the action. In some ways, separating sets often exhibit better behavior than generating sets for the ring of invariants. We investigate the leastpossible cardinality of a separating set for a given action. Our main result is a lower bound which generalizes the classical result of Serre that if the ring of invariants is polynomial, then the group action must be generated by pseudoreflections. We find these bounds to be sharp in a wide range of examples. This is based on joint work with Emilie Dufresne. -
Maximals: Cesar Lecoutre (Kent) - A Poisson Gelfand-Kirillov problem in positive characteristic
27th January 2015, 3:00pm to 5:00pm JCMB 4325B -- Show/hide abstractAbstract: We study a problem of birational equivalence for polynomial Poisson algebras over a field of arbitrary characteristic. More precisely, the quadratic Poisson Gelfand-Kirillov problem asks whether the field of fractions of a given polynomial Poisson algebra is isomorphic (as a Poisson algebra) to a Poisson affine field, that is the field of fractions of a polynomial algebra (in several variables) where the Poisson bracket of two generators is equal to their product (up to a scalar). We answer positively this question for a large class of polynomial Poisson algebras and their Poisson prime quotients. For instance this class includes Poisson determinantals varieties. -
Maximals: Robert Laugwitz (Oxford) - Braided Drinfeld and Heisenberg doubles and TQFTs with defects
20th January 2015, 3:00pm to 5:00pm JCMB 4325B -- Show/hide abstractAbstract: Abstract: A uniform categorical description for both the Drinfeld center and a Heisenberg analogue called the Hopf center of a monoidal category (relative to a braided monoidal category) is presented using morphism categories of bimodules. From this categorical definition, one obtains a categorical action as well as a definition of braided Drinfeld and Heisenberg doubles via braided reconstruction theory. In examples, this categorical picture can be used to obtain a categorical action of modules over quantum enveloping algebras on modules over quantum Weyl algebras. Moreover, certain braided Drinfeld doubles give such an action on modules over rational Cherednik algebras using embeddings of Bazlov and Berenstein of these algebras into certain braided Heisenberg doubles which can be thought of as versions of the Dunkl embeddings. We argue that the corresponding braided Drinfeld doubles can serve an quantum group analogues in the setting of complex reflection groups. Finally, the categorical description can be extended naturally to give TQFTs with defects using recent work of Fuchs-Schaumann-Schweigert. -
Maximals: Richard Hepworth (Aberdeen) - A homology theory for graphs
13th January 2015, 3:00pm to 5:00pm JCMB - 4325B -- Show/hide abstractAbstract: Tom Leinster recently introduced an invariant of graphs called the magnitude. In this talk I will define a homology theory for graphs that categorifies the magnitude, in the sense that the magnitude of a graph can be recovered from its homology by taking the Euler characteristic. (Thus this is categorification in the same sense that Khovanov homology categorifies the Jones polynomial.) Important properties of the magnitude can then be seen as shadows of properties of homology. For example, magnitude satisfies an inclusion exclusion formula that can be recovered from a Mayer-Vietoris theorem in homology. The talk will (hopefully) be accessible for anybody who knows what graphs and chain complexes are, and I will try to illustrate it with lots of pictures and examples. -
MAXIMALS: Bin Shu (East China Normal)
2nd December 2014, 4:10pm to 6:10pm JCMB 6311 -- Show/hide abstractAbstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS -
MAXIMALS: Martin Kalck (Edinburgh): Spherical subcategories and new invariants for triangulated categories
25th November 2014, 3:00pm to 5:00pm JCMB 6311 -
MAXIMALS: Rupert Yu (Reims)
18th November 2014, 4:00pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS -
MAXIMALS: David Pauksztello (Manchester)
18th November 2014, 3:00pm to 4:00pm JCMB 6311 -- Show/hide abstractAbstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS -
MAXIMALS: Arend Bayer (Edinburgh)
11th November 2014, 4:00pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS -
Karel Casteels (Kent)
4th November 2014, 3:00pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS -
Christian Lomp and Paula Carvalho
28th October 2014, 3:00pm to 5:00pm JCMB 4319a -
Evgeny Feigin
30th September 2014, 3:00pm to 5:00pm JCMB Lecture Theatre A -
NBSAN meeting
21st July 2014, 1:00pm to 5:30pm Appleton 2.14 -- Show/hide abstractAbstract: http://www.ma.hw.ac.uk/~ndg/nbsan.html -
Andre LeRoy (University of Artois)
27th May 2014, 2:30pm to 3:30pm -- Show/hide abstractAbstract: http://hodge.maths.ed.ac.uk/tiki/MAXIMALS -
Maria Chlouveraki (Versailles)
27th May 2014, 1:00pm to 2:00pm -- Show/hide abstractAbstract:- Noah Snyder (IU Bloomington)
16th May 2014, 1:00pm to 3:00pm JCMB 6206- Carl Mautner (MPIM Bonn)
13th May 2014, 3:00pm to 4:00pm JCMB 6311- Dorette Pronk (Dalhousie)
22nd April 2014, 3:00pm to 5:00pm- MAXIMALS: Alexey Sevastyanov (Aberdeen)
8th April 2014, 3:00pm to 5:00pm -- Show/hide abstractAbstract: Click here for title and abstract.- MAXIMALS: Elisa Gorla (Basel)
1st April 2014, 3:00pm to 5:00pm -- Show/hide abstractAbstract: Click here for title and abstract.- Claudio Sibilia (ETH Zurich)
25th March 2014, 4:00pm to 5:00pm- MAXIMALS: Iordan Ganev (UT Austin)
25th March 2014, 3:00pm to 4:00pm- Hodge seminar: Ben Webster (U Virginia/Paris 6)
18th March 2014, 3:00pm to 5:00pm- MAXIMALS: Felipe Rincon (Warwick)
4th March 2014, 3:00pm to 5:00pm -- Show/hide abstractAbstract: Click here for title and abstract.- MAXIMALS: Johanna Hennig (UCSD)
25th February 2014, 3:00pm to 5:00pm -- Show/hide abstractAbstract: Click here for title and abstract.- Erez Sheiner (Bar Ilan)
14th February 2014, 12:00pm to 1:00pm -- Show/hide abstractAbstract: See the maximals webpage for detalis.- MAXIMALS: Greg Ginot (Paris 6)
11th February 2014, 3:00pm to 5:00pm -- Show/hide abstractAbstract: Click here for title and abstract.- MAXIMALS: Sian Fryer (Manchester)
4th February 2014, 3:00pm to 5:00pm -- Show/hide abstractAbstract: Click here for title and abstract.- MAXIMALS: Michael Wemyss (Edinburgh)
28th January 2014, 4:00pm to 5:00pm -- Show/hide abstractAbstract: Click here for title and abstract.- MAXIMALS: Jean-Marie Bois (Kiel)
21st January 2014, 3:00pm to 5:00pm -- Show/hide abstractAbstract: Click here for title and abstract.- MAXIMALS: David Gepner (Regensburg)
7th January 2014, 3:00pm to 5:00pm -- Show/hide abstractAbstract: Click here for title and abstract.- Ghislain Fourier (Glasgow)
6th December 2013, 4:00pm to 5:00pm- Anne Thomas (Glasgow)
6th December 2013, 3:00pm to 4:00pm- Ines Henriques (Sheffield)
26th November 2013, 4:00pm to 5:00pm- Emily Dufresne (Basel)
26th November 2013, 3:00pm to 4:00pm- Tom Lenagan (Edinburgh)
19th November 2013, 3:00pm to 5:00pm- David Jordan (Sheffield)
13th November 2013, 4:00pm to 5:00pm- David Evans (Cardiff)
28th October 2013, 3:00pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: For title and abstract, click here- Chris Dodd (Toronto)
25th October 2013, 4:00pm to 5:00pm JCMB 4312 -- Show/hide abstract- Chris Dodd (Toronto)
24th October 2013, 4:00pm to 5:00pm JCMB 4312 -- Show/hide abstract- Chris Dodd (Toronto)
23rd October 2013, 4:00pm to 5:00pm JCMB 4312 -- Show/hide abstract- Chris Dodd (Toronto)
23rd October 2013, 2:00pm to 3:00pm JCMB 6311 -- Show/hide abstract- Nick Gurski (Sheffield)
22nd October 2013, 3:00pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: Click here for title and abstract.- Joseph Chuang (City University London)
15th October 2013, 3:00pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: For title and abstract, click here- Beeri Grenfeld (Bar Ilan)
8th October 2013, 4:00pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: Click here for title and abstract.- Uzi Vishne (Bar Ilan)
8th October 2013, 3:00pm to 4:00pm JCMB 6311 -- Show/hide abstractAbstract: Click here for title and abstract.- Natalia Iyudu (Edinburgh)
1st October 2013, 4:00pm to 5:00pm JCMB 6311- David Andrew Jordan (Edinburgh)
1st October 2013, 3:00pm to 4:00pm JCMB 6311- Gwyn Bellamy (Glasgow)
24th September 2013, 3:00pm to 5:00pm JCMB 6311- Stefan Kolb (Newcastle)
17th September 2013, 3:00pm to 5:00pm JCMB 6311- Francois Petit (Edinburgh)
9th September 2013, 5:00pm to 6:00pm -- Show/hide abstractAbstract: Click here for title and abstract.- Hendrik Suess (Edinburgh)
9th September 2013, 4:00pm to 5:00pm -- Show/hide abstractAbstract: Click here for title and abstract.- Qendrim Gashi (Pristina, Kosovo) - Maximals
4th September 2013, 3:00pm to 5:00pm TBA -- Show/hide abstractAbstract: Click here for title and abstract.- Vassily Gorbounov (Aberdeen) - joint with Geometry&Topology: Quantum cohomology and quantum groups
19th March 2013, 4:00pm to 5:00pm 6311 JCMB -- Show/hide abstractAbstract: We describe the recent research started by Nekrasov Shatashvilly Braverman Maulik Okounkov on correspondence between the quantum cohomology of the quiver varieties and the quantum integrable systems. Our main example will be the cotangent spaces to partial flag varieties.- MAXIMALS: Julien Bichon (Clermont-Ferrand): Hochschild homology of Hopf algebras and free Yetter-Drinfeld modules
12th March 2013, 4:00pm to 5:00pm 6311 JCMB -- Show/hide abstractAbstract: We explain how one can relate the Hochschild (co)homologies of Hopf algebras having equivalent tensor categories of comodules, in case the trivial module over one of the Hopf algebras admits a resolution by free Yetter-Drinfeld modules. This general procedure is applied to the quantum group of a bilinear form, for which generalizations of results by Collins, Hartel and Thom in the orthogonal case are obtained. It also will be shown that the Gerstenhaber-Schack cohomology of a cosemisimple Hopf algebra completely determines its Hochschild cohomology. Basic facts and definitions about Hopf algebras will be recalled first.- MAXIMALS: Alexander Engström (Aalto University, Helsinki, Finland): Powers of ideals
26th February 2013, 4:00pm to 5:00pm 6311 JCMB -- Show/hide abstractAbstract: I will discuss properties of resolutions of powers of ideals. This will be done in the framework of Betti diagrams and their polyhedral structure. At the end a conjecture regarding monomial ideals will be stated.- MAXIMALS: Joseph Grant (Leeds): Derived autoequivalences and braid relations
19th February 2013, 4:00pm to 5:00pm 6311 JCMB -- Show/hide abstractAbstract: I will talk about symmetries of derived categories of symmetric algebras, known as spherical twists, and explain when they satisfy braid relations. Using a more general collection of symmetries, known as periodic twists, I will explain how lifts of longest elements of symmetric groups to braid groups act on the derived category. This was first described in a special case by Rouquier and Zimmermann and, if time permits, I hope to present a new proof of their result.- MAXIMALS: Zoe O'Connor (Heriot-Watt): multiple Conjugacy Search Problem in Limit Groups
12th February 2013, 4:00pm to 5:00pm 6311 JCMB -- Show/hide abstractAbstract: The multiple CSP is the following: Given two conjugate lists of elements A=[a_1,a_2,...,a_n] and B=[b_1,b_2,...,b_n], can we find a conjugator x such that x^{-1}a_{i}x=b_i ? We show that one can put a linear upper bound on the geodesic length of a shortest conjugator.- MAXIMALS: Damien Calaque (ETH Zurich): Lie theory of closed embeddings
5th February 2013, 4:00pm to 5:00pm ICMS (15 South College Street) -- Show/hide abstractAbstract: I will explain how some nice Lie structures appear when one is trying to compute Ext and Tor of a closed subvariety. I will use it as an excuse to introduce some nice concepts of derived geometry. I'll end the talk with a striking analogy between two great results: one in Lie theory and the other one in algebraic geometry.- MAXIMALS: Charlie Beil
10th December 2012, 4:00pm to 5:00pm 6206 JCMB -- Show/hide abstractAbstract: Title: Morita equivalences from Higgsing toric superpotential algebras Abstract: Let A and A' be superpotential algebras of brane tiling quivers, with A' cancellative and A non-cancellative, and suppose A' is obtained from A by contracting, or 'Higgsing', a set of arrows to vertices while preserving a certain associated commutative ring. A' is then a Calabi-Yau algebra and a noncommutative crepant resolution of its prime noetherian center, whereas A is not a finitely generated module over its center, often not even PI, and its center is not noetherian and often not prime. I will present certain Morita equivalences that relate the representation theory of A with that of A'. I will also describe the Azumaya locus of A, and relate it to the Azumaya locus of A'. Along the way, I will introduce the notion of a non-local algebraic variety, and show how this notion is intimately related to these algebras.- Kobi Kremnitzer (Oxford)
4th December 2012, 4:30pm to 5:30pm ICMS (15 South College Street) -- Show/hide abstractAbstract: Title: Beilinson-Drinfeld factorization algebras and QFT. Abstract: I will explain what are factorization algebras and how they can be defined in very general settings of geometries with a good notion of D-modules. I will then talk about applications of this theory. In particular I will discuss the differentiable case and its relations to quantum field theory.- Uli Kraehmer (Glasgow)
4th December 2012, 3:00pm to 4:00pm ICMS -- Show/hide abstractAbstract: Dirac Operators on Quantised Hermitian Symmetric Spaces In this joint work with Matthew Tucker-Simmons (U Berkeley) the \bar\partial-complex of the quantised compact Hermitian symmetric spaces is identified with the Koszul complexes of the quantised symmetric algebras of Berenstein and Zwicknagl. This leads for example to an explicit construction of the relevant quantised Clifford algebras. The talk will be fairly self-contained and begin with three micro courses covering the necessary classical background (one on Dirac operators, one on symmetric spaces, one on Koszul algebras), and then I'll explain how noncommutative geometry and quantum group theory lead to the problems that we are dealing with in this project.- ARTIN meeting: Wendy Lowen (Antwerp)
1st December 2012, 12:00pm to 1:00pm JCMB, Lecture Theatre C -- Show/hide abstractAbstract: Wendy Lowen (Antwerp) On compact generation of deformed schemes. We discuss a theorem which allows to prove compact generation of derived categories of Grothendieck categories, based upon certain coverings by localizations. This theorem follows from an application of Rouquier's cocovering theorem in the triangulated context, and it implies Neeman's Result on compact generation of quasi-compact separated schemes. We give an application of our theorem to non-commutative deformations of such schemes.- ARTIN meeting: Michele D'Adderio (University Libre de Bruxelles)
1st December 2012, 11:00am to 12:00pm JCMB, Lecture Theatre C -- Show/hide abstractAbstract: Michele D'Adderio (University Libre de Bruxelles) A geometric theory of algebras. I will introduce some classical notions of geometric group theory (like growth and amenability) in the setting of associative algebras, and I will show how they interact with other classical invariants (like the Gelfand-Kirillov dimension and the lower transcendence degree).- ARTIN meeting: Oleg Chalykh (Leeds)
1st December 2012, 9:30am to 10:30am JCMB, Lecture Theatre C -- Show/hide abstractAbstract: Oleg Chalykh (Leeds) Calogero-Moser spaces for algebraic curves. I will discuss two existing definitions of Calogero-Moser spaces for curves: one in terms of Cherednik algebras, another - in terms of deformed preprojective algebras, the link between them, and explain how one can compute geometric invariants of these spaces, such as the Euler characteristic and Deligne-Hodge polynomial.- ARTIN meeting: Adrien Brochier (Edinburgh)
30th November 2012, 4:30pm to 5:30pm JCMB, Lecture Theatre B -- Show/hide abstractAbstract: Adrien Brochier (Edinburgh) On finite type invariants for knots in the solid torus. Finite type knot invariants are those invariants vanishing on the nth piece of some natural filtration on the space of knots. This notion was introduced by Vassiliev and it turns out that most of known numerical invariants are of finite type. Kontsevich proved the existence of a "universal" invariant, taking its values in some combinatorial space, of which every finite type invariant is a specialization. This result involves some complicated integrals, but can be made combinatorial using the theory of Drinfeld associators. We will review this construction and explain why the naive generalization of this theory for knot in thickened surfaces fails. We will suggest a general way of overcoming this obstruction, and prove an analog of Kontsevich theorem in this framework for the case M=C^*, i.e. for knots in a solid torus. Time permitting, we will give an explicit construction of specializations of our invariant using quantum groups.- ARTIN meeting: Stephane Launois (Kent)
30th November 2012, 3:30pm to 4:30pm JCMB, Lecture Theatre B -- Show/hide abstractAbstract: Stephane Launois (Kent) Efficient recognition of totally nonnegative cells. In this talk, I will explain how one can use tools develop to study the prime spectrum of quantum matrices in order to study totally nonnegative matrices.- Vladimir Bavula (Sheffield)
20th November 2012, 4:30pm to 5:30pm ICMS (15 South College Street) -- Show/hide abstractAbstract: Title: Characterizations of left orders in left Artinian rings. Abstract: Small (1966), Robson (1967), Tachikawa (1971) and Hajarnavis (1972) have given different criteria for a ring to have a left Artinian left quotient ring. In my talk, three more new criteria are given.- Joe Karmazyn (Edinburgh): Quivers with superpotentials and their deformations
20th November 2012, 3:00pm to 4:00pm ICMS, 15 South College St -- Show/hide abstractAbstract: Path algebras with relations constructed from a superpotential were studied by Bocklandt, Schedler and Wemyss in 'Superpotentials and Higher Order Derivations'. I consider when deformations of these algebras have relations given by an inhomogenous superpotential. This encompasses many interesting examples, such as deformed preprojective algebras and symplectic reflection algebras.- MAXIMALS: Tom Leinster (Edinburgh): The eventual image
13th November 2012, 4:10pm to 5:00pm 6206 JCMB -- Show/hide abstractAbstract: An endomorphism T of an object can be viewed as a discrete-time dynamical system: perform one iteration of T with every tick of the clock. This dynamical viewpoint suggests questions about the long-term destiny of the points of our object. (For example, does every point eventually settle into a periodic cycle?) A fundamental concept here is the "eventual image". Under suitable hypotheses, it can be defined as the intersection of the images of all the iterates T^n of T. I will explain its behaviour in three settings: one set-theoretic, one algebraic, and one geometric. I will then present a unifying categorical framework, using it to explain how the concept of eventual image is a cousin of the concepts of spectrum and trace.- MAXIMALS:Martina Balagovic (York)
6th November 2012, 4:30pm to 5:30pm ICMS (15 South College Street) -- Show/hide abstractAbstract: Title: Chevalley restriction theorem for vector-valued functions on quantum groups Abstract: For a simple finite dimensional Lie algebra g, its Cartan subalgebra h and its Weyl group W, the classical Chevalley theorem states that, by restricting ad-invariant polynomials on g to its Cartan subalgebra, one obtains all W-invariant polynomials on h, and the resulting restriction map is an isomorphism. I will explain how to generalize this statement to the case when a Lie algebra is replaced by a quantum group, and the target space of the polynomial maps is replaced by a finite dimensional representation of this quantum group. I will describe all prerequisites for stating the theorem and sketch the idea of the proof, most notably the notion of dynamical Weyl group introduced by Etingof and Varchenko.- MAXIMALS:Spela Spenko (Ljubljana)
23rd October 2012, 4:30pm to 5:30pm ICMS (15 South College Street) -- Show/hide abstractAbstract: Lie superhomomorphisms of superalgebras. The relationship between the associative and Lie structure of an associative algebra was studied by Herstein and some of his students in the 1950's and 1960's. After some initial partial results the complete classification of Lie isomorphisms was obtained by Matej Bre\vsar in 1993. It now seems natural to continue the investigation of Lie homomorphisms in the setting of superalgebras. Let $A=A_0\oplus A_1$ be an associative superalgebra over a field $F$ of characteristic not $2$. By replacing the product in $A$ by the superbracket $[\cdot,\cdot]_s$, $A$ becomes a Lie superalgebra. Recall that $[\cdot,\cdot]_s$ is defined for homogeneous elements $a,b\in A$ as $[a,b]_s=ab-(-1)^{|a||b|}ba$. A bijective linear map $\phi:A\to A$ is a Lie superautomorphism of $A$ if $\phi(A_i)=A_i$, $i\in \mathbb{Z}_2$, and $\phi([a,b]_s)=[\phi(a),\phi(b)]_s$ for all $a,b\in A$. We will present a characterization of Lie superautomorphisms of simple associative superalgebras, obtained in a joint work with Yuri Bahturin and Matej Bre\vsar.- Maximals: Gwyn Bellamy (Glasgow)
9th October 2012, 4:30pm to 5:30pm ICMS (15 South College Street) -- Show/hide abstractAbstract: Title: Rational Cherednik algebras and Schubert cells Abstract: I will recall the connection between rational Cherednik algebras, the Calogero-Moser space and the adelic Grassmannian. Then I will try to explain how one can interoperate Schubert cells, and conjecturally their intersection, in the Grassmannian in terms of the representation theory of the rational Cherednik algebra.- MAXIMALS: Christian Ballot (Caen): The story of a congruence
25th September 2012, 4:00pm to 5:00pm 5215 JCMB -- Show/hide abstractAbstract: We will tell some of the story around a classical elementary congruence due to Wolstenholme (1862) that deal with prime numbers. Like for the classical Fermat and Wilson congruences various generalizations were soon discovered. Surprisingly, yet another generalization was discovered only very recently. It involves Lucas sequences, which are a generalization of the Fibonacci numbers.- Maximals: Osamu Iyama (Nagoya)
19th September 2012, 4:00pm to 5:00pm 6311 JCMB -- Show/hide abstractAbstract: Title: $\tau$-tilting theory Abstract: Mutation is a basic operation in tilting theory, which covers reflection for quiver representations, APR tilting modules and Okuyama-Rickard construction of tilting complexes. In this talk we introduce the notion of (support) $\tau$-tilting modules, which `completes' tilting modules from viewpoint of mutation in the sense that any indecomposable summand of a support $\tau$-tilting module can be replaced in a unique way to get a new support $\tau$-tilting module. Moreover, for any finite dimensional algebra we show that there exist bijections between (1) support $\tau$-tilting modules, (2) functorially finite torsion classes, and (3) two-term silting complexes. Moreover if the algebra comes from a 2-Calabi-Yau triangulated category, (4) cluster-tilting objects also correspond bijectively. This is a joint work with Takahide Adachi and Idun Reiten.- Maximals: Stephen Harrap (York)
21st May 2012, 4:00pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: Title: 'The mixed Littlewood Conjecture'.- Maximals: Tom Bridgeland
28th March 2012, 4:10pm to 5:10pm JCMB 5215 -- Show/hide abstractAbstract: Title : Quadratic differentials as stability conditions Abstract : This talk is about how spaces of quadratic differentials on Riemann surfaces arise as stability conditions on certain CY3 categories. These categories are defined by quivers with potential but can also be viewed (heuristically?) as Fukaya categories of symplectic manifolds. I will try to explain what all this means, and give the main idea of the construction. This is joint work with Ivan Smith, inspired by a paper of physicists Gaiotto, Moore and Neitzke.- Maximals: Osamu Iyama (Nagoya)
20th March 2012, 4:00pm to 5:00pm JCMB 5215 -- Show/hide abstractAbstract: Title: n-representation infinite algebras Abstract: We introduce a distinguished class of finite dimensional algebras of global dimension n which we call n-representation infinite. For the case n=1, they are path algebras of non-Dynkin quivers. Taking (n+1)-preprojective algebras, they correspond bijectively with (n+1)-Calabi-Yau algebras of Gorenstein parameter 1. I will discuss 3 important classes of modules, preprojective, preinjective and regular as an analogue of the classical case n=1. This is a joint work with Martin Herschend and Steffen Oppermann.- Maximals: James Mitchell (St Andrews)
13th March 2012, 4:30pm to 5:30pm ICMS -- Show/hide abstractAbstract: Title: The lattice of subsemigroups of the semigroup of all mappings on an infinite set- Maximals: Christian Korff
13th March 2012, 3:00pm to 4:00pm ICMS -- Show/hide abstractAbstract: Speaker: Christian Korff Title: Quantum cohomology via vicious and osculating walkers Abstract: We relate the counting of rational curves intersecting Schubert varieties of the Grassmannian to the counting of certain non-intersecting lattice paths on the cylinder. These lattice paths can be described in terms of the combinatorial R-matrix of Kirillov-Reshetikhin crystal graphs. Crystal graphs are special directed, coloured graphs which are combinatorial objects encoding the representation theory of quantum algebras. Using the path description we identify the quantum Kostka numbers of Bertram, Ciocan-Fontanine and Fulton with the cardinality of a special subset of vertices in these graphs. Speaker: James Mitchell Title: The lattice of subsemigroups of the semigroup of all mappings on an infinite set Abstract: In this talk I will review some recent results relating to the lattice of subgroups of the symmetric group and its semigroup theoretic counterpart the lattice of subsemigroups of the full transformation semigroup on an infinite set. As might be expected, these lattices are extremely complicated. I will discuss several results that make this comment more precise, and shed light on the maximal proper sub(semi)groups in the lattice. I will also discuss a natural related partial order, introduced by Bergman and Shelah, which is obtained by restricting the type of sub(semi)groups and considering classes of, rather than individual, (semi)groups. In the case of the symmetric group, this order is very simple but in the case of the full transformation semigroup it is again very complex.- Maximals: Harry Braden (Edinburgh)
6th March 2012, 4:00pm to 5:00pm James Clerk Maxwell Building 5215 -- Show/hide abstractAbstract: Title: "Spectral Curves ans Number Theory" Abstract: The modern approach to integrable systems typically proceeds via a curve, the parameters of the curve encoding the actions and its Jacobian (or possibly some related Prym) encoding the angles. Physically relevant families of curves are often described by fixed relations amongst differentials on the curve. We shall look at number theoretic properties of these curves. For many integrable systems the curves are transcendental. I shall review W\"ustholz's Analytic subgroup theorem giving simple examples before applying this in the spectral curve context.- Maximals: Toby Stafford (Manchester)
28th February 2012, 4:30pm to 5:30pm ICMS, Swanston Room -- Show/hide abstractAbstract: Title: "Classifying Noncommutative surfaces: Subalgebras of the Sklyanin algebra" Abstract: Noncommutative projective algebraic geometry aims to use the techniques and intuition of (commutative) algebraic geometry to study noncommutative algebras and related categories. A very useful intuition here is that (the category of coherent sheaves over) a noncommutative projective scheme is simply the category of finitely generated graded modules modulo those of finite length over a graded algebra R. One of the major open problems here is to classify the noncommutative projective irreducible surfaces aka noncommutative graded domains of Gelfand-Kirillov dimension three. After surveying some of the known result on this question I will describe some very recent work of Rogalksi, Sierra and myself describing the subalgebras of the Sklyanin algebra.- Maximals: Colva Roney-Dougal (St Andrews)
28th February 2012, 3:30pm to 4:30pm ICMS, Swanston Room -- Show/hide abstractAbstract: Title: Generation of Finite Groups- Maximals: Ali Craw (Glasgow)
21st February 2012, 4:00pm to 5:00pm James Clerk Maxwell Building 6311 -- Show/hide abstractAbstract: Title: On the categorification of Reid's recipe Abstract: For a finite abeilan subgroup G of SL(3,C), Reid's recipe is a combinatorial cookery that describes very simply the relations between tautological line bundles on the G-Hilbert scheme. Building on results of Cautis-Logvinenko, I'll describe joint work that reveals the importance of this cookery for the derived category of the G-Hilbert scheme.- Maximals: Sue Sierra (Edinburgh)
14th February 2012, 4:30pm to 5:30pm ICMS -- Show/hide abstractAbstract: Title: A family of 4-dimensional algebras Abstract: We construct an interesting family of algebras of global dimension and GK-dimension 4, and show that the general member of this family is noetherian and birational to P2 (in the appropriate sense). Such algebras were conjectured not to exist by Rogalski and Stafford. We show also that these algebras have counterintuitive homological properties: in particular, the Auslander-Buchsbaum formula fails for them. This is joint work with Rogalski.- Maximals: John Mackay (Oxford)
14th February 2012, 3:00pm to 4:00pm ICMS -- Show/hide abstractAbstract: Title: "Geometry of random groups"- Maximals: Jean-Baptiste Gramain (Aberdeen)
31st January 2012, 4:30pm to 5:30pm ICMS -- Show/hide abstractAbstract: Title: Height zero characters of covering groups. Abstract: The characters of height 0 of finite groups are the object of numerous theorems and conjectures. If G is a finite group, and p a prime, we write Irr_0(G) for the set of characters of p-height 0 of G. The Alperin-Mckay Conjecture states that, if B is a p-block of G with defect group D, and with Brauer correspondent b in N_G(D), then |Irr_0(B)|=|Irr_0(b)|. In 2002, Isaacs and Navarro formulated a refinement of this conjecture. For any integer 0 < k < p, we denote by M_k(B) the set of height 0 characters of B whose degree has a p'-part congruent to ± k modulo p. The Isaacs-Navarro Conjecture then states that |M_{ck}(B)|=|M_k(b)|, where c is the p'-part of the index of N_G(D) in G. In this talk, I want to present (an idea of) the proof of this result in the Schur extensions of the symmetric and alternating groups. As in the symmetric groups, it is in this case possible to exhibit an explicit bijection, by using the combinatorics that describes the characters and blocks. I also show how these groups fit within the frame of a recent conjecture by Malle and Navarro on nilpotent blocks. Finally, I want to conclude with some related results about the combinatorics we use, in particular about hooks in partitions and bars in bar-partitions.- Maximals: Collin Bleak (St Andrews)
31st January 2012, 3:00pm to 4:00pm ICMS -- Show/hide abstractAbstract: Title: "Automorphisms of generalized R. Thompson groups via dynamics"- Maximals: Chris Smyth (Edinburgh)
24th January 2012, 4:00pm to 5:00pm James Clerk Maxwell Building 5215 -- Show/hide abstractAbstract: Title: "Conjugacy of algebraic numbers with rational parameters" Abstract: "We consider algebraic numbers having either rational real part, rational imaginary part or rational modulus, and discuss the question of whether such numbers can share their minimal polynomial. To answer this question, we apply some Galois theory and group theory." This work is joint with Karl Dilcher and Rob Noble.- Maximals: Chris Spencer (Edinburgh)
17th January 2012, 4:00pm to 5:00pm James Clerk Maxwell Building 5215 -- Show/hide abstractAbstract: Title: "Harish-Chandra bimodules of rational Cherednik algebras" Abstract: "Harish-Chandra bimodules are a class of bimodules defined for rational Cherednik algebras that have attracted much recent research interest. In this talk, I will attempt to explain some of the motivation behind this interest and then move on to present some results regarding Harish-Chandra bimodules of rational Cherednik algebras, with particular emphasis on the case of cyclic groups."- Maximals: Kiriko Kato (Osaka Furitsu)
29th November 2011, 4:00pm to 5:00pm James Clerk Maxwell Building 6311 -- Show/hide abstractAbstract: Title: Symmetric Auslander and Bass categories Abstract: We define the symmetric Auslander category $\sA^{\s}(R)$ to consist of complexes of projective modules whose left- and right-tails are equal to the left- and right-tails of totally acyclic complexes of projective modules. The symmetric Auslander category contains $\sA(R)$, the ordinary Auslander category. It is well known that $\sA(R)$ is intimately related to Gorenstein projective modules, and our main result is that $\sA^{\s}(R)$ is similarly related to what can reasonably be called Gorenstein projective homomorphisms. Namely, there is an equivalence of triangulated categories \[ \underline{\GMor}(R) \stackrel{\simeq}{\rightarrow} \sA^{\s}(R) / \sK^{\bounded}(\Prj\,R) \] where $\underline{\GMor}(R)$ is the stable category of Gorenstein projective objects in the abelian category $\Morph(R)$ of homomorphisms of $R$-modules. This result is set in the wider context of a theory for $\sA^{\s}(R)$ and $\sB^{\s}(R)$, the symmetric Bass category which is defined dually. This is joint work with Peter Jorgensen.- Maximals: Alexander Premet (Manchester)
22nd November 2011, 4:00pm to 5:00pm James Clerk Maxwell Building 6311 -- Show/hide abstractAbstract: Title: On 1-dimensional representations of finite W-algebras. Abstract: 1-dimensional representations of finite W-algebras enable one to construct completely prime primitive ideals with a prescribed associated variety and quantise coadjoint nilpotent orbits. A few years ago I conjectured that all finite W-algebras admits such representations. In my talk I am going to discuss the current status of this conjecture.- Jean-Eric Pin, ICMS
21st November 2011, 3:00pm to 4:00pm -- Show/hide abstractAbstract: The abstract notion of recognition: algebra, logic and topology (Joint work with M. Gehrke and S. Grigorieff) We propose a new approach to the notion of recognition, which departs from the classical definitions by three specific features. First, it does not rely on automata. Secondly, it applies to any Boolean algebra (BA) of subsets rather than to individual subsets. Thirdly, topology is the key ingredient. We prove the existence of a minimum recognizer in a very general setting which applies in particular to any BA of subsets of a discrete space. Our main results show that this minimum recognizer is a uniform space whose completion is the dual of the original BA in Stone-Priestley duality; in the case of a BA of languages closed under quotients, this completion, called the syntactic space of the BA, is a compact monoid if and only if all the languages of the BA are regular. For regular languages, one recovers the notions of a syntactic monoid and of a free profinite monoid. For nonregular languages, the syntactic space is no longer a monoid but is still a compact space. Further, we give an equational characterization of BA of languages closed under quotients, which extends the known results on regular languages to nonregular languages. Finally, we generalize all these results from BAs to lattices, in which case the appropriate structures are partially ordered.- Maximals: Guillaume Pouchin (Edinburgh)
15th November 2011, 4:00pm to 5:00pm Room 6311, James Clerk Maxwell Building, Edinburgh. -- Show/hide abstractAbstract: Title: Higgs algebra of weighted projective lines and loop crystals. Abstract: In this talk we contruct enveloping algebras of loop Lie algebras via geometry, considering constructible functions on the space of Higgs bundles on a weighted projective line. The geometry of this space then leads to nice elements in the algebra, which forms a basis called the semicanonical basis. Another interested feature coming from geometry is the construction of a loop crystal, which is an analog of a crystal in the loop case.- Maximals: Maria Chlouveraki (Edinburgh)
1st November 2011, 4:00pm to 5:00pm James Clerk Maxwell Building 6311 -- Show/hide abstractAbstract: Title: Modular representation theory of the Ariki-Koike algebra in characteristic 0. Abstract: The Ariki-Koike algebra is a natural generalisation of the Iwahori-Hecke algebras of types A and B. Much of its representation theory is controlled by the Schur elements, which are Laurent polynomials attached to its irreducible representations. We will give a new, pretty formula for these elements, and study the applications of our result to the representation theory of the Ariki-Koike algebra in characteristic 0- Maximals: Gwyn Bellamy (Manchester)
25th October 2011, 4:00pm to 5:00pm James Clerk Maxwell Building 5326 -- Show/hide abstractAbstract: Title: Rational Cherednik algebras in positive characteristic. Abstract: In this talk I will describe some of the basic features of rational Cherednik algebras in positive characteristic. There is a close relationship between the representation theory of these algebras and the geometry of their centres. I will show how their representation theory can be used to determine when the centre of the algebra is a regular ring. This is based on joint work with M. Martino.- Maximals: Tomoyuki Arakawa (RIMS, Kyoto)
18th October 2011, 4:00pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: Title: Affine W-algebras Abstract: Affine W-algebras may be considered as a generalization of infinite dimensional Lie algebras such as Kac-Moody algebras and the Virasoro algebra. They may be also regarded as an affinization of finite W-algebras, but affine W-algebras are introduced earlier than finite W-algebras in physics literature. Affine W-algebras are related with conformal field theories, integrable systems, quantum groups, the geometric Langlands program, and 4 dimensional gauge theories. In my talk I will discuss about their structure and describe their representation theory by focusing on type A cases.- Maximals: Michael Collins (Oxford)
18th October 2011, 2:30pm to 3:30pm JCMB 6311 -- Show/hide abstractAbstract: Title: Finite Subgroups of the Classical Groups Abstract: A theorem of Jordan (1878) states that there is a function f on the natural numbers such that if G is a finite subgroup of GL(n,C), then G has an abelian normal subgroup of index at most f(n). Several years ago, I determined the optimal value for f(n), and I will talk about this and recent work that extends the result to the finite subgroups of all classical groups, both real and complex.- Maximals: Liam o'Carroll
11th October 2011, 4:00pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: Title: "J. Sally's question and a conjecture by Y. Shimoda" Abstract: In 2007, Y. Shimoda, in connection with an open question of J. Sally from 1978, conjectured that a Noetherian local ring, such that all its prime ideals different from the maximal ideal are complete intersections, has Krull dimension at most two. This talk surveys the results that have been obtained to date concerning this conjecture. First we indicate that we can reduce to the case of dimension three, and that the conjecture has a positive answer if the ring is either regular, or is complete with infinite residue field and multiplicity at most three. Finally we consider the case of the appropriate analogue of the conjecture for standard graded rings, and indicate how a mix of algebraic and geometrical methods yields a definite answer in this setting. (Joint work with S. Goto and F. Planas-Vilanova)- Maximals: Tom Lenagan
4th October 2011, 4:00pm to 5:00pm JCMB 6311 -- Show/hide abstractAbstract: Title: Totally nonnegative matrices Abstract: A real matrix is totally nonnegative if each of its minors is nonnegative (and is totally positive if each minor is positive). The talk will survey elementary properties of these matrices and present new results which have surprising links with the theory of quantum matrices.- Maximals: Alvaro Nolla de Celis
27th September 2011, 4:00pm to 5:00pm JCMB 5326 -- Show/hide abstractAbstract: Title: Flops and mutations of polyhedral singularities Abstract: Let G be a finite subgroup of SO(3) and consider the so called polyhedral singularity C^3/G. It is well known that the G-Hilb is a distinguished crepant resolution which plays a central role in the so called McKay correspondence. I will explain in the talk how every crepant resolution of C^3/G is a moduli space of quiver representations showing that there exists a 1-to-1 correspondence between between flops of G-Hilb and mutations of the McKay quiver. This is a joint work with Y. Sekiya.- Maximals: Hokuto Uehara (Tokyo Met)
23rd August 2011, 4:00pm to 5:00pm 6311 JCMB -- Show/hide abstractAbstract: Title: Fourier--Mukai partners of elliptic surfaces. Abstract: For smooth projective varieties X and Y, when their derived categories are equivalent we say that X is a Fourier--Mukai partners of Y. We study the set of isomorphism classes of Fourier--Mukai partners of elliptic surfaces with negative Kodaira dimensions.- Maximals: Vikraman Balaji
31st May 2011, 4:10pm to 5:10pm JCMB 4312 -- Show/hide abstractAbstract: Title: "Parahoric torsors and Parabolic bundles on compact Riemann surfaces and representations of Fuchsian groups." Abstract: Let X be an irreducible smooth projective algebraic curve of genus g ≥ 2 over the ground field of complex numbers and let G be an arbitrary semisimple simply connected algebraic group. The aim of the talk is to introduce the notion of a semistable and stable parahoric torsor under certain Bruhat-Tits group schemes and construct the moduli space of semistable parahoric G –torsors and identify the underlying topological space of this moduli space with spaces of homomorphisms of Fuchsian groups into a maximal compact subgroup of G. The results give a complete generalization of the earlier results of Mehta and Seshadri on parabolic vector bundles. The talk is on a joint work with C.S. Seshadri.- Maximals: Alexander Young
13th May 2011, 3:00pm to 4:00pm JCMB 6206 -- Show/hide abstractAbstract: Title: "Slow but not too Slow: Nil Algebras and Growth"- MAXIMALs: Jan Grabowski (Oxford)
3rd May 2011, 4:10pm to 5:00pm JCMB 4312 -- Show/hide abstractAbstract: Title: Some quantum analogues of properties of Grassmannians Abstract: The classical coordinate ring of the Grassmannian has many nice structural properties and one expects these to carry over to its quantum analogue. We will discuss two properties for which this does indeed happen, namely a cluster algebra structure (recent work with Launois, quantizing work of Scott) and an action of the dihedral group (work with Allman, extending a recent construction of Launois and Lenagan). We will also mention an extension in a different direction, namely to infinite Grassmannians (work with Gratz).- in room S01 Colin Maclaurin Bldg, HW University Lisa Frenkel will speak on `Regular sets and counting in free groups'
16th March 2011, 4:15pm to 5:15pm room S01 Colin Maclaurin Bldg, HW University Lisa Frenkel will speak on `Regular sets and counting in free groups'- Maximals: Sue Sierra
8th March 2011, 4:10pm to 5:10pm JCMB 6311 -- Show/hide abstractAbstract: Title: Canonical birationally commutative factors of noetherian graded algebras Abstract: It is known that if a graded k- algebra R is strongly noetherian (that is, it remains noetherian upon commutative base-change), then there is a canonical map from R to a twisted homogeneous coordinate ring on some projective scheme. We show this can be generalized to algebras that are merely noetherian, and the resulting factor satisfies a universal property. Further, we show that under suitable conditions on the geometry of the Hilbert schemes of point modules over R, this canonical factor is a naive blowup algebra, in the sense of Keeler-Rogalski-Stafford.- Maximals: Yann Palu (Leeds)
15th February 2011, 4:10pm to 5:10pm JCMB 6311 -- Show/hide abstractAbstract: Title: Mutation of rigid objects and partial triangulations. Abstract: By several results (due to Amiot, Fomin--Shapiro--Thurston, Labardini-Fragoso and Keller--Yang) a cluster category can be associated with any compact Riemann surface with boundaries and marked points. The triangulations of the marked Riemann surface correspond to the so-called cluster-tilting objects of the cluster category. These objects are of particular interest since they categorify the clusters of Fomin--Zelevinsky's cluster algebras. In particular, they have a nice theory of mutation. This mutation turns out to be the categorical analogue of the flip of triangulations. Brustle--Zhang proved that some more general objects, the rigid objects, categorify the partial triangulations of the surface. In this talk, based on a joint paper with Robert Marsh, I will explain how both flips and mutations can be generalised to this situation. Our main tool is a result showing that Iyama--Yoshino reduction for cluster categories correspond to cutting along an arc the associated Riemann surface. All statements and results will be illustrated with some (small) geometric examples.- Maximals/Geometry: Sinan Unver (Koc)
10th February 2011, 4:10pm to 5:10pm JCMB 5326 -- Show/hide abstractAbstract: Title: Additive polylogarithms Abstract: In this talk we will define additive polylogarithms and describe how they are related to motivic cohomology over the dual numbers of a field of characteristic zero. In the characteristic p case, and in weight 2, we will also describe how the additive dilogarithm is related to Kontsevich's logarithm.- MAXIMALs: Stefan Kolb (Newcastle)
1st February 2011, 4:10pm to 5:10pm 6311 JCMB -- Show/hide abstractAbstract: Title: Braid group actions on quantum symmetric pair coideal subalgebras Abstract: It was noted recently by Molev and Ragoucy, and idependently by Chekhov, that the nonstandard quantum enveloping algebra of so(N) allows an action of the Artin braid group. We interpret and generalize this action within the theory of quantum symmetric spaces.- MAXIMALs: Natalia Iyudu (Bonn)
18th January 2011, 4:10pm to 5:10pm JCMB 5327 -- Show/hide abstractAbstract: Title: "Quadratic algebras: the Anick conjecture on Hilbert series, Koszulity, NCCI and RCI" Abstract: We present several results on the Anick conjecture which asserts that the lower bound for the Hilbert series, known as the Golod-Shafarevich estimate is attained on generic quadratic algebra. The technique (due to Anick), allowing to write down precisely the formula for the Hilbert series will be demonstrated. We will discuss also related questions of Koszulity and being noncommutative complete intersection (NCCI). Connections to the latter property on the level of finite dimensional representations, namely, introduced by Ginsburg and Etingof notion of representational complete intersection (RCI) will be considered and some examples given.- MAXIMALs: Vladimir Bavula (Sheffield)
(Open in Google Calendar)
13th January 2011, 4:10pm to 5:10pm JCMB 6311 -- Show/hide abstractAbstract: Title: "An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators''Talks in past semesters
Apologies, the calendar for Spring 2015 is temporarily down.Tuesday 20 January 3pm Robert Laugwitz (Oxford) Braided Drinfeld and Heisenberg doubles and TQFTs with defects A uniform categorical description for both the Drinfeld center and a Heisenberg analogue called the Hopf center of a monoidal category (relative to a braided monoidal category) is presented using morphism categories of bimodules. From this categorical definition, one obtains a categorical action as well as a definition of braided Drinfeld and Heisenberg doubles via braided reconstruction theory.
In examples, this categorical picture can be used to obtain a categorical action of modules over quantum enveloping algebras on modules over quantum Weyl algebras. Moreover, certain braided Drinfeld doubles give such an action on modules over rational Cherednik algebras using embeddings of Bazlov and Berenstein of these algebras into certain braided Heisenberg doubles which can be thought of as versions of the Dunkl embeddings. We argue that the corresponding braided Drinfeld doubles can serve an quantum group analogues in the setting of complex reflection groups.
Finally, the categorical description can be extended naturally to give TQFTs with defects using recent work of Fuchs-Schaumann-Schweigert.Tuesday 13 January 3pm Richard Hepworth (Aberdeen) A homology theory for graphs Tom Leinster recently introduced an invariant of graphs called the magnitude. In this talk I will define a homology theory for graphs that categorifies the magnitude, in the sense that the magnitude of a graph can be recovered from its homology by taking the Euler characteristic. (Thus this is categorification in the same sense that Khovanov homology categorifies the Jones polynomial.) Important properties of the magnitude can then be seen as shadows of properties of homology. For example, magnitude satisfies an inclusion-exclusion formula that can be recovered from a Mayer-Vietoris theorem in homology.
The talk will (hopefully) be accessible for anybody who knows what graphs and chain complexes are, and I will try to illustrate it with lots of pictures and examples.
Talks from Fall 2014:
Tuesday 2 December 3pm Bin Shu (East China Normal) Generic property and conjugacies of Borel subalgebras for restricted Lie algebras For a finite-dimensional restriicted Lie algebra $g$ over an algebraically closed field of prime characteristic, we introduce the notion "generic property", saying that a restricted Lie algebra satisfies such a property if it admits generic tori introduced in BFS. A Borel subalgebra of $g$ is defined as a maximal solvable subalgebra containing a maximal torus of $g$, which is further called generic if additionally containing a generic torus. In this talk, we first verify that a conjecture of Premet on regular Cartan subalgebras for $g$ is valid when $g$ satisfies the generic property, and show that his conjecture does not hold whenever $g$ does not satisfy the generic property. We finally classify the conjugacy classes of Borel subalgebras of the restricted simple Lie algebras $g=W(n)$ under $Aut(g)$-conjugation when $p>3$, and present the representatives of these classes. We also describe the closed connected solvable subgroups of $G$ associated with those representative Borel subalgebras. Tuesday 25 November 3pm Martin Kalck (Edinburgh) Spherical subcategories and new invariants for triangulated categories Motivated by examples arising in algebraic geometry, we study objects of k-linear triangulated categories with two-dimensional graded endomorphism algebra. Given such an object, we show that there is a unique maximal triangulated subcategory, in which the object is spherical, i.e. a Calabi-Yau object. In many examples, both from representation theory and geometry these spherical subcategories admit explicit descriptions. Furthermore, the collection of all spherical subcategories ordered by inclusion yields a new invariant for triangulated categories. We derive coarser invariants like height, width and cardinality of this poset. This talk is based on joint work with A. Hochenegger & D. Ploog. Tuesday 18 November 3pm David Pauksztello (University of Manchester) An introduction to co-t-structures and co-stability conditions In this talk we introduce the ideas of co-t-structures and co-stability conditions and compare and contrast with t-structures and stability conditions. We show that the space of co-stability conditions on a triangulated category forms a complex manifold, and give some examples. Part of this talk is joint work with Peter Jorgensen (Newcastle-upon-Tyne). Tuesday 18 November 4pm Rupert Yu (University of Reims) Jet schemes of nilpotent orbit closures In this talk, we investigate jet schemes of nilpotent orbit closures in semisimple Lie algebras. For the regular nilpotent orbit, its closure is the nilpotent cone, and their jet schemes are always irreducible. This was conjectured by Eisenbud and Frenkel, and was proved as a special case of a result of Mustata in a more general setting. We shall see that for a non regular and non zero nilpotent orbit, the jet schemes of its closure are not irreducible in general, and we may obtain from this information on certain geometric properties of nilpotent orbit closures.
This is a joint work in progress with Anne Moreau.Tuesday 11 November 4pm Arend Bayer (Edinburgh) The space of stability conditions on abelian threefolds (and a few CY 3folds) I will explain recent joint work with Emanuele Macri and Paolo Stellari, in which we describe the space of Bridgeland stability conditions on three-dimensional abelian varieties, and on (crepant resolutions of) their quotients by finite group actions. As I will explain, a lot of the structure is described by some elementary real algebraic geometry (configurations of points with respect to quadratic forms and components of the space of self-maps of the real projective line). Understanding this geometry allows us to strengthen and generalize the results by Antony Maciocia and Dulip Piyaratne (who were first to construct stability conditions on abelian varieties of Picard rank one with principal polarization). Tuesday 4 November 3pm Karel Casteels (University of Kent) Combinatorial Models of Quantum Matrix Algebras Some so-called "quantum matrix algebras" that are often defined by generators and relations (e.g, quantum matrices, the quantum special linear group, the quantum grassmannian, quantum symmetric and skew-symmetric matrices) can be embedded into a quantum torus by way of a certain directed graph. We are then able to "see" the generators and relations quite naturally, and, perhaps more importantly, we can use these models to study the prime and primitive spectra. Some of the work to be discussed is based on discussions with Stephane Launois and Tom Lenagan. Tuesday 28 October 3pm Christian Lomp (University of Porto) Semisimple Hopf algebra actions We review Hopf algebra actions on rings with particular emphasis on a question raised by Miriam Cohen in 1985 whether the smash product A # H of a semisimple Hopf algebra H acting on a semiprime algebra A is itself a semiprime ring. This question is open until now. In my talk I give a survey on known results concerning Cohen's question. Tuesday 28 October 4pm Paula Carvalho (University of Porto) On the injective hulls of simple modules over Noetherian Rings The Jacobson's conjecture is an open problem in ring theory and asks whether the intersection of the powers of the Jacobson radical of a two-sided Noetherian ring is zero. Jategaonkar answered the conjecture in the affirmative for a Noetherian ring R under an additional assumption (called FBN) which in particular implies that any finitely generated essential extension of a simple left R-module is Artinian. The latter condition, denoted by $( \diamond )$, is a sufficient condition for a positive answer to the Jacobson's conjecture.
In this talk we will consider some Noetherian algebras and study if or when they satisfy $(\diamond)$. In particular, we will be interested in the case of differential operator rings $R\theta; d$ with $R$ a commutative Noetherian ring and $d$ a derivation.Tuesday 30 September 3pm Evgeny Feigin (National Research University, Russia) PBW filtration and nonsymmetric Macdonald polynomials We discuss the recently conjectured connection between the nonsymmetric Macdonald polynomials and the Poincaré–Birkhoff–Witt filtration on Demazure modules of affine Kac-Moody Lie algebras. The conjecture has a surprising consequence relating PBW degrees of the extremal vectors in finite dimensional representations of simple Lie algebras with the extremal part of Macdonald polynomials. We also describe the connection with the Kostant q-partition function. The talk is based on joint work with I.Cherednik and I.Makedonskyi.
Talks take place in the James Clerk Maxwell Building, room 6311.Not part of MAXIMALS but related:
From 15-19 December 2014, there will be a workshop on Homological Interactions between Representation Theory and Singularity Theory at the University of Edinburgh.
Talks 2013-2014:
WARNING: Any error in the notes of the talks is probably due to the note-taker.Tuesday May 27th 2:30pm Andre Leroy, (University of Artois) Euclidean pairs , quasi Euclidean rings and continuant polynomials Starting with problem of decomposition of 2 by 2 singular matrices we will introduce Euclidean pairs and quasi Euclidean rings. We will characterise these rings in different ways and show that unit-regular rings are quasi-Euclidean. We will present relations with constant polynomials and other families of polynomials defined by linear recurrence relations. Tuesday May 27th 1:00pm Maria Chlouveraki (Versailles) Yokonuma-Hecke and Yokonuma-Temperley-Lieb algebras Yokonuma-Hecke algebras were introduced by Yokonuma in the 60's as generalisations of Iwahori-Hecke algebras. They have recently attracted the interest of topologists, because they naturally give rise to invariants for framed and classical knots. In this talk we will introduce and study the Yokonuma-Hecke algebras of type A and the Yokonuma-Temperley-Lieb algebras, which are the generalisations of classical Temperley-Lieb algebras in this case, mainly from the algebraic point of view. Friday May 16th 1:00pm JCMB 6206 Noah Snyder (IU Bloomington) Local Topological Field Theory and Fusion Categories Topological field theories give a close relationship between topology and algebra. Traditionally the main application has been from algebra to topology: using algebraic constructions like quantum groups to produce topological invariants. However, you can also run the applications the other way, using topology to arrange and clarify your knowledge about algebra. The goal of this talk is to explain one such application. More specifically, a fusion category is a category that looks like the category of representations of a finite group: it has a tensor product, duals, is semisimple, and has finitely many simple objects. A somewhat mysterious fact about fusion categories (generalizing a theorem of Radford's about Hopf algebras) is that the quadruple dual functor is canonically isomorphic to the identity functor. I will explain this mystery by showing that it follows directly from the Dirac belt trick.
The main technique in this proof is the construction of a local topological field theory attached to any fusion category. Topological field theories are invariants of manifolds which can be computed by cutting along codimension 1 boundaries. Local topological field theories allow cutting along lower codimension boundaries. Since manifolds with corners can be glued together in many different ways, this can be formalized using the language of n-categories. Using Lurie's version of the Baez-Dolan cobordism hypothesis, we describe local field theories with values in the 3-category of tensor categories. This is joint work with Chris Douglas and Chris Schommer-Pries. I will not assume prior familiarity with fusion categories or n-categories.Tuesday, May 13th 3:00pm Carl Mautner (MPIM Bonn) Modular Representation Theory and Parity Sheaves In the 1920's, Weyl proved a formula for the characters of the irreducible representations of reductive groups (e.g., the general linear group) over the complex numbers. In 1979, George Lusztig announced a conjectural character formula for representations of reductive groups over fields of positive characteristic. I will briefly discuss a history of the problem and explain the role of geometric objects called parity sheaves in recent advances. Tuesday, April 22nd 3:00pm Dorette Pronk (Dalhousie) Weakly globular double categories - a new model for weak 2-categories. In this talk I will discuss a new model for weak 2-categories. We will first show that there is more than one way to make 2-categories weak. Bicategories are obtained from 2-categories by relaxing the associativity and unit conditions, and requiring that they only hold up to coherent isomorphisms. But they are strict in the sense that we require them to have a set (or, class) X_0 of objects. To introduce the notion of weakly globular double category, we we will relax this requirement: rather, than requiring that we have a discrete set of objects (we call that the globularity condition), we will require that X_0 be a posetal groupoid (i.e., a set with equivalence relation). This is the weak globularity condition. The arrows in X_0 are distinct from the other arrows in the category, so we have now two classes of arrows, and this leads us to the notion of a double category. Weakly globular double categories are strict double categories that satisfy the weak globularity condition and an additional lifting condition on its double cells. I will discuss these conditions in detail and the show that this is sufficient to model weak 2-categories in the sense that there is a biequivalence between the 2-category of bicategories, homomorphisms and icons, and the 2-category of weakly globular doublce categories, pseudo functors and vertical transformations. This biequivalence factors through the 2-category of Tamsamani weak 2-categories. If there is enough time I will discuss the construction of a weakly globular double category of fractions and its universal properties. Tuesday, Apr 8th 3pm Alexey Sevastyanov (Aberdeen) A proof of De Concini-Kac-Procesi conjecture In 1992 De Concini, Kac and Procesi observed that isomorphism classes of irredicible representations of a quantum group at odd primitive root of unity m are parameterized by conjugacy classes in the corresponding algebraic group. They also conjectured that the dimensions of irreducible representations corresponding to a given conjugacy class O are divisible by \(m^{1/2 dim O}\). In this talk I shall prove an improved version of this conjecture and derive some important consequences of it related to q-W algebras. Tuesday, Apr 1st 3pm Elisa Gorla (Basel) Universal Groebner bases for ideals of maximal minors In 1993, Bernstein, Sturmfels, and Zelevinsky proved that the maximal minors of a matrix of variables form a universal Groebner basis. We present a very short proof of this result, along with a broad generalization to matrices with multi homogeneous structures. Our main tool is a rigidity statement for radical Borel fixed ideals in multigraded polynomial rings. This is joint work with A. Conca ed E. De Negri (University of Genoa). Tuesday, Mar 25th 4pm Claudio Sibilia (ETH Zürich) Chen homological connection for G- spaces. Chen homological connections are flat connections on a trivial bundle that can be build on any topological smooth manifold. The goal of the talk is to explain the relation between the Chen homological connection and the KZ equations in genus 0, and extends this formalism to compute some formal connections in genus 1 that are very close to the KZB connection. Tuesday, Mar 25th 3pm Iordan Ganev (Univ. Texas) Quantizations of multiplicative hypertoric varieties Multiplicative hypertoric varieties are symplectic analogues of toric varieties related to symplectic resolutions, hyperplane arrangements, and geometric representation theory. We construct quantizations, depending on a parameter $q$, of multiplicative hypertoric varieties using an algebra of difference operators on affine space. Furthermore, when $q$ is a root of unity, we show that the quantization acquires a large center and defines a matrix bundle (i.e. Azumaya algebra) over the multiplicative hypertoric variety. This is joint work with David Jordan. Tuesday, Mar 18th 3pm Ben Webster (U Virginia/Paris 6) Hodge seminar Tuesday, Mar 4th 3pm Felipe Rincon (Warwick) Positroids and the totally nonnegative Grassmannian Positroids are combinatorial objects that can be used to index the different strata in certain well-behaved decompositions of the Grassmannian and its positive part. In this talk I will present joint work with Federico Ardila and Lauren Williams, in which we study some of the combinatorial properties of positroids. As an application, we prove da Silva's 1987 conjecture that any positively oriented matroid is representable over the field of rational numbers. In particular, this implies that the positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a closed ball. Wendesday, Feb 26th 1:10pm Adam Chapman (Belgium) Clifford Algebras of Binary Cubic Forms in Characteristic 3 his talk is based on a joint work with Jung-Miao Kuo. The structure of Clifford algebras of binary cubic forms in characteristic different from 3 has been described in full detail by Haile. Kuo has generalized his results for the algebra associated to a ternary cubic curve in characteristic different from 3. Here we shall talk about the analogous results in case of characteristic 3. Tuesday, Feb 25th 3pm Johanna Hennig (UCSD) Locally finite Lie algebras in positive characteristic For finite dimensional Lie algebras, there is the well-known Ado’s theorem: Every finite dimensional Lie algebra embeds into a finite dimensional associative algebra. Bahturin, Baranov, and Zalesski proved an infinite dimensional version of Ado’s theorem for a simple, locally finite Lie algebra L over a field of characteristic zero: L embeds into a locally finite associative algebra if and only if L is isomorphic to the commutator of skew-symmetric elements of a locally finite, associative algebra with involution. We extend this result to fields of positive characteristic—we provide two structure theorems which reduce to Bahturin, Baranov, Zalesski’s result in characteristic zero and also generalize classical structure theorems for finite dimensional Lie algebras in characteristic p. Friday, Feb 14th 3pm Erez Sheiner (Bar Ilan) Exploded Layered Tropical Algebra ELT algebra is an extension of the max-plus algebra, which is a lesser degeneration of algebraic varieties. Using the layered structure we define roots of polynomials and singularity of matrices, despite the lack of an additive inverse. I will not assume familiarity with tropical geometry. Tuesday, Feb 11th 3pm Greg Ginot (Paris 6) Factorization algebras and applications to centralizers and Bar construction. in the first part of the talk, we will discuss the notion of centralizers of En-algebras maps which shall can be thought as "higher" generalizations of Hochschild cohomology and we will explain how one can see describe them in terms of factorization algebras. Factorization algebras will be detailled in the second part of the talk. They can be thought of as a "non-abelian" version of cosheaves. We will also give an analogue of the Bar constructions for Factorization algebras. Tuesday, Feb 4th 3pm Sian Fryer (Manchester) The q-Division Ring and its Fixed Rings The q-division ring (denoted here by D) is one of the easiest examples of noncommutative infinite dimensional division rings to define, but answering even fairly basic questions concerning the structure of its automorphism group or its sub-division rings of finite index is still quite difficult. The second question in particular is of interest due to its connections with Artin's conjectured classification of surfaces in non-commutative algebraic geometry. I will describe the structure of the fixed rings of D for a certain class of finite groups and use this to construct some rather unexpected examples of homomorphisms on D, including a conjugation automorphism which is not inner and a conjugation homomorphism which is not even bijective. Tuesday, Jan 28th 4pm Michael Wemyss (Edinburgh) Noncommutative deformations and applications I will mainly talk on my joint work with Will Donovan (1309.0698) which gives new invariants to any contractible curve using noncommutative deformation theory. I will explain a little about our motivation, and why studying noncommutative deformations is strictly necessary. In the setting of 3-fold flops, we obtain many new examples of finite dimensional self-injective algebras, and I will give some of their properties. If I have time, I will say a bit about how this helps to run the minimal model program in dimension three. Tuesday, Jan 21st 3pm Jean-Marie Bois (Kiel) Weyl groups for restricted lie algebras, and the Chevalley restriction theorem Tuesday, Jan 7th 3pm David Gepner (Regensburg) Brauer groups of commutative ring spectra We define and study Azumaya algebras over a commutative (or E_\infty) ring spectrum R, and study the homotopy groups of the resulting Brauer space (of Azumaya R-algebras up to Morita equivalence). Our main technical tool is an etale-local triviality result for Azumaya algebras over connective derived schemes (in the sense of J. Lurie, and inspired by a similar result of B. Toen) and a local-to-global result for the compact generation of R-linear \infty-categories with descent. The result is a spectral sequence for the homotopy groups of the Brauer space whose E_2 term consists entirely of ordinary etale cohomology groups, and which is often entirely computable. This is joint work with Ben Antieau. December 6th, 3pm @ICMS Joint w/ Heriot-Watt Anne Thomas (Glasgow) Quasi-isometry of right-angled Coxeter groups A group G with a finite generating set S can be considered as a metric space by endowing it with the word metric with respect to S. Up to quasi-isometry, this metric on G does not depend on S. A major theme in geometric group theory is to classify all finitely generated groups up to quasi-isometry. We investigate the quasi-isometric classification of right-angled Coxeter groups using divergence of geodesics and topological features of their boundaries at infinity. This is joint work with Pallavi Dani. December 6th, 4pm @ICMS Joint w/ Heriot-Watt Ghislain Fourier (Glasgow) Fusion products and symmetric functions Fusion products for current algebras have been introduced fifteen years ago. Roughly speaking, they make use of the natural grading of the polynomial ring to form graded tensor products of simple modules for a simple complex Lie algebra. These fusion products play a crucial role in the study of finite-dimensional modules for current or loop algebras, for instance they recover Weyl modules and Demazure modules to name but a few. Although intensively studied, various fundamental questions are not answered yet, for example about defining relations or graded character formulas.
The current state of art and as well as the strong connection to conjectures about Schur positivity of symmetric functions (and recent results here) will be presented.November 26, 3pm Emily Dufresne (Basel) Separating invariants and local cohomology The study of separating invariants is a new trend in Invariant Theory and a return to its roots: invariants as a classification tool. For a finite group acting linearly on a vector space, a separating set is simply a set of invariants whose elements separate the orbits of the action. Such a set need not generate the ring of invariants. In this talk, we give lower bounds on the size of separating sets based on the geometry of the action. These results are obtained via the study of the local cohomology with support at an arrangement of linear subspaces naturally arising from the action. (joint with Jack Jeffries) November 26, 4pm Ines Henriques (Sheffield) F-thresholds and Test ideals for determinantal ideals of maximal minors. Test ideals first appeared in the theory of tight closure, and reflect the singularities a ring of positive characteristic. Motivated by multiplier ideals in characteristic zero, N. Hara and K. Yoshida defined (generalized) test ideals as their characteristic p analogue.
Whereas multiplier ideals are defined geometrically, using log resolutions, test ideals are defined algebraically using the Frobenius morphism.
The test ideals of an ideal I form a non-increasing, right continuous family, {τ(c . I)}, parametrized by a positive real parameter c. The points of discontinuity in this parametrization, are called F-thresholds of I and form a discrete subset of the rational numbers (Blickle-Mustaţă-Smith, Hara, Takagi-Takahashi, Schwede-Takagi, Katzman-Lyubeznik-Zhang).
We consider ideals generated by maximal minors of a matrix of indeterminates, in its polynomial ring over a field of positive characteristic. Using an algebraic approach, we identify their F-thresholds and test ideals.November 19, 3pm Tom Lenagan (Edinburgh) Totally nonnegative matrices A real matrix is totally nonnegative if each of its minors is nonnegative, and is totally positive if each minor is greater than zero. We will outline connections between the theory of total nonnegativity and the torus invariant prime spectrum of the algebra of quantum matrices, and will discuss some new and old results about total nonnegativity which may be obtained using methods derived from quantum matrix methods. Most of the material is joint work with Stephane Launois and Ken Goodearl. NOTE: much of the material to be presented has been included in earlier seminars in Edinburgh. November 13 (Wednesday!), 4pm David Alan Jordan (Sheffield) Connected quantized Weyl algebras Connected quantized Weyl algebras are algebras with a PBW basis in which any two generators either q-commute or satisfy a quantum Weyl relation xy-qyx=1-q. The connectedness condition ensures, in some sense, enough quantum Weyl relations. The talk will begin with some discussion of how the motivating examples arose from work of Fordy and Marsh on periodic quiver mutation and Poisson algebras and then proceed to a classification of the connected quantized Weyl algebras and the determination of their prime ideals. October 25 (Friday!), 4-5pm Chris Dodd (Toronto) Cycles of Algebraic D-modules in positive characteristic I will explain some ongoing work on understanding algebraic D-moldules via their reduction to positive characteristic. I will define the p-cycle of an algebraic D-module, explain the general results of Bitoun and Van Den Bergh; and then discuss a new construction of a class of algebraic D-modules with prescribed p-cycle. October 28 (Monday!), 3pm David Evans (Cardiff) The search for the exotic - subfactors and conformal field theory Subfactor theory provides a framework for studying modular invariant partition functions in conformal field theory, and candidates for exotic modular tensor categories. I will describe work with Terry Gannon on the search for exotic theories beyond those from symmetries based on loop groups, Wess-Zumino-Witten models and finite groups. October 23 2-3pm, 4-5pm, 24th 4-5pm, (WTh!) Chris Dodd (Toronto) Quantizations of Conic Symplectic Varieties and Representation theory I will describe some recent progress- joint with G. Bellamy, K. McGerty, and T. Nevins- in understanding modules over algebraic quantizations of certain nice classes of symplectic varieties. In particular, I will explain in a leisurely fashion how quantizations come up in representation theory, how one can use the presence of a torus action to study modules over these quantizations, and the types of results (geometric and representation-theoretic) that come from this way of thinking. October 22, 3pm Nick Gurski (Sheffield) Categorical operads Operads are a convenient tool for encoding certain kinds of algebraic structures, and they are in heavy use in algebraic topology and homological algebra. There are some special features of operads in Cat, the (2-)category of categories, as well as a number of features shared with operads in other categories. I will review the basics of the general theory, and then talk about a couple of things special to the case of Cat. In the second hour, I plan to focus on how equivariance plays an important role in this story. October 15, 3pm Joe Chuang (City University London) Algebra with surfaces Frobenius algebras give rise to topological invariants of surfaces. I will review this idea (two-dimensional topological field theory) and describe joint work with Andrey Lazarev on a similar construction. October 8, 3pm Uzi Vishne (Bar Ilan) p-central elements and subspaces in central simple algebras The major open question on central simple algebras is the cyclicity problem: are all algebras of prime degree cyclic? Any cyclic algebra of degree p has p-central elements: non-central elements whose p-power is central. The cyclicity problem can thus be studied using subspaces of p-central elements, which will be the main topic of the lecture. We will discuss such subspaces from various points of views: chain lemmas, the symbol length problem, connections with elementary number theory, and a new construction in nonassociative algebra. October 8, 4pm Be'eri Greenfeld (Bar Ilan) Unions over chains of prime ideals In a commutative ring, the union over a chain of prime ideals is prime. This is not true in the general case: there exist counterexamples from many classes, including nil rings, locally finite rings, affine algebras with polynomial growth and rings satisfying a polynomial identity (PI).
In the latter case, though, the maximal number of non-prime unions of subchains of a chain of prime ideals is (tightly) bounded by the PI-class ,hence finite; this is far from being true in general.
In this talk we discuss several examples with additional properties (e.g. primitivity), positive results and suggestions for further research directions.
This is based on joint work with Louis Rowen and Uzi Vishne.October 1, 3pm David Andrew Jordan (Edinburgh) Quantum differential operators and the torus \( T^2 \) Abstract: The algebra \( D_q(G) \) is a \(q \)-deformation of the algebra \( D(G) \) of differential operators on a semi-simple algebraic group. In this talk, I will explain an intimate relationship between \(D_q(G)\) and the torus \(T^2 \): namely, \( D_q(G) \) carries an action by algebra automorphisms of the torus mapping class group \(SL_2(\mathbb{Z}) \), and also yields representations of the torus braid group extending the well-known action of the planar braid group on tensor powers of quantum group representations. Finally, the so-called Hamiltonian reduction of \( D_q(G) \) quantizes the moduli space \( Loc_G(T^2) \) of \(G\)-local systems on \(T^2\), or equivalently, homomorphisms \(\pi_1(T^2)\to G\),and this observation allows us to generalize the construction of \(D_q(G)\) to quantize \(Loc_G(\Sigma_{g,r})\), for an arbitrary surface with genus g and r punctures.
Time permitting, I will outline work in progress with David Ben-Zvi and Adrien Brochier putting all of the above into the context of topological field theories.October 1, 4pm Natalia Iyudu (Edinburgh) A proof of the Kontsevich conjecture on noncommutative birational transformations I will talk about our recent proof (arXiv1305.1965) of the Kontsevich conjecture dated back at 1996, and mentioned at the 2011 Arbeitstagung talk on 'Noncommutative identities' (arXiv1109.2469). This conjecture says that certain transformations given by matrices over free noncommutative algebra with inverses ('free field' due to P.Cohn) are periodic, on the level of orbits of the left/right diagonal action. Namely, let \( (M_{ij})_{1 \leq i,j \leq 3} \) be a matrix, whose entries are independent noncommutative variables. Let us consider three 'birational involutions'
$$ I_1: \,\, M \to M^{-1} $$ $$I_2: \,\, M_{ij} \to (M_{ij})^{-1}, \,\, \forall i,j $$ $$I_3: \,\, M \to M^t$$
Then the composition \(\Phi = I_1 \circ I_2 \circ I_3 \) has order three.September 24, 3pm Gwyn Bellamy (Glasgow) Generalizing Kashiwara's equivalence to conic quantized symplectic manifolds Abstract: Kashiwara's equivalence, saying that the category of D-modules on a variety X support on a smooth, closed subvariety Y is equivalent to the category of D-modules on Y, is a key result in the theory of D-modules. In this talk I will explain how one can generalize Kashiwara's result to modules for deformation-quantization algebras on a conic symplectic manifold. As an illustrative application, one can use this result to calculate the additive invariance such as the K-theory and Hochschild homology of these module categories. This is based on joint work C. Dodd, K. McGerty and T. Nevins. September 17, 3pm Stefan Kolb (Newcastle) Radial part calculations for affine sl2 Abstract: In their seminal work in the 1970s Olshanetsky and Perelomov used radial part calculations for symmetric spaces to prove integrability of the Calogero-Moser Hamiltonian for special parameters. In this talk, restricting to affine sl2, we will explore what happens if one extends their argument to Kac-Moody algebras. I will try to explain how this leads to a blend of the KZB-heat equation and Inozemtsev's extension of the elliptic Calogero-Moser Hamiltonian. September 9 (Monday!), 5pm Francois Petit (Edinburgh) Fourier-Mukai transform in the quantized setting Abstract: After reviewing some elements of the theory of Deformation Quantization modules (DQ-modules), I will show that a coherent DQ-kernel induces an equivalence between the derived categories of coherent DQ-modules if and only if the graded commutative kernel associated to it induces an equivalence between the derived categories of coherent O-modules. September 9 (Monday!), 4pm Hendrik Suess (Edinburgh) Equivariant vector bundles on T-Varieties Abstract: By Klyachko's work there is an equivalence of categories between equivariant vector bundles on toric varieties and families of vector space filtrations. In this talk I will discuss an generalization of this equivalence to bundles on varieties with smaller torus actions. Now, vector space filtrations are replaced by filtrations of vector bundles on some quotient space. This description comes with a nice splitting criterion and allows to prove that vector bundles of low rank on projective space, which are equivariant with respect to special subtori of the maximal acting torus must split. September 4th (Wednesday!), 3pm Qendrim Gashi (Pristina) Mazur's inequality, its converse and generalizations Abstract: We will study the classical version of Mazur's inequality, comparing Newton and Hodge polygons, and a converse thereof (due to Kottwitz and Rapoport). We will then discuss group-theoretic generalizations of these results and implications for affine Deligne-Lusztig varieties. We will conclude with some results from root theory and toric geometry. - Noah Snyder (IU Bloomington)
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