Edinburgh Hodge Institute

The EDGE seminar and the MAXIMALS seminar are combined to be the Edinburgh Hodge seminar this year.

The Edinburgh Hodge seminar is held every Wednesday, 2:00-3:00pm in Zoom, and sometimes there is a 30 minutes pre-talk starting at 1:15pm. Please check the schedule for details. The Zoom link for each meeting will be sent via the mailing list below. The seminar is organised by all faculty working in the Hodge institute, and currently coordinated by Sjoerd Beentjes, Cheuk Yu Mak, Brian Williams and Fei Xie.

Mailing Lists. Since most of our seminars will occur together with MAXIMALS 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.

The seminar is named after William Edge (1904-1997), who is known for example for his work on finite geometry, and worked at the University of Edinburgh for over 40 years (1932-1975). Related seminars are Topology, MAXIMALS, EMPG, GLEN, COW.

Current Semester

Upcoming Web Hodge talks:

Web Hodge: Tudor Dimofte (Edinburgh/UC Davis)-3d mirror symmetry and HOMFLY-PT homology
28th October 2020, 1:15pm to 3:00pm -- Show/hide abstract
Abstract: 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.
Web Hodge: Georg Oberdieck (University of Bonn)-Moduli spaces in equivariant categories (date: 04/11; time: TBC, will be different from usual)
4th November 2020, 1:15pm to 2:45pm -- Show/hide abstract
Abstract: 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: Giulia Gugiatti (Edinburgh)-title: TBC
11th November 2020, 2:00pm to 3:00pm
Web Hodge: Pieter Belmans (Bonn/Antwerp)-An introduction to semiorthogonal decompositions
18th November 2020, 1:15pm to 1:45pm -- Show/hide abstract
Abstract: 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.
Web Hodge: Pieter Belmans (Bonn/Antwerp)-Moduli of semiorthogonal decompositions in families
18th November 2020, 2:00pm to 3:00pm -- Show/hide abstract
Abstract: 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.
Web Hodge: Cheuk Yu Mak (Edinburgh)-title: TBC
25th November 2020, 1:15pm to 3:00pm
Web Hodge: Yu Zhao (MIT)-title: TBC
2nd December 2020, 1:15pm to 3:00pm
Web Hodge: Fei Xie (Edinburgh)-title: TBC
9th December 2020, 2:00pm to 3:00pm
Web Hodge: Eugene Gorsky (UC Davis)-title: TBC
16th December 2020, 1:15pm to 3:00pm

(Open in Google Calendar)

Previous Web Hodge talks:

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 abstract
Abstract: 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 abstract
Abstract:
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. Pre-talk slides Main talk slides
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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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

(Open in Google Calendar)

Previous EDGE talks:

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 abstract
Abstract: 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 abstract
Abstract:
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 abstract
Abstract:
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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract:
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 abstract
Abstract: 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 abstract
Abstract: 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 abstract

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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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 abstract
Abstract: 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
24th September 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
Abstract: 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

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