Another Sandbox

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experiment with tables:

In my talk I will construct a functor from some class of skew fields to some class of commutative fields with \( h_3 \) map
\( Q C x, x \to Q C x, y \)
\( Q C[x, \partial_x] \to Q C[x, y] \) as the main example. I will also provide some link between this construction and some known results and conjectures on skew fields.

SEMCAL experiments

• Geometry Seminar - Professor Sergey Yuzvinsky,
  31st January 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: SPECIAL FIBERS OF PENCILS OF CURVES AND TOPOLOGY - Given a one-dimensional linear system (pencil) of complex plane curves of degree d>1 with the generic fiber irredicuble we study the set of its completely reducible (i.e. unions of lines) fibers. A surprising result says that the number of these fibers has a harsh upper bound independent of d. Another surprise is that this set can be completely characterized either by certain topological properties of the complement of the union of lines or by the combinatorics of their intersections. We will give lots of examples.
• Geometry Seminar - Dr. Dmitri Alekseevky
  7th February 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: HOMOGENEOUS PARA-KAHLER EINSTEIN MANIFOLDS
• Geometry Seminar - Dr. Jonathan Nimmo
  14th February 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: DARBOUX TRANSFORMATIONS, QUASIDETERMINANTS AND NONCOMMUTATIVE INTEGRABLE SYSTEMS - Darboux transformations offer a standard technique to construct solutions of commutative integrable systems, leading to formulae involving determinants. Their use for noncommutative integrable also leads naturally expressions in terms of the quasideterminants of Gelfand and Retakh. This talk will review some recent results in this area.
• Geometry Seminar - Prof. Fran Burstall
  21st February 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: PARABOLIC SUBMANIFOLD GEOMETRY - I will describe joint work with David Calderbank which gives a uniform approach to the invariants of submanifolds of real flag manifolds and their integrability conditions (Gauss-Codazzi-Ricci equations). The theory applies to submanifolds of the conformal sphere, hypersurfaces in projective space, CR-submanifolds of the CR-sphere and much more. Unexpected ingredients include Lie algebra homology and a curved A_\infty algebra of differential operators extending the linear Bernstein-Gelfand-Gelfand operators. The talk will not assume familiarity with any of the above.
• Geometry Seminar - Prof. Michael Singer
  28th February 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: MOMENT MAPS AND EXTREMAL KAEHLER METRICS - I will give an introduction to the theory of extremal Kaehler (eK) metrics, moment maps, and the existence theory of eK metrics on the blow-up of a manifold which already has an eK metric.
• Geometry Seminar - Dr. Wilhelm Klingenberg
  6th March 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: FIBRATIONS EUCLIDEAN 3-SPACE BY GEODESICS - We characterize regular fibrations by geodesics as being spacelike surfaces in the their moduli space, namely the Klein quadric, which is equipped with a neutral Kaehler metric. The latter is invariant under the induced action of the Euclidean group and admits a rich geometry.
• Geometry Seminar - Dr. Eugene Ferapontov
  13th March 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: GEOMETRY AND INTEGRABILITY OF SECOND ORDER QUASILINEAR PDES IN 2+1 DIMENSIONS.
• Geometry Seminar - Ms. Fatima Araujo
  3rd April 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: TBA
• Geometry Seminar - Dr. Ian Strachan
  10th April 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: THE GEOMETRY OF BI-HAMILTONIAN SYSTEMS
• Geometry Seminar - Dr. Ian McIntosh
  17th April 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: TBA
• Geometry Seminar - Dr. Misha Feigin
  24th April 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: TBA
• Geometry Seminar - Dr. Eugenie Hunsicker
  1st May 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: TBA
• Geometry Seminar - Herbert Gangl
  25th September 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: POLYGONAL COMBINATORICS FOR ALGEBRAIC CYCLES - Starting from comparably explicit objects (algebraic cycles), Bloch and Kriz have given a tentative definition of a small yet rich category of motives (mixed Tate motives), at least over a field. They also exhibited a distinguished class of cycles corresponding to polylogarithms. One can also find _multiple_ polylogarithms as algebraic cycles, and it turns out that their differential structure can be conveniently described with the help of combinatorics of polygons. This leads to a coproduct on polygons which is a variant of the Connes-Kreimer coproduct on rooted trees.
• Geometry Seminar - Ivan Cheltsov
  2nd October 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: EXCEPTIONAL LOG DEL PEZZO SURFACES - Exceptional log Fano varieties are defined by Shokurov for inductive approach to study singularities and log Fano varieties. They have been studied by Shokurov, Prokhorov, McKernan, Keel and Kollar. Implicitly they have been studied by Tian, Phong, Sturm, Boyer and Rubinstein. We discuss some facts about exceptional log del Pezzo surfaces.
• Geometry Seminar - Tamas Hausel
  10th October 2008, 2:00pm to 2:50pm 4312 JCMB -- Show/hide abstract
  Abstract: TOPOLOGY OF THE HITCHIN MAP AND THE ARITHMETIC OF THE CHARACTER VARIETY - We suggest that the topology of the Hitchin map reflects the arithmetic of the character variety. This is a joint project with Mark de Cataldo and Luca Migliorini.
• Geometry Seminar - Elizabeth Gasparim
  16th October 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: RELATING ALGEBRAIC AND SYMPLECTIC GEOMETRY VIA INSTANTON COUNTING - Report on joint paper with Melissa Liu on the Nekrasov conjecture.
• Geometry Seminar - Alastair Craw
  23rd October 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: TILTING BUNDLES ON MULTILINEAR SERIES - The multilinear series of a collection of line bundles on a (toric) variety is a multigraded version of projective space. To further the analogy, I will construct a tilting bundle on each multilinear series, generalising Beilinson's celebrated theorem to the multigraded case.
• Geometry Seminar - Alexei Gorinov
  30th October 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: DIVISION THEOREMS FOR THE COHOMOLOGY OF DISCRIMINANT COMPLEMENTS AND APPLICATIONS TO AUTOMORPHISM GROUPS. - The linear group $\mathrm{GL}_{n+1}(\mathbb{C})$ acts in an natural way on the space of equations of smooth complete intersections of given dimension and multidegree in the complex projective space of dimension $n$. We show that the rational cohomology Leray spectral sequence of the corresponding quotient map degenerates in the second term. The proof uses a cohomological analogue of the linking number of curves in the 3-space (the invariant which allows one, in particular, to show a physical chain can't be undone without tearing or breaking). As a by-product of the proof, we obtain explicit expressions divisible by the order the automorphism group of any smooth projective hypersurface of given dimension and degree >2. We shall also discuss some other applications and possible generalisations of these results.
• Geometry Seminar - Dmitri Panov
  6th November 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: SIMPLY CONNECTED CALABI-YAU MANIFOLDS VIA HYPERBOLIC GEOMETRY - Classically Calabi-Yau manifold are algebraic manifolds with a non-vanishing holomorphic volume form, they are both symplectic and complex manifolds. A well known idea from mirror symmetry is to consider separately symplectic and complex structures. So we can introduce a notion of Symplectic Calabi Yau and Complex nonalgebraic Calabi Yau. While in real dimension 4 these manifolds are expected to be very close to algebraic Calabi Yaus, in real dimension 6 the divergence is huge (and intriguing). We will show how to construct simply connected Symplectic Calabi Yaus with b_3=0 out of 4-dimensional hyperbolic geometry and how to construct Complex non algebraic 3-folds with trivial canonical class out of hyperbolic knots in S^3. This is a joint work with Joel Fine.
• Geometry Seminar - Claire Voisin
  13th November 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: HODGE STRUCTURES ON COHOMOLOGY ALGEBRAS AND GEOMETRY - We study restrictions on cohomology algebras of Kaehler compact manifolds, not depending on the h^{p,q} numbers or the symplectic structure. To illustrate the effectiveness of these restrictions, we give a number of examples of compact symplectic manifolds satisfying the Lefschetz property but not having the cohomology algebra of a compact Kaehler manifold. We also prove the stability of these restrictions under products.
• Geometry Seminar - Mario Micallef
  20th November 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: HARMONIC MAPS BETWEEN COMPLETE, NONCOMPACT NEGATIVELY CURVED MANIFOLDS. - The construction of a harmonic extension of a map between the ideal boundaries of two complete noncompact manifolds of strictly negative curvature involves two parts: (i) the construction of an extension of the boundary map to the whole space with bounds on the energy density and the tension fields, (ii) deformation of the extension in (i) to a harmonic map. I will describe recent progress due to Fotiadis, Markovic and myself concerning step (ii) which enables one to allow the boundary map to be considerably more singular than previously allowed by Li-Tam and others. I will also describe a proof by Fotiadis, different from the Teichmuller proof of Markovic, of the existence of a harmonic diffeomorphism of the hyperbolic plane which extends a symmetric map of the ideal circle.
• Geometry Seminar - John Bolton
  27th November 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: THE MODULI SPACE OF HARMONIC 2-SPHERES IN ROUND SPHERES - I would like to report on recent work by Luis Fernandez and myself on the above moduli space M. Firstly, Fernandez has proved a conjecture of myself and Lyndon Woodward concerning the dimension of M. Secondly, in the case when the target is the unit 4-sphere, Fernandez and myself have considered the regularity of M, and we have also investigated elements of M possessing some symmetry.
• Geometry Seminar - Yang-Hui He
  4th December 2008, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: QUIVERS AND PLETHYSTICS - We present some recent progress in the study of quiver theories in the context of string and gauge theory. In particular, we study gauge theories which arise from D-branes on Calabi-Yau spaces and advocate a ``Plethystic Programme'' which provides a simple bridge between (1) the defining equation of the Calabi-Yau, (2) the generating function of single-trace operators and (3) the counting of multi-trace operators. Mathematically, fascinating and intricate inter-relations between quiver gauge theories, algebraic geometry and combinatorics exhibit themselves in the form of plethystics and syzygies.
• Geometry Seminar - Gabriel Paternain
  15th January 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: TRANSPARENT CONNECTIONS OVER NEGATIVELY CURVED SURFACES - A unitary connection is said to be transparent if its parallel transport along every closed geodesic is the identity. We show that transparent connections over a closed negatively curved surface can be understood in terms of Backlund transformations and that the trivial connection is locally unique. The proof also uses the non-abelian Livsic theorem and a suitable Fourier analysis on the unit tangent bundle of the surface.
• Geometry Seminar - Eugenie Hunsicker
  29th January 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: PSEUDODIFFERENTIAL OPERATORS FOR LOCALLY SYMMETRIC SPACES - This talk presents the first part of an ongoing collaboration between the speaker and D. Grieser on the construction of a pseudodifferential operator calculus for locally symmetric spaces and structurally similar space. The work to date is on generalizations of Q-rank 1 spaces, that is, manifolds with boundary where the boundary has a multiple fibration structure. The general philosophy is that of the Melrose b-calculus. The talk will focus on geometric motivations, the philosophy of the construction and statements of the main results.
• Geometry Seminar - Andrew Ranicki
  5th February 2009, 4:10pm to 5:00pm 5215 JCMB (note unusual location) (note location) -- Show/hide abstract
  Abstract: MICHEL KERVAIRE'S WORK IN SURGERY AND KNOT THEORY - Michael Kervaire (1927-2007) was a French topologist who worked in Switzerland and the United States. Kervaire made major contributions to our understanding of the topology of high-dimensional manifolds, using homotopy theory and quadratic forms. The talk is (an abbreviated) dress rehearsal for the 2 talks I am giving at the Kervaire memorial symposium in Geneva February 10-13 (http://www.unige.ch/math/folks/kashaev/symposiumMK.html) My talk will cover Kervaire's work in the period 1955-1965, dealing with the following highlights from his oeuvre: 1. Curvatura integra and the Kervaire semicharacteristic 2. The J-homomorphism 3. The classification (with Milnor) of exotic spheres 4. The Kervaire PL manifold without differentiable structure 5. High-dimensional knot theory. These topics have a common feature: the use of stably trivialized vector bundles in topology and quadratic forms in algebra.
• Geometry Seminar - Leo Butler
  12th February 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: EXOTIC TORI AND COMPLETE INTEGRABILITY - In hamiltonian mechanics, a key role is played by completely integrable systems. A priori, the existence of such a system is a smooth invariant of the configuration space. This talk will demonstrate that this invariant is non-trivial. We will show that, if a smooth manifold homeomorphic to the n-torus is the configuration space of a completely integrable convex hamiltonian, then it is actually diffeomorphic to the standard n-torus. On the other hand, we will exhibit topological 7-manifolds each of whose 28 smooth structures is the configuration space of a completely integrable convex hamiltonian. A key ingredient is Viterbo's work on vanishing Maslov cycles.
• Geometry Seminar - Gilles Carron
  25th February 2009, 4:10pm to 5:00pm 5215 JCMB (note unusual date and location) (note location) -- Show/hide abstract
  Abstract: QALE METRICS ON THE HILBERT SCHEME OF N POINTS ON C2. - We show that the Hyperkahler metric introduced by H. Nakajima on the Hilbert Scheme of n points on C2 is QALE (Quasi Asymptotically Locally Euclidean).
• Geometry Seminar - Mikhail Feigin
  26th February 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: SPECIAL REPRESENTATIONS OF CHEREDNIK ALGEBRAS AND QUANTUM INTEGRABLE SYSTEMS - We investigate radical ideals in the polynomial ring which are submodules of the polynomial representation of the rational Cherednik algebra associated with a Coxeter group. These ideals are supported on certain parabolic strata that is on the orbits of special intersections of mirrors of the Coxeter group. Dunkl operators can be restricted to these strata which leads to known and new generalisations of integrable systems of Calogero-Moser type. Similar construction exists for the double affine Hecke algebras which leads to the generalized Ruijsenaars-Macdonald systems - joint work with A. Silantyev.
• Geometry Seminar - Chris Athorne
  5th March 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: A NEW LOOK AT \WP-FUNCTIONS - We use some elementary representation theory to develop new machinery for deriving and classifying differential identities satisfied by \wp-functions of plane algebraic curves.
• Geometry Seminar - Tom Bridgeland
  12th March 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: DONALDSON-THOMAS CURVE-COUNTING INVARIANTS. - Donaldson-Thomas invariants are numbers that count curves on a three-dimensional Calabi-Yau variety. Conjecturally they are closely related to Gromov-Witten invariants. I will start from the beginning and explain a little bit about how such invariants are defined. It will be useful to know what a Hilbert scheme is. Then I will state a recent result which shows that the generating functions for such invariants have certain rationality properties.
• Geometry Seminar - Susan Cooper
  19th March 2009, 3:10pm to 4:00pm 4312 JCMB -- Show/hide abstract
  Abstract: IN SEARCH OF EXACTNESS - Certain data about a finite set of distinct, reduced points in projective space can be obtained from its Hilbert function. It is well known what these Hilbert functions look like, and it is natural to try to generalize this characterization to non-reduced schemes. In particular, we consider a fat point scheme determined by a set of distinct points (called the support) and non-negative integers (called the multiplicities). In general, it is not yet known what the Hilbert functions are for fat points with fixed multiplicities as the support points vary. However, if the points are in projective 2-space and the number of support points is 8 or less, we can write down all of the possible Hilbert functions for any given set of multiplicities (due to Guardo-Harbourne and Geramita-Harbourne-Migliore). In this talk we focus on what can be said, in projective 2-space, given information about what collinearities occur among the support points. Using this information we measure how far related sequences can be from being exact on global sections. Doing so, we obtain upper and lower bounds for the Hilbert function of the fat point scheme. Moreover, we give a simple criterion for when the bounds coincide yielding a precise calculation of the Hilbert function in this case. This is joint work with B. Harbourne and Z. Teitler.
• Geometry Seminar - Gavin Brown
  19th March 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: DIPTYCH VARIETIES AND THE EQUATIONS OF MORI FLIPS - A Mori flip is a particular class of surgery operations that can be applied to a complex 3-folds: roughly speaking, a flip excises one curve and recompactifies with another, making the line bundle of rational 'volume' forms more positive and the singularities of the 3-fold more tame in the process. Their existence was proved by Mori 20 years ago, and their classification into A,D,E types completed by Kollar-Mori soon after. I will report on work with Miles Reid that computes the equations of flips of type A, extending more recent work of Mori describing a Euclidean-style algorithm in a certain group of line bundles. Our constructions are built on a new class of nearly-symmetric spaces called 'diptych varieties'. I will explain their role and the relation to the Mori-Euclidean algorithm.
• Geometry Seminar - Joel Fine
  1st April 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: CALABI FLOW AND PROJECTIVE EMBEDDINGS - Let X subset CP^N be a smooth projective variety. I will describe a flow, called balancing flow, which tries to move the embedding of X until it has zero centre of mass in CP^N. I will then explain how this flow is related to Calabi flow. Calabi flow is a parabolic flow for a Kahler metric which tries to deform the metric until it has constant scalar curvature. If L->X is a Kahler manifold with ample line bundle, high powers L^k of L give embeddings X->CP^N in larger and larger projective spaces. For each k, we evolve the embedding via balancing flow and restrict the Fubini-Study metric to X to get a sequence of metric flows. Prorvided the initial embeddings are correctly chosen, this sequence of flows converges to Calabi flow.
• Geometry Seminar - Jens Kroeske
  16th April 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: CARTAN GEOMETRIES AND INVARIANT MULTILINEAR DIFFERENTIAL OPERATORS - For a differential operator in Euclidean space (or better on the sphere) the notion of being invariant under conformal transformations is well understood. Recent research has extended this notion for linear differential operators to conformal manifolds and more generally to a wide class of Cartan Geometries. The classification of such invariant linear differential operators has led to many interesting results. This talk will give a brief introduction to the concept of Cartan Geometries, discuss the general notion of invariance and motivate the study of multilinear differential operators. General classification results and the ideas behind it that go back to Penrose's Twistor Theory will be presented.
• Geometry Seminar - Simon Gindikin
  30th April 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: HOROSPHERICAL GEOMETRY OF SYMMETRIC SPACES AND ITS APPLICATION TO INTEGRAL GEOMETRY
• Geometry Seminar - Hugh Griffiths
  1st October 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: CONSTRUCTING SELF-DUAL MANIFOLDS WITH TORUS SYMMETRY - I will give a a brief background on what self-dual metrics are and why they are interesting. A result of Joyce reduces the problem of finding such a metric to a pair of linear PDEs by making an extra assumption of T^2 symmetry, and gives many solutions to these equations. Later results have shown that, subject to a few extra assumptions, this method in fact constructs all the compact toric self-dual metrics. I will then discuss a result from my thesis which makes use of Joyce's method to give a new family of metrics on non-compact spaces and can be used to perturb his original metrics on suitably small regions. If there is time I will briefly discuss some future directions in which I would like to advance this work.
• Geometry Seminar - Dominic Wright
  8th October 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: TORIC 4-ORBIFOLDS AND THE QUATERNIONIC QUOTIENT - My talk is about the construction of compact toric anti-self-dual orbifolds using the quaternionic quotient and how this leads to a quotient construction for asymptotically locally Eucliden scalar-flat Kaehler 4-orbifolds. This will involve explaining a classification of compact toric anti-self-dual orbifolds in terms of their twistor spaces which is used to check that these quotients are indeed the orbifolds we are looking for.
• Geometry Seminar - Richard Melrose
  15th October 2009, 4:10pm to 5:00pm 6301 JCMB (note location) -- Show/hide abstract
  Abstract: RESOLUTIONS OF COMPACT G-ACTIONS AND COHOMOLOGY - I will describe a precise form of the `Folk Theorem' that the action of a compact Lie group on a compact manifold can be resolved, by iterated blow up, to have unique isotropy type. One application of this joint work with Pierre Albin is the resolution of equivariant cohomology which in turn leads to a `delocalized' equivariant cohomology, extending the work of Baum, Brylinski and MacPherson in the abelian case. This cohomology theory is the natural target for the Chern-Atiyah-Hirzebruch isomorphism from rationalized equivariant K-theory.
• Geometry Seminar - Victor Przyjalkowski
  22nd October 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: MIRROR SYMMETRY APPROACH TO GEOMETRY OF FANO VARIETIES - Given a smooth Fano variety, Mirror Symmetry predicts the existence of a so called Landau--Ginzburg model --- a pencil, whose symplectic geometry reflects the algebraic geometry of the Fano variety, and viceversa. We discuss this relation for mirror symmetry conjecture of Hodge structure variations that translates this relation to a quantitative level. We discuss (mostly for threefolds) how, given Landau--Ginzburg model, predict numerical invariants of Fano variety (Gromov--Witten invariants, Hodge numbers, characteristic numbers) and its birational type.
• Geometry Seminar - Radu Ionas
  29th October 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: QUATERNIONIC KAHLER SPACES WITH LARGE TORIC SYMMETRY - Hyperkahler metrics with a free toric action of rank equal to their quaternionic dimension can be described locally by a generalized Gibbons-Hawking Ansatz. Using the device of Swann bundles I will show that a similar description exists for 4n-dimensional quaternionic Kahler metrics with a free toric action of rank n+1. In four dimensions, corresponding to the case of self-dual Einstein manifolds with two commuting Killing vector fields, the resulting formulas coincide with those obtained through a different approach by Calderbank and Pedersen.
• Geometry Seminar - Ivan Arzhantsev
  5th November 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: FACTORIAL ALGEBRAIC GROUP ACTIONS AND CATEGORICAL QUOTIENTS - Given an action of an affine algebraic group with only trivial characters on a factorial quasiaffine variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of a finitely generated algebra of invariants. As an application, we provide a combinatorial GIT-type construction of categorial quotients for actions on, e.g. complete, varieties with finitely generated Cox ring via lifting to the universal torsor. (This is joint work with D. Celik and J. Hausen)
• Geometry Seminar - Daniele Sepe
  12th November 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: CLASSIFICATION OF LAGRANGIAN FIBRATIONS - Lagrangian fibrations are of interest in many fields of mathematics, from affine geometry to dynamical systems. In this talk I shall construct the topological characteristic classes of such fibrations with the aid of some universal bundles. These invariants have also been constructed in the past by Duistermaat, Dazord, Delzant and Zung using different techniques and arose as the obstructions to defining global action-angle coordinates in the total space of a Lagrangian fibration (such coordinates always exist locally by the Liouville-Mineur-Arnold theorem). Time permitting, I will also describe a family of torus fibrations which are topologically indistinguishable from Lagrangian fibrations but are not Lagrangian.
• Geometry Seminar - Wendy Lowen
  19th November 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: HOCHSCHILD COHOMOLOGY AND NON-COMMUTATIVE DEFORMATIONS - I will explain the relation between the Hochschild cohomology of a scheme and its ``non-commutative deformations'' using associated presheaves. Along the way, I will discuss a new cohomology comparison theorem which clarifies certain aspects of Gerstenhaber and Schack's original theorem.
• Geometry Seminar - Alex Quintero Velez
  26th November 2009, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: MCKAY CORRESPONDENCE FOR LANDAU-GINZBURG MODELS - The McKay correspondence is a principle that relates the geometry of a resolution of singularities of a quotient variety M/G and the equivariant geometry of the group action. In this talk, we discuss an analogue of the McKay correspondence for Landau-Ginzburg models. If time permits I will also discuss how is this related to the Calabi-Yau/Landau-Ginzburg correspondence (after Orlov, Hori-Herbst-Paige) and Homological Mirror Symmetry.
• Geometry Seminar - Teru Thomas
  14th January 2010, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: GEOMETRIFYING GLOBAL HECKE ALGEBRAS - I will begin with a brief introduction to the Langlands correspondence for curves X over finite fields. From this the "global Hecke algebra" emerges as a basic object of study (Langlands roughly says that its spectrum is the set of l-adic local systems on X). This algebra has very explicit generators, so the mystery lies in its relations. In this talk I present work in progress to analyse these relations in terms of the geometry of the moduli space of vector bundles on X. The goal (fulfilled in some examples) is to construct some kind of "motivic" moduli space of local systems.
• Geometry Seminar - Timothy Hollowood
  21st January 2010, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: PENROSE LIMITS AND THE REFRACTIVE INDEX OF CURVED SPACETIME
• Geometry Seminar - Brian Smithling
  4th February 2010, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: INTRODUCTION TO LOCAL MODELS - Local models are schemes, defined in their greatest generality by Rapoport and Zink, which classify certain moduli problems of a purely linear-algebraicnature. (Rapoport and Zink introduced these to study integral models of certain Shimura varieties over p-adic fields, but I will assume no knowledge about Shimuravarieties in this talk.) This talk will be a general introduction to the subject, focusing on the simplest examples throughout.
• Geometry Seminar - Timothy Logvinenko
  18th February 2010, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: DERIVED FUNCTORS BETWEEN COTANGENT BUNDLES OF FLAG VARIETIES - This is a joint work with Rina Anno (UChicago). We show how to construct a network of functors which correspond to `generalized braid diagrams', between derived categories of coherent sheaves on cotangent bundles of full and partial flag varieties. For a subclass of these diagrams (which includes all the ordinary braids) we prove that isotopic diagrams correspond to isomorphic functors. We then outline our strategy for proving the general case.
• Geometry Seminar - David Wraith
  4th March 2010, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: MODULI SPACES OF RICCI POSITIVE METRICS - Given a manifold admitting a smooth Ricci positive metric, we can study the moduli space of such metrics on the manifold. This moduli space is obtained by considering the space of all Ricci positive metrics and and forming the quotient under the action of the diffeomorphism group. In general, very little is known about the topology of such moduli spaces. In this talk, we investigate the connectedness of this moduli space for a large class of homotopy spheres.
• Geometry Seminar - Richard Thomas
  11th March 2010, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: KHOVANOV HOMOLOGY FROM SURFACE SINGULARITIES - Khovanov homology is an invariant of knots and links coming from algebra (Khovanov), symplectic geometry (Seidel-Smith) or algebraic geometry (Cautis-Kamnitzer). I will describe another construction coming from algebraic geometry that is simpler than that of Cautis-Kamnitzer, being essentially two dimensional.
• Geometry Seminar - Alexander Veselov
  12th March 2010, 3:45pm to 4:45pm 4312 JCMB -- Show/hide abstract
  Abstract: SPECTRA OF SOL-MANIFOLDS, OR CAN ONE HEAR AN INDEFINITE BINARY QUADRATIC FORM ? - I will explain how the quantisation of the integrable geodesic flows on Sol-manifolds leads to the classical number-theoretic problem about the values of a given indefinite binary quadratic form. This talk is part of the Integrability and Number Theory meeting. More information is available at http://www.maths.ed.ac.uk/~hwb/iant.html.
• Geometry/Topology Seminar - Vicente Munoz
  18th March 2010, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: HODGE STRUCTURES ON MODULI SPACES - In this talk we will focus on moduli spaces of vector bundles over a complex curve, and on moduli spaces of pairs formed by a vector bundle and a global section. There is a concept of stability for pairs which depends on a real parameter. These moduli spaces suffer a birational transformation when the parameter crosses certain critical values. We shall see how properties of the algebraic structure of the mentioned moduli spaces (such as Hodge numbers, Hodge structures, Torelli theorems) can be deduced from a careful study of these birational transformations.
• Geometry/Topology Seminar - Jean-Paul Brasselet
  25th March 2010, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: PERVERSE SHEAVES AND TORIC VARIETIES - This a joint work with Gottfried Barthel, Karl-Heinz Fieseler and Ludger Kaup. We investigate minimal extension sheaves on arbitrary (possibly non-rational) fans as an approach toward a combinatorial ``virtual" intersection cohomology. As in ``classical" intersection cohomology, minimal extension sheaves are models for the pure objects of a ``perverse category". A decomposition theorem holds.
• Geometry Seminar - Rosa Sena-Dias
  6th May 2010, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: SCALAR-FLAT KAHLER METRICS ON NON-COMPACT TORIC SURFACES - In this talk we will discuss a new construction of scalar-flat Kahler toric metrics on non-compact 4-manifolds. The construction actually gives a new perspective on Gibbons-Hawking's so called gravitational instantons. Perhaps more interestingly, it allows us to write down some new examples of complete scalar-flat Kahler metrics on some important non-compact toric varieties. We will give some background on such metrics and show how symplectic toric geometry is well suited to tackle them. This is joint work with Miguel Abreu.
• EGA-AGS - Jihun Park (POSTECH)
  14th October 2010, 2:10pm to 3:00pm 6311 (note location) -- Show/hide abstract
  Abstract: SLOPES OF SMOOTH CURVES ON FANO MANIFOLDS
• EGA-AGS - Vladimir Guletskii (Liverpool)
  14th October 2010, 3:10pm to 4:00pm 6311 (note location) -- Show/hide abstract
  Abstract: NON-TRIVIAL ELEMENTS IN THE ABEL-JACOBI KERNELS OF HIGHER DIMENSIONAL VARIETIES
• Geometry Seminar - Geoff Robinson
  14th October 2010, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: LINEAR GROUPS, FINITE AND INFINITE
• Geometry Seminar - Roger Bielawski (Leeds)
  25th October 2010, 11:10am to 12:00pm 4312 -- Show/hide abstract
  Abstract: THE GEOMETRY OF HYPERBOLIC MONOPOLES
• Geometry Seminar - André Arroja Neves
  28th October 2010, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: RIGIDITY AND NON-RIGIDITY RESULTS FOR MANIFOLDS WITH POSITIVE SCALAR CURVATURE - The positive mass theorem, a cornerstone in Differential Geometry, says that metrics which are Euclidean outside a compact set and have nonnegative scalar curvature must be flat. I will talk about the state of the art regarding extensions of this result for metrics with positive scalar curvature. id: ugqAJep
• Geometry Seminar - Jason Lotay
  4th November 2010, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: ASSOCIATIVE SUBMANIFOLDS OF THE 7-SPHERE - In the 7-sphere, there is a distinguished class of minimal 3-dimensional submanifolds called associative 3-folds. They are connected with the G_2 geometry of S^7 and examples include Hopf lifts of holomorphic curves in CP^3 and minimal Legendrian submanifolds. I will describe the basic properties of associative submanifolds of the 7-sphere and discuss some classification results. id: MIF0Ed7p
• Geordie Williamson (Oxford) - Algebra+Geometry Seminar
  25th November 2010, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: INTRODUCTION TO SOERGEL BIMODULES - I will explain what Soergel bimodules are, and what they are useful for. This will cover some representation theory, link homology, and will be used to illustrate the emerging phenomenon of "categorification". If time permits I will describe work in progress with Ben Elias which presents the monoidal category of Soergel bimodules by generators and relations. id: ugFb0vhr
• Geometry Seminar - Florian Pokorny
  2nd December 2010, 3:10pm to 4:00pm JCMB Theatre C (note location) -- Show/hide abstract
  Abstract: THE BERGMAN KERNEL ON TORIC POLARIZED KAEHLER MANIFOLDS - I will try to explain why the Bergman Kernel is an interesting object to study and how its asymptotics link topological and geometric information about polarized Kaehler manifolds. I will talk about some toric results that I have been working on and will try to illustrate why "Bergman Kernels with vanishing" are of interest. id: 02-12-2010-3pm
• Geometry Seminar - Alcibiades Rigas
  2nd December 2010, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: EXAMPLES OF EXOTIC ACTIONS. - We exhibit simple formulas related to the Gromoll - Meyer construction of an exotic 7 sphere, that is, free actions of groups on very common spaces, such as spheres and Lie groups, whose quotient manifolds are not diffeomorphic / homeomorphic to the ones of standard such actions. We discuss some geometric considerations behind the motivation, such as geodesics and curvature, as well as elementary topological data that imply ample existence of such actions. Related problems will be discussed. id: 02-12-2010
• Geometry-Topology seminar- Tony Pantev (UPenn)
  24th February 2011, 4:10pm to 5:10pm JCMB -- Show/hide abstract
  Abstract: Title:Mirror symmetry for del Pezzo surfaces Abstract: I will discuss the general mirror symmetry question for del Pezzo surfaces in a setup that goes beyond the Hori-Vafa ansatz. I will describe the mirror map explicitly and will describe non-trivial consequences of homological mirror symmetry that can be proven directly. This is a joint work with Auroux, Katzarkov and Orlov.
• Geometry Seminar - Alexey Bolsinov
  11th March 2011, 4:10pm to 5:00pm 4312 JCMB -- Show/hide abstract
  Abstract: TBA id: 11-03-2011
• Geometry Seminar: Jacopo Stoppa
  12th May 2011, 4:30pm to 5:30pm Room 4312 JCMB -- Show/hide abstract
  Abstract: On the stability of extremal metrics
• Geometry Seminar - Hajo Hein
  27th May 2011, 12:00pm to 1:00pm -- Show/hide abstract
  Abstract: Speaker: Hans-Joachim Hein (Imperial College London) Title: Gravitational instantons from rational elliptic surfaces Abstract: Gravitational instantons are complete Ricci-flat Kaehler metrics on noncompact algebraic surfaces. I will show how to construct a family of such metrics on the complement of any fiber in any rational elliptic surface, and how to analyze their Riemannian geometry. This involves Kodaira's classical theory of singular fibers, as well as some new analysis for the complex Monge-Ampere equation.
• Geometry Seminar: Kaloghiros
  27th May 2011, 3:00pm to 4:00pm
• Geometry Seminar, Sergey Galkin ( IMPU, Tokyo)
  29th September 2011, 4:00pm to 5:00pm Room 5215 JCMB -- Show/hide abstract
  Abstract: Title: More symmetries of K3 Abstract: I will discuss different natural extensions of the holomorphic symmetry group of K3 surface (using moduli and derived category) and will show new interesting examples.
• Geometry Seminar: Michael Wemyss (Edinburgh)
  6th October 2011, 4:10pm to 5:10pm Room 5215 JCMB -- Show/hide abstract
  Abstract: Title: Noncommutative Minimal Models and applications to geometry. Abstract: I will try and explain the ideas behind noncommutative minimal models (=MMAs) and why they should not only reprove parts of the MMP in dimension three, but also give us extra information that currently the geometry does not "see". This extra information (in the form of a quiver) should then allow us to run aspects of the MMP in a much easier way. The talk will mainly be example based, but as an application of the homological techniques, I will give (in the dimension three Gorenstein setting) a characterization of the Q-factorial property in terms of derived categories. This is joint with Iyama.
• Geometry Seminar: Victor Przyjalkowski (Moscow+Vienna).
  13th October 2011, 4:10pm to 5:00pm 5215 JCMB -- Show/hide abstract
  Abstract: Title: Laurent Polynomials in Mirror Symmetry. Abstract: We discuss quantitative properties of Mirror Symmetry correspondence for Fano varieties. Laurent polynomials naturally appear in this picture. They describe (the essential part of) dual Landau--Ginzburg models for Fanos. They are related to toric degenerations of the initial Fano varieties. Their relative compactifications are candidates for Landau--Ginzburg models from the Homological Mirror Symmetry point of view. We consider our main example --- Fano threefolds. We discuss why Landau--Ginzburg model (for given Fano variety) represented by Laurent polynomial is unique and why it is not unique.
• Geometry Seminar: Sue Sierra (Edinburgh)
  20th October 2011, 4:10pm to 5:10pm Room 5215 JCMB -- Show/hide abstract
  Abstract: Title: Moduli spaces in graded ring theory Abstract: Let R be a noetherian N-graded algebra, generated in degree 1, over the complex numbers. A {\em point module} is a cyclic R-module with Hilbert series 1/(1-s). If R is strongly noetherian --- that is, it remains noetherian upon base extension --- then its point modules are parameterized by a projective scheme X, and this induces a canonical map from R to a twisted homogeneous coordinate ring on X. This technique has been very important in ring theory; for example, it was crucial in the analysis of noncommutative P^2's (regular algebras of dimension 3). We study an important non-strongly noetherian case: the noncommutative Rees rings known as naive blowup algebras. We show that there is a stack that represents (is a fine moduli space for) point modules, and that a certain equivalence relation on point modules is corepresented by a projective scheme. We then show that this geometry characterises naive blowup algebras, in a sense we make precise. This is joint work with Tom Nevins.
• Geometry Seminar: Costya Shramov (Moscow)
  27th October 2011, 4:10pm to 5:10pm Room 5215 JCMB -- Show/hide abstract
  Abstract: Title: Rationality of quotients by p-groups. Abstract: Let G be a p-group and V its complex representation. I'll survey the known results on rationality and stable rationality of V/G, in particular the obstructions to rationality. In addition to this seminar we will have 2 more talks at JCMB: See http://www.maths.gla.ac.uk/~anc/highlandcow/ 5:10-6:00PM, Diane Maclagan (Warwick), Room 5326 JCMB. Title: Tropical bounds on effective cones Abstract: Tropical geometry associates a polyhedral complex, called the tropical variety, to a subvariety X of a torus. The tropical variety is a "combinatorial shadow" of X which preserves many invariants of X and of a good compactification Y of X. In this talk I will explain how tropical techniques can give bounds on the nef and effective cones of a projective variety Y, and more generally on the cones of effective cycles of arbitrary dimension. 6:10-7:00PM, Will Donovan (Edinburgh), Room 5326 JCMB. Title: Spherical twists and brane transport Abstract: Interest in derived categories has been driven in recent years by connections with theoretical physics and birational geometry. For instance, given a suitable variety X we can perform a "flop" (a particular sort of birational map) to a new variety X', and we hope that this will correspond to an equivalence of the derived categories of the varieties X and X'. We will explain how such an equivalence can be seen in terms of "brane transport" between related string theories, and use this to produce new equivalences and derived automorphisms for the variety X, including spherical twists and their natural generalizations. After the last talk we will have an informal wine reception for those who survived all three talks :).
• Geometry Seminar: Ed Segal (Imperial)
  3rd November 2011, 4:10pm to 5:10pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Window-shifts and Grassmannian twists Abstract: 'Window-shifting' is a new technique for constructing equivalences and symmetries of derived categories of sheaves. I'll warm up by discussing how we apply it in the case of the standard flop, and show that it produces the Siedel-Thomas spherical twist. I'll then go on to our main example which is a vector bundle over a Grassmannian, in this case we can produce an autoequivalence which seems to be completely new. I'll then describe a more geometric construction of this autoequivalence. This is joint work with Will Donovan.
• Geometry Seminar: ANDRIY HAYDYS (Bielefeld)
  10th November 2011, 4:00pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Higher dimensional gauge theory and Fueter maps. Abstract: I will describe a generalization of the anti-self-duality equations for manifolds with exceptional holonomies due to Donaldson and Thomas. Under certain circumstances, higher dimensional instantons can degenerate to Fueter maps, which constitute a class of harmonic maps between hyperKaehler manifolds. I will describe some properties of Fueter maps and their relation to the compactified moduli space of higher dimensional instantons.
• Geometry Seminar: WILL DONOVAN (Edinburgh)
  17th November 2011, 4:10pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Derived autoequivalences of hyper-Kähler manifolds from Heisenberg categorification Abstract: The derived autoequivalence group of a variety is a much-studied object with links to representation theory, physics and the minimal model program, but little is known about its structure in general. We will discuss ongoing work to understand this group for hyper-Kähler manifolds, taking Hilbert schemes of K3 surfaces as an important special case. Recently Cautis-Licata exhibited a categorical Heisenberg algebra action on the derived category of the Hilbert scheme of points on certain surfaces, categorifying an action of Nakajima-Grojnowski on homology. The surfaces in question were the minimal resolutions of simple singularities: we will show how their work can be used as a starting point for the construction of interesting derived autoequivalences for Hilbert schemes of more general surfaces.
• Geometry Seminar: Dmitry Sakovich
  24th November 2011, 4:10pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Weakly-exceptional quotient singularities Abstract: The classification of A-D-E singularities on surfaces and their relation to Platonic solids and regular polygons is a very well-known classical result. I will discuss one of the possible generalizations of this idea into higher dimensions and some recent results on the classification of such singularities.
• Geometry Seminar: Joan Simon (Edinburgh)
  1st December 2011, 4:10pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title : Topology from cosmology Abstract : I will informally discuss how the Euler number of Calabi-Yau 3-folds (CY) can be constrained due to cosmological inflationary measurements in certain string theory compactification set-ups. The emphasis of the talk will be on the links between seemingly different disciplines rather than in a precise theoretical presentation.
• Geometry Seminar, Alex Kasprzyk (Imperial)
  8th December 2011, 4:10pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Small polygons and toric codes Abstract: I'll describe a special class of linear codes called toric codes, and sketch an attempt to systematically generate new "champion" codes from polygons. This is joint work with Gavin Brown.
• Edge Seminar: Rafael Torres (University of Oxford)
  19th January 2012, 4:10pm to 5:10pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Constructions of generalized complex structures in dimension four. Abstract: Recent constructions of exotic smooth structures on small 4-manifolds can be canonically used to expand our understanding of generalized complex structures. This talk will be an exposition of such an enterprise, whose produce include unbosoming unexpected phenomena on the number of type change loci of a generalized complex structure, and their construction on a myriad of 4-manifolds. http://www.maths.ed.ac.uk/cheltsov/seminar
• Edge Seminar: Sergey Grigorian (Stony Brook)
  26th January 2012, 4:10pm to 5:10pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Deformations of G2-structures with torsion. Abstract: We consider non-infinitesimal deformations of G2-structures on 7-dimensional manifolds and derive a closed expression for the torsion of the deformed G2-structure. We then specialize to the case where the deformation lies in the seven-dimensional representation of G2 and is hence defined by a vector v. In this case, we explicitly derive the expressions for the different torsion components of the new G2-structure in terms of the old torsion components and derivatives of v. In particular this gives a set of differential equations for the vector v which have to be satisfied for a transition between G2-structures with particular torsions. For some specific torsion classes we then explore the solutions of these equations.
• Edge Seminar: Damiano Testa (Warwick)
  2nd February 2012, 4:10pm to 5:10pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: The surface of cuboids and Siegel modular threefolds Abstract: A perfect cuboid is a parallelepiped with rectangular faces all of whose edges, face diagonals and long diagonal have integer length. A question going back to Euler asks for the existence of a perfect cuboid. No perfect cuboid has been found, nor it is known that they do not exist. In this talk I will first compute the Picard group of the space of cuboids (joint with M. Stoll). Then, I will show that the space of cuboids is a divisor in a Siegel modular threefold, thus allowing to translate the existence of a perfect cuboid to the existence of special torsion structures in abelian surfaces defined over number fields.
• Edge Seminar: Antony Maciocia (Edinburgh)
  9th February 2012, 4:10pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Computing the Walls for Bridgeland stability. Abstract: We show how the walls associated to Bridgeland stability for sheaves on surfaces can be constrained on certain planes in the stability manifold and we give some applications to answering finiteness problems for the family of walls and to describing moduli spaces of Bridgeland stable objects.
• Edge Seminar: Jesus Martinez-Garcia (Edinburgh)
  16th February 2012, 4:00pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Degeneration of del Pezzo surfaces after Hacking and Prokhorov. Abstract: This is a survey talk on the paper "Smoothable del Pezzo surfaces with quotient singularities" by Paul Hacking and Yura Prokhorov. See http://arxiv.org/abs/0808.1550
• Edge seminar: José Miguel Figueroa-O'Farrill (Edinburgh)
  1st March 2012, 4:00pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Why we like homogeneous manifolds Abstract: A homogeneous manifold is a manifold admitting a transitive action of a Lie group. They are some of the nicest manifolds one could work with, as on them differential geometry reduces to algebra. Geometers like them because they are natural manifolds on which to construct interesting (e.g., Einstein) metrics. Physicists like them for much the same reasons: algebraic equations are easier to solve than PDEs. I will give a general overview of the emergence of homogeneous geometries in one area of research close to my heart: supergravity, an extension of Einstein's general theory of relativity tracing its origins to early attempts (circa 1975) to unify gravity and quantum mechanics. I only mention this to put the theory in context: I will not assume any prior knowledge of physics.
• Edge Seminar: Kuzma Khrabrov (Moscow)
  8th March 2012, 4:00pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Graded rings and Q-Fano threefolds with exotic Weil class group. Abstract: Kuzma is an undergraduate from Moscow State University. He is our prospective postgraduate student. He will speak about his fifth year project at Moscow State. This talk can be considered as a part of his interview.
• Edge Seminar: Michael Singer (University of Edinburgh)
  29th March 2012, 4:10pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Partial Bergman kernels and toric K-stability Abstract: In the study of Kaehler metrics of constant scalar curvature, the basic conjecture is that the existence of such metrics should be equivalent to the algebraic-geometric condition of K-(poly)stability on the underlying projective variety M. The `easy' version of this conjecture (cscK => K-stable) has been proved by Donaldson and Stoppa (at least if there are no non-zero holomorphic vector fields on M) while the converse direction is a hard unsolved problem. In recent work with Florian Pokorny, we have obtained a new proof of cscK => K-stable for toric projective varieties. Our proof puts into practice an initial suggestion of Richard Thomas to use partial Bergman kernels to provide a bridge between the worlds of differential geometry and algebraic geometry.
• Edge Seminar: Julius Ross (Cambridge)
  25th May 2012, 2:30pm to 3:20pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Local Moment Maps associated to Okounkov bodies Abstract: The Okounkov body can be thought of as an generalisation of the polytope associated to a projective toric variety. Starting with a line bundle L on an n-dimensional variety X and a flag of smooth subvarieties in X, the Okounkov body is a convex subset Delta in R^n, whose volume (with respect to the Lebesgue measure) is the volume of X (with respect to the line bundle L). This property forms the basis of work by Lazarsfeld-Mustata who use Okounkov bodies to study the volume functional on the space of big divisors. In this talk I will start with a general discussion of Okounkov bodies, along with some examples. I shall then describe ongoing work with David Witt-Nystrom which constructs a local version of a moment map in this setting connecting with the complex monge ampere equation.
• Edge Seminar: Hugues Auvray (Paris 6)
  31st May 2012, 4:10pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Uniqueness and obstructions to existence of constant scalar curvature Kähler metrics: the quasi-projective case Abstract: Let X be a compact Kähler manifold, and D a divisor with simple normal crossings in X. We consider on X\D a class of Kähler metrics with cusp singularities along D. We solve the equation of geodesics joining metrics in this class. We apply this resolution to a uniqueness result of a possible metric with constant scalar curvature among the considered class. We will moreover evocate some topological constraints involved by the existence of a constant scalar curvature metric in the class we consider, related to a conjecture by G. Székelyhidi and his notion of K-stability for the pair (X,D).
• GEOMETRY/TOPOLOGY SEMINAR (M.Hering, A.Bayer)
  20th September 2012, 4:00pm to 6:00pm Room 4312, JCMB -- Show/hide abstract
  Abstract: Milena Hering (Edinburgh) "Cox rings of toric bundles" Abstract: Section rings of arbitrary line bundles on toric varieties are polytopal semigroup rings and thus always finitely generated. A related question is whether the section ring of the Serre line bundle on the projectivization of a toric vector bundle is always finitely generated. It turns out that this is not the case. We show this by finding toric vector bundles whose Cox ring is isomorphic to the Cox ring of the blow up of points in projective space. The latter is well known not to be finitely generated in general. This is joint work with José González, Sam Payne and Hendrik Süss. Arend Bayer (Edinburgh) "Wall-crossing and birational geometry".
• Edge Seminar: Jason Lotay (UCL)
  4th October 2012, 4:10pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Uniqueness of Lagrangian self-expanders Abstract: In Mean Curvature Flow an important class of solutions are the self-expanders, which move simply by dilations under the flow. Self-expanders provide models for smoothing of singular configurations and are analogues of minimal submanifolds. I will show that two-dimensional Lagrangian self-expanders asymptotic to pairs of planes are unique. This is joint work with André Neves.
• Edge Seminar: Jonathan Pridham (Cambridge), "Semiregularity and reduced obstruction theories"
  11th October 2012, 4:10pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Semiregularity and reduced obstruction theories Abstract: Obstruction theories measure the singularities of a moduli space, giving rise to notions of virtual dimension, and virtual numbers of points. Bloch defined a semiregularity map on the obstruction space of any LCI embedding in a smooth scheme X. I will indicate how this map is related to the Goodwillie-Jones Chern character in K-theory. As a consquence, it annihilates all obstructions, giving smaller obstruction theories.
• Edge Seminar: Andrei Trepalin (HSE Moscow)
  18th October 2012, 4:10pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Rationality of quotients of the projective plane over non-algebraically closed fields Abstract: TBA.
• Edge Seminar: Charles Vial (Cambridge)
  25th October 2012, 4:10pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Non-commutative resolutions and Grothendieck groups Abstract: This is joint work with Hailong Dao, Osamu Iyama and Ryo Takahashi. A finitely generated module M over a commutative noetherian ring R is said to give a non-commutative resolution (NCR) of R if M is faithful and End_R(M) has finite global dimension. The aim of this talk is to discuss the relevance of such a definition and to give necessary conditions for the existence of NCRs. These conditions focus on the Grothendieck group of the category of finitely generated modules over R and its subcategories. This group is related, via Riemann-Roch, to the group of so-called algebraic cycles. I will explain how methods from the theory of algebraic cycles can be used in that setting and I will show that a standard graded Cohen-Macaulay algebra R over a field of zero characteristic with only rational singularities outside the irrelevant ideal has a NCR only if R has rational singularities
• Edge Seminar: Alastair Craw (Glasgow)
  1st November 2012, 4:10pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: Mori Dream Spaces as fine moduli of quiver representations Abstract: Mori Dream Spaces provide a relatively large class of examples in algebraic geometry that nevertheless exhibit many beautiful properties, the most obvious being that all operations of Mori Theory can be carried out by variation of GIT quotient. In particular, every Mori Dream Space can be constructed as a GIT quotient in a natural (though not canonical) way. In this talk I will introduce alternative GIT constructions using noncommutative algebra. As an application, we refine a result of Bergman-Proudfoot by showing that del Pezzo surfaces can be reconstructed from a tilting bundle
• EDGE seminar: Dave Anderson (IMJ, Université Paris 7)
  8th November 2012, 4:10pm to 5:10pm JCMB 4312 -- Show/hide abstract
  Abstract: Title: Pfaffian formulas for symplectic degeneracy loci Abstract: Many interesting varieties arise as degeneracy loci: the set of points where a map of vector bundles drops rank, or equivalently, the set of points where two vector bundles intersect more than necessary. The problem of finding formulas for the cohomology classes of these loci dates to the 19th century, but has experienced a surge of interest in the last few decades. The answer will be a universal polynomial in the Chern classes of the vector bundles involved, and is closely related to the equivariant classes of Schubert varieties in G/B, where G is semisimple algebraic group. I'll describe recent progress in understanding these polynomials in classical types, including joint work with William Fulton.
• No Edge Seminar (Colloquium)
  22nd November 2012, 4:00pm to 5:00pm
• Edge Seminar: Yuji Odaka (Imperial)
  27th November 2012, 4:10pm to 5:00pm 6206 JCMB -- Show/hide abstract
  Abstract: Title: Towards algebro-geometric understanding of K-stability of Fano varieties. Abstract: The K-stability of Fano manifold was introduced by Tian in 90s and reformulated, generalized in more algebraic way by Donaldson. It is a (conjectural) counterpart of existence of Kahler-Einstein metrics which gives a "unique" way to regard Fano variety as Kahler (thus Riemannian) manifold, which the speaker has been expected to give application to construction of projective moduli variety of Fano varieties and even more general polarized varieties ("K-moduli") as well. However, for a given Fano manifold, it is actually hard to see it is K-stable or not in general, even for simple examples. This is the main focus of the talk. Reviewing the developments so far, I would like to discuss for future developments.
• Edge Seminar: Timothy Logvinenko (Warwick)
  29th November 2012, 4:10pm to 5:00pm Room 4312 JCMB -- Show/hide abstract
  Abstract: Title: TBA Abstract: TBA.
• EDGE seminar: Victor Lozovanu
  7th February 2013, 4:10pm to 5:10pm JCMB 4213 -- Show/hide abstract
  Abstract: Title: An extension of Kodaira vanishing in arbitrary codimension Abstract: One of the most celebrated theorems in complex algebraic geometry is Kodaira vanishing, together with its extension due to Kawamata and Viehweg. In this talk I will discuss about a joint work with Greg G. Smith, where we generalize Kodaira vanishing to arbitrary codimension. I will present a few applications to this work; by giving answers to questions of projective normality and bounding the multigraded regularity. I will also discuss some future projects that derive nicely from this work.
• EDGE seminar: Tom Bridgeland (Oxford)
  28th February 2013, 4:10pm to 5:10pm JCMB 4213 -- Show/hide abstract
  Abstract: Title : Cluster varieties and stability conditions Abstract : I will begin in a very elementary way by explaining the operation of mutation of quivers, and using it to define two complex manifolds, one of them being the cluster variety, and the other being a baby version of the space of stability conditions. I will then describe what these spaces look like for a class of quivers related to triangulations of surfaces, and point out the connection with moduli spaces of Higgs bundles and the Hitchin map. I will finish by discussing some ideas of physicists Gaiotto, Moore and Neitzke which suggest that these examples should fit into a more general framework.
• EDGE seminar: John Ottem (Cambridge): Ample subschemes and partially ample line bundles
  7th March 2013, 4:10pm to 5:10pm JCMB 4213 -- Show/hide abstract
  Abstract: Title: Ample subschemes and partially ample line bundles Abstract: We introduce a notion of ampleness for subschemes of any codimension. We show that this gives a satisfactory generalization of ample divisors by proving analogues of classical theorems for ample divisors, e.g., the Lefschetz hyperplane theorems and numerical positivity. Using these properties, we also construct a counterexample to the converse of the Andreotti-Grauert vanishing theorem.
• Emma Praviato (Boston University)
  12th March 2013, 2:00pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Title: The cavalcade of Poncelet's theorem Abstract: This talk will review results of Poncelet type that involve special functions and Hamiltonian systems.
• EDGE seminar: Ugo Bruzzo (Trieste)
  14th March 2013, 4:10pm to 5:00pm JCMB 4213 -- Show/hide abstract
  Abstract: Title: Noether-Lefschetz theory for hypersurfaces in toric 3-folds Abstract: We prove a sufficient criterion for a very general hypersurface in simplicial toric 3-fold to have the same Picard number as the ambient variety. We also give some results about the estimate of the codimension of the locus where the Picard number is bigger.
• EDGE seminar: Daniel Greb (Bochum)
  14th March 2013, 5:10pm to 6:00pm JCMB 4213 -- Show/hide abstract
  Abstract: Title: Compact moduli spaces for slope-semistable sheaves Abstract: While the variation of moduli spaces of H-slope/Gieseker-semistable sheaves on surfaces under change of the ample polarisation H is well-understood, research on the corresponding question in the case of higher-dimensional base manifolds revealed a number of pathologies. After presenting these, I will discuss recent joint work with Matei Toma (Nancy), which resolves some of these pathologies by looking at curves instead of divisors. This naturally leads to the question whether higher-dimensional analogues of the Donaldson-Uhlenbeck compactification exists, and I will discuss our construction of such a compactification.
• Vassily Gorbounov (Aberdeen): Quantum cohomology and quantum groups.
  19th March 2013, 4:00pm to 5:00pm JCMB 6311 -- Show/hide abstract
  Abstract: This is a joint Maximals-Geometry seminar. Title: Quantum cohomology and quantum groups. Abstract: We describe the recent research started by Nekrasov Shatashvilly Braverman Maulik Okounkov on correspondence between the quantum cohomology of the quiver varieties and the quantum integrable systems. Our main example will be the cotangent spaces to partial flag varieties.
• EDGE seminar: Michael Thaddeus (Columbia)
  4th April 2013, 4:10pm to 5:00pm JCMB 4213 -- Show/hide abstract
  Abstract: Title: A tale of two compactifications Abstract: We will examine two compactifications of reductive groups with very attractive properties, namely the wonderful compactification of PGL(n,C) and the Kausz compactification of GL(n,C). We will explore their connections to each other and to moduli problems in several corners of mathematics: spaces of broken flows in Morse theory, spaces of stable maps to Grassmannians, and spaces of coherent sheaves on singular Riemann surfaces. Then we will see how parts of this circle of ideas have been generalized, in joint work with Johan Martens, to arbitrary reductive groups.
• Maria Angelica Cueto (Frankfurt)
  8th May 2013, 4:00pm to 5:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Title: Mixed discriminants Abstract: The mixed discriminant of n Laurent polynomials in n variables with fixed support is the irreducible polynomial in the coefficients which vanishes whenever two of the roots coincide. It represents the variety of ill-posed systems. By means of the Cayley trick, we can express the mixed discriminant as an A-discriminant in the sense of Gelfand, Kapranov and Zelevinski. Our goal is to characterize its degree. I will discuss in detail the case of two plane curves where an explicit degree formula can be provided. In the case of two dense polynomials, this formula recovers the classical tact invariant of Salmon. Finally, inspired by the tropical approach to computing A-discriminants, I will show that the degree of the mixed discriminant is a piecewise linear function in the Plücker coordinates of a mixed Grassmannian. This is joint work with E. Cattani, A. Dickenstein, S. Di Rocco and B. Sturmfels.
• Nuno M Romao (HIM, University of Bonn)
  9th May 2013, 3:00pm to 4:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Title: "Supersymmetric quantum mechanics on vortex moduli spaces" Abstract: I will report on an ongoing project aiming at uncovering fundamental features of N=(2,2) supersymmetric quantum mechanics on moduli spaces of vortices on compact Riemann surfaces, in analogy with the spectrum of quantum dyon-monopole bound states that emerged in connection with Sen's S-duality conjectures in the 1990s. My focus in this talk will be on the (topological A-twisted) supersymmetric Abelian Higgs model coupling to local systems, with both linear and nonlinear targets; the corresponding ground states can be investigated by means of the theory of L^2-invariants. I shall explain why the quanta of such Abelian gauge theories can nontrivially realize non-Abelian statistics, and motivate a conjecture regarding the nonlinear superposition of ground states.
• Workshop on "Integrability and Hyperbolic Monopoles"
  10th May 2013, 10:00am to 5:00pm ICMS -- Show/hide abstract
  Abstract: http://www.maths.ed.ac.uk/~hwb/ihm.html
• Paul Xin Zhou (Michigan)
  4th June 2013, 4:00pm to 5:00pm JCMB 4312 -- Show/hide abstract
  Abstract: Title: Schur asymptotics of Veronese syzygies Abstract: The syzygies of Veronese embeddings are functors in the underlying vector space. We are interested in their decompositions as Schur functors. We are able to show that they have very rich structures in various asymptotic situations. In particular, by a convex geometric inspired study of the decomposition of \otimes^p Sym^d and various plethysms, we give an exact estimate of the scale of growth of the number of distinct Schur functors and the total number counting multiplicities when we look at a fixed syzygy functor and let the degree of the embedding increase. In this talk, I will discuss this case, the Schur asymptotics of the p-th syzygy functor fixing p and letting degree of the Veronese go to infinity. We will deduce the conclusion from the numerics of asymptotic plethysms.
• Alasair Craw (Bath)
  4th June 2013, 5:00pm to 6:00pm JCMB 4312 -- Show/hide abstract
  Abstract: Title: Geometric Reid's recipe for dimer models Abstract: A consistent dimer model on a real 2-torus determines a CY3 algebra A whose centre defines a Gorenstein toric threefold X. Results of Ishii-Ueda show that every projective crepant resolution of X can be constructed as a fine moduli space of A-modules such that the tautological bundle T on the moduli space is a tilting bundle. I'll explain what is known about the image of the vertex simple A-modules under a derived equivalence induced by T for the moduli space defined by a special stability parameter. When A is the skew group algebra of a finite abelian subgroup G in SL(3,C), this moduli space is the G-Hilbert scheme, and our results extend work of Cautis-Logvinenko and Logvinenko on 'Geometric Reid's recipe'. Our proof uses variation of GIT quotient and is rather different from the original work of Cautis and Logvinenko. This is joint work with Raf Bocklandt and Alex Quintero Velez.
• EDGE seminar: Hendrik Süß (Edinburgh) - joint with MAXIMALS
  9th September 2013, 4:00pm to 5:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Title: Equivariant vector bundles on T-Varieties Abstract: By Klyachko's work there is an equivalence of categories between equivariant vector bundles on toric varieties and families of vector space filtrations. In this talk I will discuss an generalization of this equivalence to bundles on varieties with smaller torus actions. Now, vector space filtrations are replaced by filtrations of vector bundles on some quotient space. This description comes with a nice splitting criterion and allows to prove that vector bundles of low rank on projective space, which are equivariant with respect to special subtori of the maximal acting torus must split.
• EDGE seminar: Francois Petit (Edinburgh) - joint with MAXIMALS
  9th September 2013, 5:00pm to 6:00pm JCMB 6311
• EDGE seminar: Evgeny Shinder (Edinburgh) - Exceptional collections on fake projective planes
  19th September 2013, 3:00pm to 4:00pm JCMB 5215 -- Show/hide abstract
  Abstract: See http://hodge.maths.ed.ac.uk/tiki/EDGE
• EDGE seminar: Ciaran Meachan - Derived autoequivalences of hyperkähler varietiesP functors are a natural generalisation of Huybrechts and Thomas' P objects and as such, they determine autoequivalences of the codomain category. We will report on the latest developments in this direction when the codomain category is the derived category of coherent sheaves on a hyperkähler variety
  19th September 2013, 4:00pm to 5:00pm JCMB 5215 -- Show/hide abstract
  Abstract: See http://hodge.maths.ed.ac.uk/tiki/EDGE
• Costya Shramov (HSE Moscow)
  26th September 2013, 3:00pm to 4:00pm JCMB 5326
• Maxwell colloquium on Combinatorial Algebraic Geometry
  4th October 2013, 9:30am to 6:00pm ICMS -- Show/hide abstract
  Abstract: http://www.maths.ed.ac.uk/~mhering/Maxwell/index.html
• EDGE seminar: Burt Totaro (Cambridge/UCLA)
  10th October 2013, 3:00pm to 4:00pm JCMB 6311 -- Show/hide abstract
  Abstract: See http://hodge.maths.ed.ac.uk/tiki/EDGE#Current_Semester
• EDGE seminar:Yuri Fedorov (UPC Barcelona)
  17th October 2013, 3:10pm to 4:00pm JCMB 6311 -- Show/hide abstract
  Abstract: See http://hodge.maths.ed.ac.uk/tiki/EDGE#Current_Semester
• EDGE seminar: Jihun Park (Pohang)
  24th October 2013, 3:10pm to 4:00pm JCMB 6311 -- Show/hide abstract
  Abstract: See http://hodge.maths.ed.ac.uk/tiki/EDGE
• EDGE seminar: Vasily Golyshev
  31st October 2013, 3:10pm to 4:00pm JCMB 6311 -- Show/hide abstract
  Abstract: See http://hodge.maths.ed.ac.uk/tiki/EDGE
• EDGE seminar: Gregory Sankaran (Bath)
  7th November 2013, 3:10pm to 4:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Title: Stable homology of toroidal compactifications
• EDGE seminar: Johan Martens
  14th November 2013, 3:10pm to 4:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Title: TBA
• EDGE seminar: Will Donovan
  21st November 2013, 3:10pm to 4:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Title: TBA
• EDGE seminar: Jonathan Pridham
  21st November 2013, 4:10pm to 5:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Title: TBA
• EDGE seminar: Francesco Sala (HWU)
  16th January 2014, 3:10pm to 4:00pm JCMB 4312
• EDGE seminar: Diane Maclagan (Warwick)
  23rd January 2014, 4:00pm to 5:00pm JCMB 5326 -- Show/hide abstract
  Abstract: Tropical schemes, tropical cycles, and valuated matroids
• EDGE seminar: Alastair King (Bath)
  30th January 2014, 3:10pm to 4:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Title: Grassmannian cluster categories and dimers on a disc
• EDGE/MAXIMALS seminar: Pierre Schapira (Paris VI)
  6th February 2014, 3:00pm to 5:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Title: Microlocal Euler classes and Hochschild cohomology
• EDGE seminar: Adam Boocher (Edinburgh)
  13th February 2014, 3:10pm to 4:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Title: Closures of a linear space
• EDGE seminar:Tara Holm (Cornell)
  6th March 2014, 3:10pm to 4:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Title: TBA
• EDGE seminar: Paolo Stellari (Milano)
  13th March 2014, 3:10pm to 4:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Title: TBA
• Vladimir Baranovsky (UC Irvine)
  20th March 2014, 4:00pm to 5:00pm JCMB 5325 -- Show/hide abstract
  Abstract: Title: Deformation quantization of smooth Lagrangian subvariety Abstract: See http://hodge.maths.ed.ac.uk/tiki/EDGE
• HODGE SEMINAR (ETIENNE GHYS)
  21st March 2014, 3:00pm to 4:00pm TBA
• EDGE seminar: Nicolò Sibila (MPI Bonn)
  27th March 2014, 4:00pm to 5:00pm JCMB 5325 -- Show/hide abstract
  Abstract: See http://hodge.maths.ed.ac.uk/tiki/EDGE
• EDGE: Klaus Altmann (FU Berlin)
  3rd April 2014, 3:10pm to 5:00pm JCMB 5325 -- Show/hide abstract
  Abstract: See http://hodge.maths.ed.ac.uk/tiki/EDGE
• EDGE seminar: Sam Payne (Yale)
  21st May 2014, 11:30am to 12:30pm JCMB 5205
• EDGE seminar: Jason Lo (MPI Bonn)
  27th May 2014, 2:00pm to 3:00pm JCMB 5215
• EDGE seminar: Ziyu Zhang (Bath)
  27th May 2014, 3:00pm to 4:00pm JCMB 5215
• EDGE seminar: Nathan Ilten (Berkeley)
  12th June 2014, 2:00pm to 3:00pm JCMB 6311
• EDGE seminar: Dave Anderson (IMPA)
  16th June 2014, 3:00pm to 4:00pm JCMB 6206
• EDGE seminar: Pablo Solis (Berkeley)
  18th June 2014, 3:00pm to 4:00pm JCMB 5215
• EDGE seminar: Emanuele Macri (Ohio State)
  29th July 2014, 2:00pm to 3:00pm JCMB 5215
• EDGE seminar: Ivan Cheltsov
  25th September 2014, 2:10pm to 3:00pm JCMB 6311
• EDGE seminar: Martin Kalck
  2nd October 2014, 2:10pm to 3:00pm JCMB 6311
• EDGE seminar: Brent Prym (Oxford)
  9th October 2014, 2:10pm to 3:00pm JCMB 6311
• EDGE seminar: Balazs Szendroi (Oxford)
  16th October 2014, 2:10pm to 3:00pm JCMB 6311
• EDGE seminar: Evgeny Smirnov (HSE Moscow)
  23rd October 2014, 2:10pm to 3:00pm JCMB 6311
• EDGE seminar: Chunyi Li (Edinburgh)
  30th October 2014, 2:10pm to 3:00pm JCMB 6311
• EDGE seminar: Gavin Brown (Loughborough)
  6th November 2014, 2:10pm to 3:00pm JCMB 5215 -- Show/hide abstract
  Abstract: See http://hodge.maths.ed.ac.uk/tiki/EDGE for details.
• EDGE seminar: Sue Sierra (Edinburgh)
  13th November 2014, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: See http://hodge.maths.ed.ac.uk/tiki/EDGE for details.
• EDGE seminar: Georg Oberdieck (Zurich)
  20th November 2014, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: See http://hodge.maths.ed.ac.uk/tiki/EDGE for details.
• EDGE seminar: Cristina Manolache (Imperial)
  11th December 2014, 3:00pm to 4:00pm JCMB 6206 -- Show/hide abstract
  Abstract: See http://hodge.maths.ed.ac.uk/tiki/EDGE for details.
• EDGE: Michael Wemyss
  15th January 2015, 2:10pm to 3:00pm JCMB 6311
• EDGE seminar: Mario García Fernández (ICMAT)
  5th February 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Stability data, irregular connections and tropical curves Abstract: I will give an overview of recent joint work with S. Filippini and J. Stoppa, in which we construct isomonodromic families of irregular meromorphic connections on P1, with generalized monodromy in the automorphisms of a class of infinite-dimensional Poisson algebras. Our main results concern the limits of the families as we vary a scaling parameter R. In the R → 0 “conformal limit” we recover a semi-classical version of the connections introduced by Bridgeland and Toledano Laredo (and so the Joyce holomorphic generating functions for DT invariants). In the R → infinity “large complex structure limit” the families relate to tropical curves in the plane and tropical/GW invariants. The connections we construct are a rough but rigorous approximation to the (mostly conjectural) four-dimensional tt*-connections introduced by Gaiotto–Moore–Neitzke.
• EDGE seminar: Alice Rizzardo (Edinburgh)
  12th February 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Title: An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves Abstract: Orlov’s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. We will show that this result is false without the fully faithfulness hypothesis. This is joint work with Michel Van den Bergh.
• EDGE seminar: Sergey Arkhipov (Aarhus)
  26th February 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Title: Quasi-coherent Hecke categories and affine braid group actions. Abstract: We propose a geometric setting leading to categorical braid group actions. First we consider the quasi-coherent Hecke category QCHecke(G,B) for a reductive group G with a Borel subgroup B. We show that a monoidal action of QCHecke(G,B) on a triangulated category gives rise to a categorification of degenerate Hecke algebra representation known as Demazure Descent Data. Next we replace the group G by the derived group scheme LG of topological loops with values in G and consider QCHecke(LG,LB). A monoidal action of the category QCHecke(LG,LB) gives rise to a categorical action of the affine Braid group. Trying to avoid heavy derived algebraic geometry methods, we present an example of the construction above in classical algebro-geometric terms. For a G-variety X, we construct a Braid group action on a category of equivariant matrix factorizations on the product of T∗X and the Grothendieck variety for the Lie algebra of G. The potential for matrix factorizations is provided by the moment map.
• EDGE seminar: Michael Groechenig (Cambridge)
  12th March 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Infinite-dimensional vector bundles and reciprocity Abstract: Following Beilinson, Kato, and Drinfeld we study certain infinite-dimensional vector bundles (named after Tate), through the lens of exact categories. The simplest non-trivial example is the vector space of formal Laurent series k((t)) with the t-adic topology. The K-theory space of this exact category of Tate bundles on a variety, can be identified with the K-theory of the variety up to shift, by a construction reminiscent of Kapranov's determinantal theories. As an application of this theory we give a K-theoretic treatment of higher-dimensional analogues of Contou-Carrère symbols, and use this viewpoint to establish reciprocity laws for those symbols. This is joint work with O. Braunling and J. Wolfson.
• EDGE seminar: Joseph Karmazyn (Edinburgh)
  26th March 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Moduli, McKay, and Minimal Models Abstract: Moduli spaces are often used to realise derived equivalences in algebraic geometry. I will recall the examples of the derived equivalence of 3-fold flops in the minimal model program and the derived SL2 McKay correspondence. These derived equivalences can be translated to noncommutative algebra where they have been extended to include more general settings. I will discuss how a moduli interpretation can be extended to include these derived equivalences with noncommutative algebras.
• EDGE seminar: Rebecca Tramel (Edinburgh)
  26th March 2015, 3:10pm to 4:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Bridgeland stability on surfaces with curves of negative self-intersection Abstract: I consider X a smooth projective complex surface containing a curve C whose self-intersection is negative. In 2002, Bridgeland defined a notion of stability for the objects in Db(X), which generalized the notion of slope stability for vector bundles on curves. The space of such stability conditions is a complex manifold, Stab(X). If we fix a numerical class, then we can decompose Stab(X) into open chambers where the moduli space of stable objects of this class remains constant, and codimension one walls where this moduli space may change. I consider objects in Db(X) whose numerical class is that of OX the skyscraper sheaf of a point. In the geometric chamber of Stab(X), the moduli space of stable objects of this class is X itself. In 2012 Toda showed that if C has self-intersection -1, there is a wall to the geometric chamber along which the points of C become semistable. I generalize this result to find such a wall when C has arbitrary negative self-intersection. I then describe the moduli space of stable objects whose class is that of OX after wall-crossing.
• EDGE seminar: Tyler Kelly (Cambridge)
  23rd April 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Equivalences of (Stacky) Calabi-Yaus in Toric Varieties Abstract: Given Calabi-Yau complete intersections in a fixed toric variety, there are possibly various constructions to compute its mirror. Sometimes these mirrors are isomorphic but sometimes not. Mirror symmetry predicts a relationship amongst these so-called double mirrors. In this talk, we will show that the stacky versions of these varieties are derived equivalent. In the proving of this theorem, we get some applications which involve polarisations of K3 surfaces, special degenerate families of CY hypersurfaces in toric varieties, and a generalization of the BHK mirror constructions to families.
• Structure and Symmetry Symposium: Arkady Vaintrob (Oregon)
  18th May 2015, 1:30pm to 2:30pm JCMB 6206 -- Show/hide abstract
  Abstract: Noncommutative spaces, singularities and mirror symmetry. Abstract: I will explain a point of view according to which polynomials with isolated singularities may be viewed as noncommutative spaces as nice as smooth projective varieties. This leads to interesting geometric invariants of singularities, both classical (analogs of de Rham cohomology and Chern classes) and quantum (analogs of Gromov-Witten invariants and quantum cohomology), and to new kinds of mirror symmetry. The talk is based on joint work with Sasha Polishchuk.
• Structure and Symmetry Symposium: Ivan Smith (Cambridge)
  18th May 2015, 2:30pm to 3:30pm JCMB 6206 -- Show/hide abstract
  Abstract: Title: Symplectic Khovanov cohomology Abstract: I will explain a symplectic geometry approach to Khovanov's combinatorial invariant of links in three-dimensional space, which involves studying the symplectic topology of certain spaces arising both in Lie theory and as Hilbert schemes of Milnor fibres of surface singularities. At a technical level, the geometric definition relies on a formality theorem which is a kind of non-commutative version of the classical formality of a Kahler manifold. The talk reports on joint work with Mohammed Abouzaid.
• EDGE seminar: Daniel Halpern-Leistner (Columbia)
  28th May 2015, 2:00pm to 3:00pm JCMB 6206 -- Show/hide abstract
  Abstract: Equivariant Hodge theory Abstract: Recent results have revealed a mysterious foundational phenomenon: some quotient stacks for algebraic groups G acting on non-proper schemes X still behave as if they are proper schemes. I will report on one instance of this yoga: one can consider the non-commutative Hodge-to-de-Rham sequence, from Hochschild homology to periodic cyclic homology, for the derived category of coherent sheaves. This spectral sequence degenerates on the first page for smooth and proper schemes, and it turns out that this degeneration also occurs for many "cohomologically proper" quotient stacks. With a little work, this leads to a canonical weight 0 Hodge structure on the Atiyah-Segal equivariant K-theory of the complex analytification of X. The associated graded of the Hodge filtration is the space of functions on the "derived loop space" of the stack.
• 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
• 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: 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: Dmitri Orlov (Steklov) - Derived noncommutative schemes, their geometric realizations, and quiver algebras
  8th October 2015, 3:10pm to 4:00pm JCMB 6311
• 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 categor​ies 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: 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: 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.
• 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 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.
• 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: 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: 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: 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: 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: 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.
• 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: 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.
• 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: Caucher Birkar(Cambridge)- Birational geometry of Fano varieties
  19th January 2017, 2:10pm to 3:00pm JCMB 1501
• 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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/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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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.
• 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: 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.
• 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: 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: 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: 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: Andrea Fanelli (Dusseldorf) - Del Pezzo fibrations in positive characteristic
  15th March 2018, 2:05pm to 2:55pm JCMB Seminar Room 5323 -- Show/hide abstract
  Abstract:
  

  In this talk, I will discuss some pathologies for the generic fibre of del Pezzo fibrations in characteristic p>0, motivated by the recent developments of the MMP in positive characteristic. The main application of the joint work with Stefan Schröer concerns 3-dimensional Mori fibre  spaces.

• EDGE: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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 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.
• 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
• 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: 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: 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
• 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.
• 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 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: 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: 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: 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.
• MAXIMALS Doubleheader
  9th October 2018, 2:00pm to 4:30pm
• 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.
• 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 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: 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: 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: 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.
• ICMS workshop: Constructions and obstructions in birational geometry (NO EDGE)
  6th November 2018, 3:30pm to 4:30pm Bayes Centre, 5th floor
• 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.
• 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.
• 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).
• 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: 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.
• 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: 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: 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: 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: 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.
• 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: 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.
• 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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.
• 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: 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.
• 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: 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: 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: 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.
• 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 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 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
  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
  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
  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: 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 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.
• 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: 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.
• 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: 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: 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: 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: 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: 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.
• 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.
• 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)
• 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: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: 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: 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.
• 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.
  
• 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.
• 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: 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.
• 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
• 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: 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: 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.
• 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: 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. Talk notes
• Web Hodge: Georg Oberdieck (University of Bonn)-Moduli spaces in equivariant categories
  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)-Hypergeometric functions and new mirrors of Fano varieties
  11th November 2020, 1:15pm to 3:00pm -- Show/hide abstract
  Abstract: I will outline a strategy to exhibit the hypergeometric function of a Fano weighted complete intersection X of dimension n as a period of a pencil of (n-1)-dimensional varieties. Since conjecturally this function encodes the quantum period of X, these ideas can be viewed as the basis of a new method to find Landau-Ginzburg (LG) mirrors for all Fano weighted complete intersections. I will show how to use the strategy to produce LG mirrors for the Johnson-Kollar surfaces X_{8k+4} in P(2,2k+1,2k+1,4k+1). The main feature of these surfaces, which makes the construction especially interesting, is that they have empty anticanonical linear system, and therefore fall out of any known mirror construction. This is joint work with A. Corti. Talk notes
• Web Hodge: 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. Talk notes
• 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. Talk notes
• Web Hodge: Cheuk Yu Mak (Edinburgh)-Symplectic annular Khovanov homology
  25th November 2020, 1:15pm to 3:00pm -- Show/hide abstract
  Abstract: Annular Khovanov homology is an invariant of annular links (links in a solid torus) introduced by Asaeda-Przytycki-Sikora as an analogue of Khovanov homology for links. Auroux-Grigsby-Wehrli showed that the first non-trivial piece of the annular Khovanov homology can be identified with the Hochschild homology of the Fukaya-Seidel category of A_n Milnor fibers with coefficients in braid bimodules. In this talk, we will introduce a symplectic version of annular Khovanov homology using Hochschild homology of the Fukaya-Seidel category of more general type A nilpotent slices. Building on the work of Abouzaid-Smith and Beliakova-Putyra-Wehrli, we show that the symplectic version is isomorphic to the ordinary version. Finally, we will explain how to derive a spectral sequence from the symplectic annular Khovanov homology to the symplectic Khovanov homology directly using symplectic geometry. This is based on a joint work with Ivan Smith.
  
  In the pre-talk, we will talk about the Fukaya-Seidel category of A_n Milnor fibers. Pre-talk notes Main talk notes
• Web Hodge: Yu Zhao (MIT)-Hilbert Schemes of Points on Surfaces and Heisenberg Algebra
  2nd December 2020, 1:15pm to 1:45pm -- Show/hide abstract
  Abstract: We will introduce Nakajima and Grojownoski's construction of the Heisenberg algebra action on the cohomology of Hilbert schemes of points on surfaces. If time permits, we will also introduce the generalizations like quantum toroidal algebra action by Schiffmann-Vasserot and Feigin-Tsymbaliuk and the K-theoretic Hall algebra action by Kapranov-Vasserot and the speaker on the Grothendieck groups of Hilbert schemes. Pre-talk slides
• Web Hodge: Yu Zhao (MIT)-A Weak Categorical Quantum Toroidal Action on the Derived Categories of Hilbert Schemes
  2nd December 2020, 2:00pm to 3:00pm -- Show/hide abstract
  Abstract: The quantum toroidal algebra is the affinization of the quantum Heisenberg algebra. Schiffmann-Vasserot, Feigin-Tsymbaliuk and Negut studied the quantum toroidal algebra action on the Grothendieck group of Hilbert schemes of points on surfaces, which generalized the action by Nakajima and Grojnowski in cohomology. In this talk, we will categorify the above quantum toroidal algebra action. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semi-divisorial log terminal singularities. Main talk slides
• Web Hodge: Fei Xie (Edinburgh)-Derived categories of quintic del Pezzo fibrations
  9th December 2020, 1:15pm to 3:00pm -- Show/hide abstract
  Abstract: I will discuss the quintic del Pezzo surfaces with rational Gorenstein singularities over both algebraically closed and non-closed fields. I will describe a semiorthogonal decomposition (SOD) of their derived categories and generalise it to a fibration. The SOD of the fibration can be obtained via the Homological Projective Duality (HPD) method and has a moduli space interpretation. In the pre-talk, I will give an introduction to the HPD and its relation to classical projective duality. Pre-talk notes Main talk notes
• Web Hodge: Eugene Gorsky (UC Davis)-Parabolic Hilbert schemes on singular curves and representation theory
  16th December 2020, 1:15pm to 3:00pm -- Show/hide abstract
  Abstract: I will construct representations of various interesting algebras (such as rational Cherednik algebras and quantized Gieseker varieties) using the geometry of parabolic Hilbert schemes of points on plane curve singularities. A connection to Coulomb branch algebras of Braverman, Finkelberg and Nakajima will be also outlined. The talk is based on a joint work with Jose Simental and Monica Vazirani. Talk notes
• Web Hodge: Brian Williams (Edinburgh) - Dualities, vertex algebras, and holomorphic strings
  13th January 2021, 1:15pm to 3:00pm -- Show/hide abstract
  Abstract: Abstract: Physical reasoning has been instrumental in predicting and guiding various mathematical dualities, perhaps most famously in the context of mirror symmetry. In this talk I will discuss progress towards a mathematical proof of a duality between a topological string theory, called the B-model, and a physical string theory, called the heterotic string. The duality turns out to have an elegant mathematical formulation in terms of topological vertex algebras and we describe variants and generalizations that can be understood in a similar fashion. Slides for pre talk. Slides for main talk.
• Web Hodge: Jeremy Lane (McMaster)-Gradient-Hamiltonian vector fields and applications in symplectic geometry
  20th January 2021, 2:00pm to 3:00pm -- Show/hide abstract
  Abstract: Gradient-Hamiltonian vector fields have emerged in recent years as an important bridge between algebraic geometry and symplectic geometry. As shown in the work of Nishinou-Nohara-Ueda and Harada-Kaveh, gradient-Hamiltonian vector fields allow one to construct integrable systems on smooth projective varieties from toric degenerations.
  
  In this talk I will describe how this construction can be extended to degenerations of quasi-projective varieties which are not necessarily smooth. Such degenerations are common and arise in many important contexts, such as toric degenerations of the base affine space G//N of a reductive algebraic group G associated to Lusztig's dual canonical basis. As a consequence, we are able to construct integrable systems with nice properties such as convexity on arbitrary Hamiltonian K-manifolds, for K a compact connected Lie group. Slides
• Web Hodge: Yagna Dutta (Bonn) Holomorphic 1-forms and geometry
  27th January 2021, 1:15pm to 3:00pm Zoom -- Show/hide abstract
  Abstract: In this talk I will discuss various topological and geometric consequences of the existence of zeros of global holomorphic 1-forms on smooth projective varieties. Such consequences have been indicated by a plethora of results. I will present some old and new results in this direction. One of the key points is an interesting connection between two sets of such 1-forms, one that arises out of the generic vanishing theory and the other that falls out of the decomposition theorem of the albanese morphism. This is an on-going joint work with Feng Hao and Yongqiang Liu. Notes for the pre-talk. Notes for the main talk.
• Web Hodge: Younghan Bae (ETH)-The Chow ring of the moduli stack of prestable curves
  3rd February 2021, 1:15pm to 3:00pm -- Show/hide abstract
  Abstract: I will discuss the Chow ring of the moduli stack of prestable (without stability condition) curves. This stack contains Deligne-Mumford's moduli space of stable curves as an open substack. When the genus is zero, we will compute the Chow ring explicitly. We will see how higher Chow groups play a role. This is a joint work with Johannes Schmitt. Notes for the pre-talk. Notes for the main talk.
• Web Hodge: Owen Gwilliam (UMass Amherst). Title: Spontaneous symmetry breaking, a view from derived geometry
  10th February 2021, 2:00pm to 3:00pm -- Show/hide abstract
  Abstract: We will give an overview of how physics and homological algebra have met in the setting of gauge theory, with an emphasis on how the new subject of derived geometry provides a clarifying framework. The talk's concrete aim is to explain the Higgs mechanism as a case study. Our approach will be low-tech and will emphasize the motivations; anyone familiar with notions like vector bundle and cochain complex should be able to follow.
• Pre-talk: Alexei Oblomkov (Amherst) - Matrix factorizations: derived algebraic geometry without tears.
  24th February 2021, 1:15pm to 1:45pm Online -- Show/hide abstract
  Abstract: I will give a brief introduction to the theory of matrix factorizations (due to Eisenbud, Kontsevich and Orlov). In particular, I will explain how matrix factorizations help us to understand derived categories of some important classes of singular varieties.
• Web Hodge: Alexei Oblomkov (Amherst) - Khovanov-Rozansky homology and sheaves on Hilbert scheme of points on the plane.
  24th February 2021, 2:00pm to 3:00pm Online -- Show/hide abstract
  Abstract: Talk is based on the joint work with Lev Rozansky. I will explain a construction that attaches to a $n$-stranded braid $\beta$ a two-periodic complex $S_\beta$ of $\mathbb{C}^*\times \mathbb{C}^*$-equivariant sheaves on $Hilb_n(\mathbb{C}^2)$ such that the $H^*(S_\beta)$ is a categorification of the Oceanu-Jones trace. We show the corresponding link homology coincide with the triply graded Khovanov-Rozansky link homology which categorifies of Jones construction of HOMFLYPT polynomial. We also show that $S_{\beta FT}=S_\beta \otimes L$ where $FT$ is a full twist and $L$ is a generator of the Picard group of $Hilb_n(\mathbb{C}^2)$. The natural involution of $\mathcal{C}^2$ results in Poincare duality of the Khovanov-Rozansky homology (conjectured in 2005). As an application we obtain explicit an combinatorial formula for Khovanov-Rozansky homology of torus knots.
• Web Hodge: Lenny Taelman (University of Amsterdam)-Derived equivalences between hyperkähler varieties
  3rd March 2021, 2:00pm to 3:00pm Online -- Show/hide abstract
  Abstract: We study equivalences D(X)-->D(Y) between the derived categories of coherent sheaves on complex hyperkähler varieties X and Y. An important tool is the Looijenga--Lunts--Verbitsky Lie algebra acting on the total cohomology of X. We show that this Lie algebra is preserved
  by derived equivalences, and deduce various consequences from this. Talk notes
• Web Hodge: Sam Raskin (UT Austin)- Geometric Langlands for l-adic sheaves
  10th March 2021, 1:15pm to 3:00pm Online -- Show/hide abstract
  Abstract: In celebrated work, Beilinson-Drinfeld formulated a categorical analogue of the Langlands program for unramified automorphic forms. Their conjecture has appeared specialized to the setting of algebraic D-modules: non-holonomic D-modules play a prominent role in known constructions. In this talk, we will translate their work back into a statement suitable for (certain) automorphic functions, refining the Langlands conjectures in this setting. We will deduce this result from a categorical conjecture suitable in other geometric settings, including l-adic sheaves. One of the main constructions is a suitable moduli space of local systems. This is joint work with Arinkin, Gaitsgory, Kazhdan, Rozenblyum, and Varshavsky. In the pre-talk, we will give background on the relevant parts of the arithmetic Langlands correspondence, setting the stage for the problem considered in the main talk.
• Web Hodge: Ben Davison (Edinburgh) - Finite-dimensional Jacobi algebras, flopping curves, and BPS invariants
  17th March 2021, 1:15pm to 3:00pm Online -- Show/hide abstract
  Abstract: Floppable curves in threefolds are a fundamental feature of the minimal model programme in 3 dimensions. Not only do they provide the stepping stones between different minimal models, but their local geometry itself turns out to be tremendously rich. The list of examples starts with Atiyah's flop, then through the infinite class in Reid's "Pagoda" into the zoo of (-3,1) curves. Associated to these curves are a number of enumerative invariants, amongst them the Gopakumar-Vafa invariants, which can be reinterpreted as modified counts of coherent sheaves on the threefold containing the curve. The contraction algebra, controlling the deformation theory of the structure sheaf of the curve, turns out to be a strictly more refined invariant. This is a finite-dimensional Jacobi algebra introduced by Donovan and Wemyss. In this talk I will explain how the Brown-Wemyss conjecture that all finite-dimensional Jacobi algebras arise as contraction algebras implies a strong positivity statement for BPS invariants of these algebras (while defining what this all means), and explain how to prove this positivity statement, while also explaining how the classical Gopakumar-Vafa invariants of floppable curves can be "categorified" using cohomological DT theory.
• Web Hodge: Yalong Cao (Kavli IPMU, Tokyo) - Gopakumar-Vafa type invariants for Calabi-Yau 4-folds
  24th March 2021, 10:30am to 12:00pm Online -- Show/hide abstract
  Abstract: Gopakumar-Vafa type invariants on Calabi-Yau 4-folds (which are non-trivial only for genus zero and one) are defined by Klemm-Pandharipande from Gromov-Witten theory, and their integrality is conjectured. In this talk, I will explain how to give a sheaf theoretical interpretation of them using counting invariants on moduli spaces of one dimensional stable sheaves. Based on joint works with D. Maulik and Y. Toda.
• Web Hodge: Maxim Jeffs (Harvard)-Mirror symmetry and Fukaya categories of singular varieties
  31st March 2021, 1:15pm to 3:00pm -- Show/hide abstract
  Abstract: One problem that arises when studying mirror symmetry is that even mirrors of smooth varieties are often singular, and it is difficult to make sense of the A-model for these mirrors. In this talk I will explain Auroux' definition of the Fukaya category of a singular hypersurface and two results about this definition, illustrated with some fundamental examples. The first result is analogous to Orlov's derived Knorrer periodicity theorem, relating Auroux' category to a Landau-Ginzburg model on a higher-dimensional variety; the second result is a homological mirror symmetry equivalence at certain large complex structure limits. Slides
• Web Hodge: Benedict Morrissey (Chicago)-Nonabelianization for Reductive groups
  7th April 2021, 1:15pm to 3:00pm -- Show/hide abstract
  Abstract: Nonabelianization provides coordinate charts on the moduli space of local systems on a curve. In the work of Gaiotto, Moore, and Neitzke this is done by giving a map from the moduli space of one dimensional local systems on a spectral curve to the moduli space of n-dimensional local systems on the original curve. I will describe joint work with M. Ionita which extends this construction from GL(n) to an arbitrary reductive algebraic group. Time permitting I will describe work in progress towards proving that these maps are symplectomorphisms (under certain conditions).
• Web Hodge: Dougal Davis (Edinburgh)-Elliptic quantum groups as shuffle algebras
  14th April 2021, 10:30am to 12:00pm Online -- Show/hide abstract
  Abstract: Elliptic quantum groups are the algebraic gadgets that arise from quantising functions on an elliptic curve valued in a Lie algebra, and are closely related to quantum moduli of principal bundles. They have something of the flavour of Yangians and quantum loop algebras, but due to the rich geometry of elliptic curves are much more subtle to define. In this talk, I will discuss joint work in progress with Yaping Yang and Gufang Zhao in which we give a "shuffle" realisation of the elliptic quantum group as an algebra object in an appropriate category of sheaves on a coloured Hilbert scheme. This combines a previous construction of Yang and Zhao for the positive part with a new doubling procedure to produce the full algebra. Time permitting, I will also discuss how to recover previously studied dynamical elliptic quantum groups from our sheafified one via a Fourier transform.
  
  Pre-talk: In the pre-talk, I will give a quick introduction to the theory of Yangians (where the elliptic curve is replaced with the additive group in the above story). This should serve as a useful down-to-earth example of the more abstract constructions in the main talk.
  Slides for pre-talk, Slides for main talk
• Web Hodge: Asher Auel (Dartmouth College)- Brill-Noether theory for cubic fourfolds
  21st April 2021, 1:25pm to 3:00pm Online -- Show/hide abstract
  Abstract: It is well-known that the study of special cubic fourfolds leads to beautiful connections between algebraic cycles, the rationality problem, K3 surfaces, hyperkaeler manifolds, and derived categories. In this talk, I’ll explain a connection to the theory of algebraic curves, via a notion of Brill-Noether theory for cubic fourfolds. Specifically, I’ll discuss several open problems related to Brill-Noether special K3 surfaces Hodge-theoretically associated to smooth cubic fourfolds. Talk notes
• Web Hodge: Yusuf Baris Kartal (Princeton) Algebraic torus actions on Fukaya categories
  28th April 2021, 1:15pm to 3:00pm -- Show/hide abstract
  Abstract: Given symplectic manifold M, its continuous dynamics (modulo
  Hamiltonians) is governed by its first cohomology. In particular, one
  expects an "action" of the first cohomology on invariants such as the
  Fukaya category of M. The goal of this talk is to show that under some
  hypotheses (such as negative monotonicity), this action is tame, in
  the sense that it can be extended to an algebraic action of an
  algebraic torus. We plan to show how to use this to deduce tameness
  results on the change of Lagrangian Floer homology groups under
  symplectic isotopies. Time remaining, we will discuss applications to
  mirror symmetry. Notes
• Web Hodge: Dustin Clausen (University of Copenhagen)-Solid quasicoherent sheaves
  5th May 2021, 1:15pm to 3:00pm Online -- Show/hide abstract
  Abstract: I will discuss an enlargement, called solid R-modules, of the usual category of R-modules for a commutative ring R. I'll also try to explain how solid R-modules can be used to give a clean proof of Serre duality. This is joint work with Peter Scholze.
• Web Hodge: Si Li (Tsinguha University) - Regularized integrals and quantum master equation
  12th May 2021, 2:00pm to 3:00pm -- Show/hide abstract
  Abstract: We present an effective BV quantization theory for chiral deformation of two dimensional conformal field theories. We introduce a simple procedure (which we call regularized integral) to integrate differential forms with arbitrary holomorphic poles on Riemann surfaces. This gives a geometric renormalization method for 2d chiral quantum field theories. As an application, we construct a vertex algebra analogue of the canonical trace map in algebraic index theory.
• Jeff Hicks: Symplectic to Tropical to Complex Geometry
  8th September 2021, 1:15pm to 1:45pm International Centre for Mathematical Sciences (ICMS), 47 Potterrow, Edinburgh EH8 9BT, UK -- Show/hide abstract
  Abstract: Mirror symmetry is a proposed relation between symplectic geometry and complex geometry; namely, there exists pairs of symplectic and complex spaces (mirror pairs) and a dictionary between their symplectic/complex invariants.  The Strominger-Yau-Zaslow (SYZ) conjecture proposes that these mirror pairs should be constructed as dual torus fibrations. In this pre-talk, we'll go over some first examples of SYZ fibrations, and match up some symplectic objects with complex ones.
  
  

  https://ed-ac-uk.zoom.us/j/82494257502

  Meeting ID: 824 9425 7502

• Jeff Hicks: Realizability and Unobstructedness
  8th September 2021, 2:00pm to 3:00pm Bayes Centre, The University of Edinburgh, 47 Potterrow, Edinburgh EH8 9BT, UK -- Show/hide abstract
  Abstract:

  The complex-to-tropical correspondence is the observation that, in good circumstances, complex curves in the algebraic torus can be degenerated to piecewise linear curves (tropical curves) in Euclidean space. The realization problem asks if this process can be reversed. That is, starting with the data of a tropical curve, produce a complex curve which realizes it under the log-norm map. When the tropical curve is in the real plane, then it is always possible to find a complex curve in (C*)^2 which degenerates to it; however, this is generally not the case.

   https://ed-ac-uk.zoom.us/j/82494257502

  Meeting ID: 824 9425 7502

  A similar tropical-to-symplectic correspondence exists. In contrast to the setting above, every tropical curve can be lifted to a Lagrangian submanifold inside a symplectic space. In this talk, I'll discuss some work-in-progress showing that the Lagrangian realization problem gives us insight into the complex realization problem, and look at some specific examples highlighting how homological mirror symmetry can be used to transfer results from symplectic geometry (unobstructedness) to the realization problem.
  

• Hodge Seminar: Léa Bittmann (Edinburgh)
  29th September 2021, 1:45pm to 2:45pm Bayes Centre 5.10 and Zoom -- Show/hide abstract
  Abstract:

  A Schur-Weyl duality between Double Afffine Hecke Algebras and quantum groups

• Hodge Seminar: Kostya Tolmachov (Edinburgh)
  29th September 2021, 3:00pm to 4:00pm Bayes Centre 5.10 and Zoom -- Show/hide abstract
  Abstract:

  Monodromic model for Khovanov-Rozansky homology

• Hodge Seminar: Lucien Hennecart (Edinburgh)
  7th October 2021, 1:00pm to 2:00pm Bayes Centre 5.10 and Zoom -- Show/hide abstract
  Abstract: Title: The number of isoclasses of absolutely indecomposable representations of the modular group is a polynomial Abstract: We will explain how to prove that the number of isomorphism classes of absolutely indecomposable representations of the modular group over a finite field is a polynomial (with integer coefficients) in the cardinality of the finite field. For this, we consider the stack of representations, its inertia stack and the nilpotent version of the inertia stack. By standard techniques, we reduce the question to the calculation of the number of points of the nilpotent inertia stack. This is done using a Jordan stratification and favourable homological properties. We will sketch the ideas to prove the positivity of the coefficients of the counting polynomial, which include a purity property of the representation stack. This is ongoing joint work with Fabian Korthauer.
• Hodge seminar pretalk: Shengxuan Liu (Warwick)
  13th October 2021, 1:00pm to 1:30pm -- Show/hide abstract
  Abstract:

  pre-talk title: A short introduction to stability condition

• Hodge Seminar: Shengxuan Liu (Warwick)
  13th October 2021, 1:45pm to 2:45pm Bayes Centre 5.02 and Zoom -- Show/hide abstract
  Abstract:
  

  Stability condition on Calabi-Yau threefold of complete intersection of quadratic and quartic hypersurfaces Abstract: In this talk, I will first introduce the background of Bridgeland stability condition. Then I will mention some existence result of Bridgeland stability. Next I will prove the Bogomolov-Gieseker type inequality of X_(2,4), Calabi-Yau threefold of complete intersection of quadratic and quartic hypersufaces, by proving the Clifford type inequality of the curve X_(2,2,2,4). Then this will provide the existence of Bridgeland stability condition of X_(2,4).

• Hodge Seminar: Wahei Hara (Glasgow)
  13th October 2021, 3:00pm to 4:00pm Bayes Centre 5.02 and Zoom -- Show/hide abstract
  Abstract:
  

  Global generation of instanton bundles of charge 3 on del Pezzo threefolds of degree 4

  
  Abstract: We show that any del Pezzo threefold of degree 4 admits an instanton bundle E of charge 3 such that E(1) is globally generated. This question is the most important part in our classification of weak Fano bundles on del Pezzo threefolds of degree 4. We study elliptic curves of degree 7 and show that any del Pezzo threefold of degree 4 contains such curves that are generated by quadratic equations using the deformation theory, and then we construct the desired instanton bundles by Hartshorne-Serre correspondence. This talk depends on a joint works with T. Fukuoka and D. Ishikawa.
• Hodge Seminar: Sasha Minets (Edinburgh)
  21st October 2021, 1:00pm to 2:00pm Bayes Centre 5.10 and Zoom -- Show/hide abstract
  Abstract: KLR algebras in positive characteristic and their stratifications Abstract:For a given quiver, modules over KLR algebras are used to categorify the corresponding quantum group. While for Dynkin quivers representation theory of KLR algebras is fairly well understood, it becomes much more intricate for affine quivers, especially in positive characteristic. In this talk, I will explain how to obtain some structural results in this case by studying analogues of KLR algebras associated to curves and surfaces. This is based on ongoing work with Ruslan Maksimau.
• Hodge Seminar pretalk: Franco Rota (Glasgow)
  27th October 2021, 1:00pm to 1:30pm
• Hodge Seminar: Franco Rota (Glasgow)
  27th October 2021, 1:45pm to 2:45pm Bayes Centre 5.10 and Zoom -- Show/hide abstract
  Abstract:

  Title: Motivic semi-orthogonal decompositions for abelian varieties

  Abstract: A motivic semiorthogonal decomposition is the decomposition of the derived category of a quotient stack [X/G] into components related to the "fixed-point data". They represent a categorical analog of the Atiyah-Bott localization formula in equivariant cohomology, and their existence is conjectured for finite G (and an additional smoothess assumption) by Polishchuk and Van den Bergh. I will present joint work with Bronson Lim, in which we construct a motivic semiorthogonal decomposition for a wide class of smooth quotients of abelian varieties by finite groups, using the classification by Auffarth, Lucchini Arteche, and Quezada.

• Hodge Seminar: Dougal Davis (Edinburgh)
  27th October 2021, 3:00pm to 4:00pm Bayes Centre 5.10 and Zoom -- Show/hide abstract
  Abstract:

  Title: Mixed Hodge modules, Lusztig-Vogan polynomials, and unitary representations of real reductive groups

  Abstract: Let (G, K) be the symmetric pair associated with a real reductive group G_R. In this talk, I will explain joint work in progress with Kari Vilonen concerning K-equivariant twisted mixed Hodge modules on the flag variety of G, and an application to the representation theory of G_R. The Grothendieck group of mixed Hodge modules, which enhances the Grothendieck group of G_R-modules, has two bases consisting of standard and irreducible objects. At the level of weights, the change of basis matrix was computed algorithmically by Kazhdan-Lusztig and Lusztig-Vogan in terms of Hecke algebra combinatorics. Our first main theorem upgrades this to the full Grothendieck group by adding an extra Hodge parameter to the Lusztig-Vogan polynomials. Our second main theorem is a "polarised" version of the Jantzen conjecture; following ideas of Schmid and Vilonen, it allows the signature multiplicity polynomial of Adams-van Leeuwen-Trapa-Vogan to be read off from our Hodge polynomial. These two results combined recover a key formula in the ALTV algorithm for identifying the unitary representations of G_R.

• Hodge Seminar: Umut Varolgunes (Edinburgh)
  4th November 2021, 1:00pm to 2:00pm Bayes Centre 5.10 and Zoom -- Show/hide abstract
  Abstract: Title: Trying to quantify Gromov's non-squeezing theorem
  
  Abstract: Gromov's celebrated result says (colloquially) that one cannot symplectically embed a ball of radius 1.1 into a cylinder of radius 1. I will show that in 4d if one removes from this ball a Lagrangian plane passing through the origin, then such an embedding becomes possible. I will also show that this gives the smallest Minkowski dimension of a closed subset with this property. I will end with many questions. This is based on joint work with K. Sackel, A. Song and J. Zhu.
  
  Recording :

  https://ed-ac-uk.zoom.us/rec/share/tHyPq8xqDLl_7JCFMqq01YCd1qmjaPUwHYeJiMf8I6TIQX5GM39nfg8BHbzZnYrk.kHes3x5Xj7eD2ryp?startTime=1636031090000

• Hodge Seminar: Vyjayanthi Chari (UC Riverside)
  18th November 2021, 1:00pm to 2:00pm Bayes Centre 5.10 and Zoom
• Hodge Seminar: Elie Casbi (Max Plank)
  18th November 2021, 2:15pm to 3:15pm Bayes Centre 5.10 and Zoom -- Show/hide abstract
  Abstract: Recording : 

  https://ed-ac-uk.zoom.us/rec/share/Kc2N8FurJOKmSs3vIZpIquzVMAKz0V2fw5-AglrIVyWmNXW1O9FpA7ebu2ENhAhK.yEETtkii1Hhy-omj?startTime=1637240823000

• Hodge Seminar: Wille Liu (Max Plank)
  18th November 2021, 3:45pm to 4:45pm Bayes Centre 5.10 and Zoom -- Show/hide abstract
  Abstract: Recording: 

  https://ed-ac-uk.zoom.us/rec/share/Kc2N8FurJOKmSs3vIZpIquzVMAKz0V2fw5-AglrIVyWmNXW1O9FpA7ebu2ENhAhK.yEETtkii1Hhy-omj?startTime=1637245328000

• Hodge Seminar pretalk: Junliang Shen (Yale)
  24th November 2021, 1:00pm to 1:30pm Online talk, live on Bayes Centre 5th floor and Zoom -- Show/hide abstract
  Abstract:
  
  Pretalk title: "Non-perverse" perverse filtrations  
  Abstract: This is an introduction to perverse filtrations associated with projective morphisms. These are main characters in the P=W conjecture as we will see in the main talk.
• Hodge Seminar: Junliang Shen (Yale)
  24th November 2021, 1:45pm to 2:45pm Online talk, live on Bayes Centre 5.10 and Zoom -- Show/hide abstract
  Abstract:
  
  Title: Cohomology of the moduli of Higgs bundles via positive characteristic
  
  
  Abstract: In this talk, I will explain how techniques arising from the non-abelian Hodge theory in positive characteristic provide "consistency checks" of the P=W conjecture, where the latter concerns the cohomological aspect of the non-abelian Hodge theory over the complex numbers. We will focus on two aspects: (1) the Galois conjugation, and (2) the Hodge-Tate decomposition. Based on joint work with Mark de Cataldo, Davesh Maulik, and Siqing Zhang.
  
  Recording: 

  https://ed-ac-uk.zoom.us/rec/share/RBxuXV8H9c-ItJJxbKaWofYEFXgSz1HSOpR-vtd7C8rP_asJsOMjUbxMC2qGehc.Vbw9fQ9YwF8GCtV_?startTime=1637762003000

• Hodge Seminar: Orsola Capovilla-Searle (UC Davis)
  24th November 2021, 3:00pm to 4:00pm Online talk, live on Bayes Centre 5.10 and Zoom -- Show/hide abstract
  Abstract:
  Title: Infinitely many Lagrangian Tori in Milnor fibers constructed via Lagrangian Fillings of Legendrian links
  
  Abstract: One approach to studying symplectic manifolds with contact boundary is to consider Lagrangian submanifolds with Legendrian boundary; in particular, one can study exact Lagrangian fillings of Legendrian links. There are still many open questions on the spaces of exact Lagrangian fillings of Legendrian links in the standard contact 3-sphere, and one can use Floer theoretic invariants to study such fillings. We show that a family of oriented Legendrian links has infinitely many distinct exact orientable Lagrangian fillings up to Hamiltonian isotopy. Within this family, we provide some of the first examples of a Legendrian link that admits infinitely many planar exact Lagrangian fillings. Weinstein domains are examples of symplectic manifolds with contact boundary that have a handle decomposition compatible with the symplectic structure of the manifold. Weinstein handlebody diagrams are given by projections of Legendrian submanifolds. We provide Weinstein handlebody diagrams of the 4-dimensional Milnor fibers of T_{p,q,r} singularities, which we then use to construct infinitely many Lagrangian tori and spheres in these spaces.
  
  Recording:

  https://ed-ac-uk.zoom.us/rec/share/RBxuXV8H9c-ItJJxbKaWofYEFXgSz1HSOpR-vtd7C8rP_asJsOMjUbxMC2qGehc.Vbw9fQ9YwF8GCtV_?startTime=1637768023000

• Hodge Seminar: Cancelled
  2nd December 2021, 1:00pm to 2:00pm JCMB 5323 and Zoom
• Pretalk: Matthew Pressland
  8th December 2021, 1:00pm to 1:30pm -- Show/hide abstract
  Abstract:
  Title: Grassmannian cluster algebras
  Abstract: For this pretalk, I will give an overview of the Grassmannian, its ring of functions and its positroid stratification. I will review the definition of a cluster algebra, and hint at its relationship to the Grassmannian: the main talk will make this connection more precise.
• Hodge Seminar: Matthew Pressland (University of Leeds)
  8th December 2021, 1:45pm to 2:45pm Bayes Centre 5.10 and Zoom -- Show/hide abstract
  Abstract:
  Title: Categorification of positroids and positroid varieties
  Abstract: The Grassmannian and its totally positive part have a very rich combinatorial structure, studied by many people. In particular, Postnikov has shown how the totally positive Grassmannian is stratified by positroid varieties. Recent work of Galashin and Lam shows that the coordinate ring of each (open) positroid stratum is a cluster algebra, with this structure determined by a combinatorial object called a Postnikov diagram. In this talk I will explain how the same diagram also gives rise to representation theoretic objects which can be used to (additively) categorify this cluster algebra. This is partly joint work with İlke Çanakçı and Alastair King.
  
  Recording : Hodge Seminar 8 Dec - Zoom
• Hodge Seminar: Hongdi Huang (Rice University)
  8th December 2021, 3:00pm to 4:00pm Bayes Centre 5.10 and Zoom -- Show/hide abstract
  Abstract:

  Title: Twisting of graded quantum groups and solutions to the quantum Yang-Baxter equation
  

  
  
  Let $H$ be a Hopf algebra over a field $k$ such that $H$ is $\mathbb Z$-graded as an algebra. In this talk, we introduce the notion of a twisting pair for $H$ and show that the Zhang twist of $H$ by such a pair can be realized as a 2-cocycle twist. As an application of twisting pairs, we discuss an algorithm to produce a family of solutions to the quantum Yang-Baxter equation from a given solution via the Faddeev-Reshetikhin-Takhtajan construction.
  
  Recording: Hodge Seminar 8 Dec - Zoom

  
  

• Hodge Seminar: Sukjoo Lee (Edinburgh)
  27th January 2022, 2:00pm to 3:00pm Bayes 5.10 and Zoom -- Show/hide abstract
  Abstract:
  Title: The mirror P=W conjecture from Homological Mirror Symmetry
  Abstract: The mirror P=W conjecture, recently formulated by A.Harder, L.Katzarkov and V.Przyjalkowski, is the refined Hodge number symmetry between a log Calabi-Yau mirror pair $(U, U^\vee)$. It predicts that the weight filtration on the cohomology $H^\bullet(U)$ is equivalent to the perverse filtration on the cohomology $H^\bullet(U^\vee)$ associated to the mirror Landau-Ginzburg potentials. In this talk, I will introduce the precise formulation of the conjecture and discuss how to see this from the categorical viewpoint when $U$ admits a Fano compactification.
  
  
  Zoom links:
  https://ed-ac-uk.zoom.us/rec/share/e4PdRGMyWroaRVRVXckvRqBMJ2zJxa56DydNCaVHk51kS8fqfJZNMsDg-PHkgIH7.ULFc9n30Zn2ekBBg
• Hodge Seminar pretalk: Tristan Bozec
  3rd February 2022, 1:00pm to 2:00pm
• Hodge Seminar: Tristan Bozec (IMAG)
  3rd February 2022, 2:00pm to 3:00pm Bayes 5.10 and Zoom -- Show/hide abstract
  Abstract: Title: Quiver moduli and Calabi--Yau structures.
  Abstract: In this talk I will describe a procedure, based on constrained or relative critical loci, that allows to construct lagrangian subvarieties inside symplectic quiver varieties. We will see how this works on the example of the Hilbert scheme of points on the plane. I will explain using derived geometry that the noncommutative counterpart of lagrangian strucutres is given by so-called relative Calabi--Yau ones, involving for instance generalizations of Ginzburg dg-algebras. If time permits I will give applications to multiplicative variants linked to Poisson and Hamiltonian geometries. This is a joint work with Damien Calaque and Sarah Scherotzke.
  
  Zoom recording:
  https://ed-ac-uk.zoom.us/rec/share/9avliEQhJzmOQsrqSUS5n30doWgC4hCFb7gdSu0OzNdiYyimVIAP1gc9z1P3oL1K.QILC326V3hziTyeR
• Hodge Seminar pretalk: Hamid Abban (Loughborough)
  10th February 2022, 1:00pm to 2:00pm
• Hodge Seminar: Hamid Abban (Loughborough)
  10th February 2022, 2:00pm to 3:00pm Bayes 5.10 and Zoom -- Show/hide abstract
  Abstract:
  Title: An inductive approach to K-stability via linear algebra
  Abstract: K-stability is an algebraic condition that detects the existence of Kähler-Einstein metrics on Fano manifolds. However, it has many other fruitful properties. In this talk, I will give a shortcut introduction to K-stability from an algebraic perspective. Then I discuss various techniques for checking K-stability of a given Fano variety. I will explain the main ideas behind a new method for verifying K-stability which relies on (complicated) induction and (easy) linear algebra. The method will be illustrated by several examples and results. This is a joint work with Ziquan Zhuang.
  
  
  zoom recording:
  

  https://ed-ac-uk.zoom.us/rec/share/vT9cbBU-_SkZGq52yfBbyXjNP2cmGO8CuYrsAUMbAQw3v53uY51MGmNQXoyTCfoT.pyYzWXvfrqiCIBs-

• Hodge Seminar pretalk: Vladimir Fock (IRMA)
  17th February 2022, 1:15pm to 1:45pm -- Show/hide abstract
  Abstract: Zoom recordings:
  https://ed-ac-uk.zoom.us/rec/share/Q4SU_hhV9v4-c9JBfwe79ebzEhbxrZ_8cd4DWwcVZD84NxY8bITSigqrfgRYnCSL.Fg7nMiUUYTqyi_0r?startTime=1645103961000
• Hodge Seminar: Vladimir Fock (IRMA)
  17th February 2022, 2:00pm to 3:00pm Bayes 5.10 and Zoom -- Show/hide abstract
  Abstract: Title: Hecke algebras and TQFT.
  
  Abstract: An open-closed two-dimensional topological field theory associatesvector spaces to surfaces with marked points on the boundary. The simplest examples of such theories can be constructed from finite dimensional algebras with trace. We will describe the TQFT related to Hecke algebra in details and explain its relation to counting of coverings of surfaces. In the introduction I'll try to explain basics about Hecke algebras, what is Satake correspondence and why it is interesting to generalize it to surfaces.
  
  
  Zoom recordings:
  https://ed-ac-uk.zoom.us/rec/share/Q4SU_hhV9v4-c9JBfwe79ebzEhbxrZ_8cd4DWwcVZD84NxY8bITSigqrfgRYnCSL.Fg7nMiUUYTqyi_0r?startTime=1645106572000
• Hodge Seminar: Tomasz Przezdziecki (Edinburgh)
  24th February 2022, 2:00pm to 3:00pm JCMB seminar room and Zoom -- Show/hide abstract
  Abstract:

  Title: KLR algebras and quantum symmetric pairs - categorification, combinatorics and Schur-Weyl duality

  
  

  Abstract: We consider a family of KLR-type algebras associated to the infinite linear quiver with an involution and a framing. We approach their representation theory through categorification and combinatorics. Orientifold KLR algebras are known to categorify highest weight modules over the Enomoto-Kashiwara algebra and their canonical bases. Using a new shuffle realization of these modules, together with a combinatorial construction of their PBW and canonical bases in terms of Lyndon words, we are able to classify irreducibles and prove finite global dimension.

  
  

  The second part of the talk is devoted to the connection between orientifold KLR algebras and quantum Kac-Moody symmetric pairs. The key role is played by a Schur-Weyl duality type functor, generalizing a construction due to Kang, Kashiwara and Kim. We explain the connection between the framing in the definition of the orientifold KLR algebra, parameters of a coideal subalgebra, and the poles of meromorphic K-matrices. We present some properties of this functor and formulate a conjecture that, under appropriate assumptions, it induces an equivalence of categories.

  
  

  Zoom recording: 

  https://ed-ac-uk.zoom.us/rec/share/HB9G-0oZi2sFvkmHfAB4wfS9dMexkJXvpLu5iWJGPgisfZznUgh2vJ0Fp0YBdhBr._qk1tBRJiRQsJTVh
  

• Hodge Seminar: Eirini Chavli (University of Stuttgart)
  3rd March 2022, 2:00pm to 3:00pm JCMB seminar room and Zoom -- Show/hide abstract
  Abstract: Title: The center of the generic Hecke algebra
  
  Abstract: In 1997 M. Geck and R. Rouquier described the center of the Heckealgebra for the case of Coxeter groups. Such a description for the center
  of the Hecke algebra in the complex case is still unknown, apart from
  the cases of the reflection groups $G_4$ and $G(4,1,2)$, thanks to A.
  Francis. In this talk, we give a new description of the center of the
  Hecke algebra in the real case, a description one can generalize to the
  complex case. Using GAP, we also give a basis of the center for some
  exceptional groups of rank 2 (common work with G. Pfeiffer).
  
  
  Zoom recording:
  https://ed-ac-uk.zoom.us/rec/share/hCdwQq5mZO3fynPL6W-CSZ0_QPx1z0PRo3fUPAvfBNEgNFGRB2z9L0XaSfjgqsb2.ZwspSpIoRY2tGUl7
  
• Hodge Seminar pretalk: Ryo Fujita (Université de Paris)
  10th March 2022, 1:15pm to 1:45pm
• Hodge Seminar: Ryo Fujita (Université de Paris)
  10th March 2022, 2:00pm to 3:00pm Bayes 5.10 and Zoom -- Show/hide abstract
  Abstract: Title: Deformed Cartan matrices and generalized preprojective algebras
  
  Abstract: In their study of the deformed W-algebras associated with complex simple Lie algebras, E. Frenkel-Reshetikhin (1998) introduced certain two parameter deformations of the Cartan matrices. They play an important role in the representation theory of quantum affine algebras. In this talk, we explain a representation-theoretic interpretation of these deformed Cartan matrices and their inverses in terms of the generalized preprojective algebras recently introduced by Geiss-Leclerc-Schröer (2017). We also discuss its application to the representation theory of quantum affine algebras in connection with the theory of cluster algebras. This is a joint work with Kota Murakami (arXiv: 2109.07985 ).
  
  
  Zoom recording:
  https://ed-ac-uk.zoom.us/rec/share/ku6Y00dIOuXM5pIZLzZPqd3bzDUyYMmoWKUfloXIddvWm8Q8GpklEtXCzm3nK7KO.gRCWQQ_JP6EaYDIr
  
• Hodge Seminar: Emily Norton (Kent)
  17th March 2022, 2:00pm to 3:00pm JCMB seminar room and Zoom -- Show/hide abstract
  Abstract:

  Title: Calibrated representations of cyclotomic Hecke algebras at roots of unity
  

  
  Abstract: The cyclotomic Hecke algebra is a "higher level" version of the Iwahori-Hecke algebra of the symmetric group. It depends on a collection of parameters, and its combinatorics involves multipartitions instead of partitions. We are interested in the case when the parameters are roots of unity. In general, we cannot hope for closed-form character formulas of the irreducible representations. However, a certain type of representation called "calibrated" is more tractable: those representations on which the Jucys-Murphy elements act semisimply. We classify the calibrated representations in terms of their Young diagrams, give a multiplicity-free formula for their characters, and homologically construct them via BGG resolutions. This is joint work with Chris Bowman and José Simental.

  Zoom Recording:

  https://ed-ac-uk.zoom.us/rec/share/VFg5IOEpI7MKgLD5GW8xYgJgxsruHkuK9vpnu93peFSCJPbdILzraPH_mt_1Aeeg.C6iGIodvAv5mSr0T?startTime=1647525985000
  

• Hodge Seminar: Adeel Khan (Academia Sinica)
  24th March 2022, 2:00pm to 3:00pm Bayes 5.10 and Zoom -- Show/hide abstract
  Abstract: Title: Virtual fundamental classes and categorification
  Abstract: I will discuss a derived analogue of specialization and microlocalization of sheaves (in the sense of Verdier and Kashiwara-Schapira), and explain how these can be regarded as categorifications of Kontsevich's virtual fundamental class. I will also discuss an interesting relationship with Joyce's categorification of Donaldson-Thomas theory.
  
  
  Zoom links:
  https://ed-ac-uk.zoom.us/rec/share/nT_V1MD18zmZm8RnO_Zv1GOk4VFzSGX5hqdd2ChFKiZqEy8C5cXKIIpdyOd-hT-q.s5NL8ZkJaQqqxXDK
• Hodge Seminar: Quoc Ho (HKUST)
  31st March 2022, 2:00pm to 3:00pm JCMB 5323 and Zoom -- Show/hide abstract
  Abstract: Title: Revisiting mixed geometry
  
  I will present joint work with Penghui Li on our theory of graded sheaves which provides a uniform construction of "mixed versions" or "graded lifts" in the sense of Beilinson--Ginzburg--Soergel which works for arbitrary Artin stacks. In particular, we obtain a general construction of graded lifts of many categories arising in geometric representation theory and categorified knot invariants. Our sheaf theory comes with a six-functor formalism, a perverse t-structure in the sense of Beilinson--Bernstein--Deligne--Gabber, and a weight (or co-t-)structure in the sense of Bondarko and Pauksztello, all compatible, in a precise sense, with the six-functor formalism, perverse t-structures, and Frobenius weights on ell-adic sheaves. Historically, constructions of graded lifts were done on a case-by-case basis and were technically subtle, due to Frobenius' non-semisimplicity. Our construction sidesteps this issue by semi-simplifying the Frobenius action itself. However, our categories agree with those previously constructed when they are available. For example, the monoidal DG-category of chain complexes of Soergel bimodules is equivalent to the category of constructible graded sheaves on B\G/B.
  
  
  Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/tj9yD-9_brz7_8XhSAxyLoZqGdBlncmeHvxphVCC-1eJmmPyKchYgyunI7R08Gqw.QtWE2At3kFeblzFr?startTime=1648731717000
• Hodge Seminar Pretalk: Abigail Ward (MIT)
  7th April 2022, 1:15pm to 1:45pm JCMB 5323 and Zoom -- Show/hide abstract
  Abstract: Pretalk Title: Homological mirror symmetry for log Calabi-Yau surfaces
  Abstract:
  I'll discuss work of Hacking and Keating proving homological mirror symmetry for log Calabi-Yau surfaces, as well as an alternate approach, in progress, joint with Andrew Hanlon.
• Hodge Seminar: Abigail Ward (MIT)
  7th April 2022, 2:00pm to 3:00pm JCMB 5323 and Zoom -- Show/hide abstract
  Abstract: Symplectomorphisms mirror to birational transformations of the complex plane
  
  We construct a non-finite type four-dimensional Weinstein domain M_{univ} and describe a HMS-type correspondence between certain birational transformations of the complex plane preserving a standard holomorphic volume form and symplectomorphisms of M_{univ}. The space M_{univ} is universal in the sense it admits every Liouville four-manifold mirror to a log Calabi-Yau surface as a Weinstein subdomain; our construction recovers a mirror correspondence between the automorphism group of any open log Calabi-Yau surface and the symplectomorphism group of its mirror by restriction to these subdomains. This is joint work in progress with Ailsa Keating.
  
  
  Zoom recording:
  https://ed-ac-uk.zoom.us/rec/share/oB8W1uKHfN7b6Rk4FLtOpQ2Mnm6xo3dPrP5d9zpFdc6Oc-BCVaH4Vrjcol_2_R8p.D9etHGaJTIVA0qQO
  
• Whittaker Lecture: Claire Voisin (IMJ-PRG)
  27th April 2022, 1:00pm to 2:30pm Lecture Theatre C, JCMB -- Show/hide abstract
  Abstract: Schedule:
      13:00-14:00 Lunch (room 5214) 
      14:00-15:00 Claire Voisin (first lecture)
  See https://www.maxwell.ac.uk/events/distinguished_lectures/
  for more details.
• Whittaker Lecture: Claire Voisin (IMJ-PRG), Egor Yasinsky (Paris)
  28th April 2022, 1:00pm to 5:00pm Lecture Theatre C, JCMB -- Show/hide abstract
  Abstract: Schedule:    
  13:00-14:00 Lunch (room 5214)
  14:00-15:00 Egor Yasinsky
  15:00-15:30 Coffee break (room 5214)
  15:30-16:30 Claire Voisin (second lecture)
  16:30-17:00 Wine reception (room 5214)
  
  See https://www.maxwell.ac.uk/events/distinguished_lectures/
  for more details.
• Whittaker Lecture: Claire Voisin (IMJ-PRG), Evgeny Shinder (Sheffield)
  29th April 2022, 12:30pm to 4:00pm Lecture Theatre C, JCMB -- Show/hide abstract
  Abstract: Schedule:     12:30-13:30 Lunch (room 5214) 13:30-14:30 Evgeny Shinder 14:30-15:00 Coffee break (room 5214) 15:00-16:00 Claire Voisin (third lecture) See https://www.maxwell.ac.uk/events/ and https://www.maxwell.ac.uk/events/distinguished_lectures/
• Hodge Seminar: Yin Li (Edinburgh)
  5th May 2022, 2:00pm to 3:00pm JCMB 5323 and Zoom -- Show/hide abstract
  Abstract: Title: Cyclic dilations and closed Lagrangian submanifolds
  
  Abstract: Given a closed, possibly non-exact Lagrangian submanifold L in a Liouville manifold M, one can define a Maurer-Cartan element in the (completed) string homology of L with respect to an L_\infty structure. When the first Gutt-Hutchings capacity of M is finite, and L is topologically a K(\pi,1) space, the existence of such a Maurer-Cartan element implies that L bounds a non-constant pseudoholomorphic disc of Maslov index 2. When M is 6-dimensional, this leads to a topological classification of closed, prime Lagrangian 3-fold in M, generalizing a theorem of Fukaya and Irie.
  
  
  Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/W-Y-otJBED4pQKP8jUcNE619zxxZmiTMoKivXVwHuVnpoWm1nz0fKSm-zPdXfu-v.K1-CMIv55pmeXOPr
• Hodge Seminar pretalk: Ursula Whitcher
  17th May 2022, 1:15pm to 1:45pm -- Show/hide abstract
  Abstract:

  A gentle introduction to classical mirror symmetry 
• Hodge Seminar: Ursula Whitcher (University of Michigan)
  17th May 2022, 2:00pm to 3:00pm JCMB 5323 -- Show/hide abstract
  Abstract: Talk Title: Strong arithmetic mirror and Calabi-Yau pencils
  
  Abstract: Mirror symmetry predicts surprising geometric correspondences between distinct families of algebraic varieties. In some cases, these correspondences have arithmetic consequences. For example, one can use mirror symmetry to explore the structure of the zeta function, which encapsulates information about the number of points on a variety over a finite field. We describe a factor of the zeta functions of invertible Calabi-Yau pencils and suggest strategies for predicting similar arithmetic structures in other Calabi-Yau pencils.
  
  
  Recording: https://ed-ac-uk.zoom.us/rec/share/kMg5t2ykr2OlGw_ZTZ5Saoh8GHFfK9mvmH37B6-Slt_466HXdeaHCaBEOXLd04Oc.qjn9V6mw0W_Aj3wN
• Hodge Seminar pretalk: Tobias Dyckerhoff (Hamburg)
  19th May 2022, 1:15pm to 1:45pm
• Hodge Seminar: Tobias Dyckerhoff (Hamburg)
  19th May 2022, 2:00pm to 3:00pm Zoom -- Show/hide abstract
  Abstract: Title: Complexes of derived categories
  
  Abstract: Derived categories have come to play adecisive role in a wide range of topics within algebra, geometry, and topology. Several recent developments, in particular inthe context of topological Fukaya categories, arouse the desire to study not just singlecategories, but rather complexes of categories. In this talk, we will discuss examples of suchcomplexes in algebra, topology, algebraic geometry, and symplectic geometry, along with someresults involving them.
  
  
  Zoom recording: https://ed-ac-uk.zoom.us/rec/share/TUNWdGdeJ70pbaRWqaTqDiphVV7BN2aloh82FlosJDMqDGLaU87SQygTntmBwIl3.EGvJqRTyAs8WKeKU
  
• Hodge Seminar: Pretalk
  26th May 2022, 1:00pm to 1:30pm
• Hodge Seminar: Dinakar Muthiah (Glasgow)
  26th May 2022, 1:45pm to 2:45pm JCMB 5323 and Zoom -- Show/hide abstract
  Abstract:
  Title: Fundamental monopole operators and affine Grassmannian slices
  
  Abstract:Affine Grassmannians are objects of central interest in geometric representation theory. For example, the geometric Satake correspondence tells us that their singularities carry representation theoretic information. In fact, it suffices to work with affine Grassmannian slices, which retain all of this information.Recently, Braverman, Finkelberg, and Nakajima showed that affine Grassmannian slices arise as Coulomb branches of certain quiver gauge theories. Remarkably, their construction works in Kac-Moody type as well. Their work opens the door to studying affine Grassmannians for Kac-Moody groups. Unfortunately, it is difficult at present to do any explicit geometry with the Coulomb branch definition. For example, a basic feature is that affine Grassmannian slices embed into one another. However, this is not apparent from the Coulomb branch definition. In this talk, I will explain why these embeddings are necessarily subtle. Nonetheless, I will show a way to construct the embeddings using fundamental monopole operators.This is joint work in progress with Alex Weekes.
  
  
  Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/gMmVxYoxc8I3prC1t3UskpNHQRJLF5A_bw_OpNtZ0Sbd8N6CXShHeuKH3ebtCwk._pcT5ErLeaFJFn78?startTime=1653569487000
• Hodge Seminar: Noah Snyder (Indiana)
  26th May 2022, 3:00pm to 4:00pm JCMB 5323 and Zoom -- Show/hide abstract
  Abstract:
  Title: String diagrams and explicit descriptions of homotopy groups
  
  
  
  
  
  Abstract If X is an n-connected space, then \Omega^n X can be thought of as a (grouplike) E_n-monoidal \infty-groupoid, and then one can study it using string diagrams and their generalizations the way one studies tensor categories or braided tensor categories via string diagrams.  Although the language in the last sentence is quite modern, one can think of this approach as just a generalization of the Pontryagin-Thom construction, and so it goes back to the earliest days of calculating homotopy groups.  This suggests as a natural goal not merely computing homotopy groups abstractly, but instead giving explicit string diagram descriptions of the elements and of the relations.  For example, following Pontryagin, the generator of \pi_3(S^2) is given by a "figure-8" framed unknot.  My main motivation for asking these kinds of questions is to be able to compute explicitly the SO(3)-action on the space finite tensor categories guaranteed by the cobordism hypothesis.  The main examples I'll discuss are \Omega^n S^n for small n (i.e. the homotopy groups of spheres) and \Omega^2 \Sigma RP^2 which comes up in studying BSO(3).  The former examples will touch on work of Pontryagin, a MO answer of Schommer-Pries, and a calculation by Henriques, while the latter example is relevant to SO(3) actions on tensor categories, as mentioned above, and will appear in joint work in progress with Douglas, Reutter, and Schommer-Pries.
  
  Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/gMmVxYoxc8I3prC1t3UskpNHQRJLF5A_bw_OpNtZ0Sbd8N6CXShHeuKH3ebtCwk._pcT5ErLeaFJFn78?startTime=1653573924000
• Hodge Seminar Pretalk: Andrew Hanlon
  16th June 2022, 1:15pm to 1:45pm -- Show/hide abstract
  Abstract: Title: Deformed Hermitian Yang-Mills and special Lagrangians Abstract: We will see how the deformed Hermitian Yang-Mills equation can be viewed as the mirror to the special Lagrangian equation and give a brief overview of obstructions to solutions found by Collins and Yau.
• Hodge Seminar: Andrew Hanlon (Stony Brook)
  16th June 2022, 2:00pm to 3:00pm -- Show/hide abstract
  Abstract: Title: On GIT stability in mirrors to toric varieties
  Abstract: Using geometric invariant theory, Collins and Yau have produced obstructions to the existence of solutions to deformed Hermitian Yang-Mills on line bundles on a Kahler variety. Passing to the mirror in the toric setting, we will re-interpret their obstructions as integrals over certain Lagrangians and ponder their categorical interpretation. This talk is based on joint work with Tristan Collins and Jeff Hicks.
  
  
  https://ed-ac-uk.zoom.us/rec/share/Dc0jyIqlE9wmNkw7tAhV2LTJ5l3pLHBjtvQ-z5-mPQdmQ0yQGUIsxWmoUNAsfAN4.dvH4oC3nRMg5WsdQ
  
• Hodge Seminar: Maria Chlouveraki (Versailles & Athens)
  27th June 2022, 3:30pm to 4:30pm JCMB 5323 -- Show/hide abstract
  Abstract:
  Title: The block defect of Hecke algebras
  Abstract: The complexity of a block of a symmetric algebra can be measured by the notion of defect, a numerical datum associated with each of the simple modules contained in the block. Geck showed that the defect is a block invariant for Iwahori–Hecke algebras of finite Coxeter groups in the equal parameter case, and speculated that a similar result should hold in the unequal parameter case. In a joint work with Nicolas Jacon, we have proved that the defect is a block invariant for Hecke algebras associated with the complex reflection groups of type G(r,1,n), which include the Weyl groups of type Bn in the unequal parameter case, by showing that the defect corresponds to the notion of weight in the sense of Fayers. We conjecture that the defect is a block invariant for Hecke algebras in general.
• Hodge Seminar:Ruijie Yang (Universität zu Berlin)
  30th June 2022, 12:30pm to 1:30pm -- Show/hide abstract
  Abstract:
  Title: Hodge theory of singularities of divisors
  
  Abstract: Given a holomorphic function on a complex manifold, the vanishing cycle complex of the function encodes cohomology of Milnor fibers. In projective algebraic geometry, we often encounter divisors rather than functions. In this talk, I will explain how to glue vanishing cycle complexes of local defining functions of a divisor and endow each associated graded part of the monodromy weight filtration with a structure of twisted pure Hodge module. This construction brings tools from birational geometry such as vanishing theorems to the study of vanishing cycles. Furthermore, we define a sequence of ideal sheaves, which encode Hodge-theoretic information of singularities of divisors. It turns out that these ideal sheaves are closely related to the Hodge ideals introduced by Mustata-Popa. This is based on the joint work (partially in progress) with Christian Schnell.
• Hodge Seminar: Antonella Grassi
  30th June 2022, 1:45pm to 2:45pm JCMB 5323 -- Show/hide abstract
  Abstract: Title: Stringy Kodaira and applications. Abstract:Kodaira classified fibres on relatively minimal complex elliptic surfaces. I will discuss the classification problem for complex threefolds and its application, in particular to questions from string theory.
• EDGE: Bivas Khan (IISER) - Seshadri constants of equivariant vector bundles on toric varieties
  28th September 2022, 11:00am to 12:00pm Bayes 5.46 -- Show/hide abstract
  Abstract: Abstract: Seshadri constants measure the local positivity of an ample line bundle. They were introduced by Demailly, motivated by Seshadri's ampleness criterion for line bundles. Later, Hacon generalized the notion of Seshadri constants to vector bundles. In general, Seshadri constants are not easy to compute, and a lot of research is aimed at finding good estimates. In this talk, we consider torus equivariant vector bundles on toric varieties. Assuming certain conditions on the vector bundle, we give the precise value of Seshadri constants at arbitrary points on projective spaces and Bott towers of height at most 3. This talk is based on joint work with Jyoti Dasgupta and Aditya Subramaniam.
  
  Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/Qm_fidm7ZLZXAXVDYfyzEFM0bzJ3e6LKySLlk8h3I-XMCcZfNmNaFAEIBnxSopn8.XEsjeJnQppvOSmIV
• EDGE: Mohamed El Alami (Edinburgh) - Enumerative invariants of Lagrangian embeddings in Fano varieties
  5th October 2022, 11:00am to 12:00pm Bayes 5.40 -- Show/hide abstract
  Abstract: Abstract: Let Z be a Fano variety, D an anti-canonical divisor, and L a Lagrangian submanifold in the complement of D. I will describe a counting theorem relating holomorhic discs which are tangent to D, and holomorphic discs which are transverse to D. If time permits, I will discuss applications to Lagrangian topology and Fukaya categories.
  
  
  Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/Aw3OVKBy3DiO5mMdKJTGVe7GYAnGAX0ydoLx6i4tRqwYH2WORUwfkbVglV4LeuUq.PgPe6s8yirD-VBi3?startTime=1664964165000
• EDGE: Morgan Weiler (Cornell) - Title: Infinite staircases of 4D symplectic embeddings
  12th October 2022, 11:00am to 12:00pm Bayes 5.46 -- Show/hide abstract
  Abstract:

  Abstract: The ellipsoid embedding function of a symplectic manifold measures the amount by which the symplectic form must be scaled in order to fit an ellipsoid of a given eccentricity. It generalizes the Gromov width and ball packing numbers. In 2012 McDuff and Schlenk computed the ellipsoid embedding function of the ball, showing that it exhibits a delicate piecewise linear pattern known as an infinite staircase. Since then, the embedding function of many other symplectic four-manifolds have been studied, and not all have infinite staircases. We will classify those symplectic Hirzebruch surfaces whose embedding functions have an infinite staircase, coming from the obstructive perspective. Based on work with Magill and McDuff and work in progress with Magill and Pires.

• EDGE: Nicki Magill (Cornell) - Title: Infinite Staircases and Almost Toric Fibrations
  19th October 2022, 11:00am to 12:00pm Bayes 5.10 -- Show/hide abstract
  Abstract: Abstract: The ellipsoidal embedding function of a symplectic four manifold M measures how much the symplectic form on M must be dilated in order for it to admit an embedded ellipsoid of some eccentricity. In most cases, finitely many obstructions besides the volume determine the function. If there are infinitely many obstructions determining the function, M is said to have an infinite staircase. We will focus on explaining the correspondence between the obstructions coming from exceptional classes and the constructions from almost toric fibrations for infinite staircases for Hirzebruch surfaces. We define a way to mutate triples of exceptional classes (which give obstructions to embeddings) to produce new triples of exceptional classes, which corresponds to mutations in almost toric fibrations. This is partially based on various joint work with Dusa McDuff, Ana Rita Pires, and Morgan Weiler.
  
  Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/QNo_6VtW4ciP60REtNCCgiSWMVidI0QYRpNfltwxRV3SY_0hOUpmlgOqIcSZFIF2.koiQfczPA12bgieE?startTime=1666174020000
• EDGE: Laurent Côté (Harvard)
  20th October 2022, 2:00pm to 3:00pm JCMB 5328 -- Show/hide abstract
  Abstract: Title: Holomorphic symplectic manifolds Abstract: This talk will be about symplectic manifolds of the form (X, \omega), where X is a complex manifold and \omega is the real part of a holomorphic closed and non-degenerate 2-form. I will explain why these manifolds are special, in particular with regards to their Fukaya categories. Anything new results from joint work with Chris Kuo, David Nadler and Vivek Shende.
• EDGE: Ruadhaí Dervan (Glasgow) - Infinite-dimensional moment maps in Kähler geometry
  2nd November 2022, 11:00am to 12:00pm Bayes 5.02 -- Show/hide abstract
  Abstract: Abstract: Much of Kähler geometry is motivated by the problem of linking the existence of solutions to geometric PDEs to notions of stability in algebraic geometry. The most fundamental example of such a link is the Yau-Tian-Donaldson conjecture, which predicts that the existence of constant scalar curvature Kähler metrics on a projective variety is equivalent to K-stability of the variety. This conjecture is primarily motivated by a foundational result of Donaldson and Fujiki: the scalar curvature operator can be viewed as a moment map on an infinite-dimensional symplectic manifold. In the first part of this talk I will give a new, geometric (and quite simple) proof of this fact, using tools from equivariant differential geometry. The second part of the talk will discuss an ongoing programme to develop general classes of geometric PDEs in Kähler geometry with links to algebro-geometric stability conditions. I will discuss a class of PDEs determining what I call Z-critical Kähler metrics, and will show that these PDEs can also be realised as infinite dimensional moment maps. I will briefly explain the motivation behind these results, which is to give an analytic counterpart to an analogue of Bridgeland stability conditions for varieties.
• EDGE: Woonam Lim (ETH Zurich) - Virasoro constraints and wall-crossing via vertex algebras
  9th November 2022, 11:00am to 12:00pm Bayes 5.10 -- Show/hide abstract
  Abstract: Abstract: In enumerative geometry, Virasoro constraints first appeared in the context of moduli of stable curves and maps. These constraints provide a rich set of conjectural relations among Gromov-Witten descendent invariants. Recently, the analogous constraints were formulated in several sheaf theoretic contexts; stable pairs on 3-folds, Hilbert scheme of points on surfaces, and higher rank sheaves on surfaces with only (p,p)-cohomology. In joint work with A. Bojko, M. Moreira, we extend and reinterpret Virasoro constraints in sheaf theory using Joyce's vertex algebra. This new interpretation yields proof of Virasoro constraints for curves and surfaces with only (p,p) cohomology by means of wall-crossing formulas.
  
  Zoom Recording: https://ed-ac-uk.zoom.us/rec/play/hU_R_Y_n4VrE2kjikmeQISDJeNRpiagNhl0npprON-tITkzux-4zYmgfLeGvd6WJ7OH9ThL3hIAT4cuX.WwkoTvc92rU4v88w?autoplay=true&startTime=1667991936000
• EDGE: Ilaria Di Dedda (KCL) - A symplectic interpretation of Auslander correspondence
  16th November 2022, 11:00am to 12:00pm Bayes 5.10 -- Show/hide abstract
  Abstract: Abstract: Auslander correspondence establishes a bijection between the class of algebras A of finite representation type and their corresponding Auslander algebras. This allows us to study the representation theory of the former via the homological properties of the latter. The aim of this talk is to give a symplectic interpretation to this correspondence when A is the path algebra of the quiver of Dynkin type A_n. This result relies on a realisation of Auslander algebras of type A as Fukaya-Seidel categories of a family of Lefschetz fibrations. It is a stronger symplectic realisation than the one given by previous work of Dyckerhoff-Jasso-Lekili.
  
  
  Zoom Recording :https://ed-ac-uk.zoom.us/rec/play/VY3AxDdiKg-X3fCUbYjyi6gY61VJeryTZceOqNAoDkdad5y-l1J5SmY_CWFhDt-hgzr4RJxLdsujPLJ1.0WTJWbDAD_3TF-H0?autoplay=true&startTime=1668596437000
• EDGE seminar: Baris Kartal (Edinburgh) - Frobenius operators in Floer homology
  7th December 2022, 11:00am to 12:00pm Bayes 5.10 -- Show/hide abstract
  Abstract:
  Abstract: Given prime p, one can define Frobenius operators on the commutative rings of characteristic p. This notion has generalizations in a larger class of rings and even in topological spaces and spectra. Spectra with circle actions and Frobenius operators are called cyclotomic spectra. A simple example is the free loop space. Major examples arise in algebraic and arithmetic geometry, as topological Hochschild homology of rings and categories, and many applications to these fields are found. By mirror symmetry, it is natural to expect the cyclotomic spectra to arise in symplectic topology. In this talk, we will explain how to obtain cyclotomic spectra using holomorphic cylinders in symplectic manifolds, i.e. by using Hamiltonian Floer theory. Joint work in progress with Laurent Cote.
  
  
  Zoom Recording: https://ed-ac-uk.zoom.us/rec/play/-DZ8lywXOVDX-xPCGkf_Ey4uvHGRmGgCdElAlXpQg3D9x2kPgn5b0GZ-V0-AfmFhcfiJSHX_2A8F2iSO.aNp7zclgKesVF6Y1?autoplay=true&startTime=1670411023000
• GLEN seminar Fei Xie (Edinburgh)
  9th December 2022, 1:00pm to 2:00pm JCMB 4325A -- Show/hide abstract
  Abstract: Quadric surface bundles and relative Hilbert schemes of lines
  
  
  https://www.maths.ed.ac.uk/~jmartens/GLEN.html
• GLEN Seminar Matthew Pressland (Glasgow)
  9th December 2022, 2:15pm to 3:15pm JCMB 4325A -- Show/hide abstract
  Abstract: Calabi–Yau algebras from consistent dimer models
  
  https://www.maths.ed.ac.uk/~jmartens/GLEN.html
• GLEN Seminar: Tom Wennink (Liverpool)
  9th December 2022, 4:00pm to 5:00pm JCMB 4325A -- Show/hide abstract
  Abstract: A reconstruction theorem for genus 2 Gromov-Witten invariants
  
  https://www.maths.ed.ac.uk/~jmartens/GLEN.html
• EDGE: Camilla Felisetti (University of Modena and Reggio Emilia) - P=W phenomena on singular moduli spaces
  14th December 2022, 11:00am to 12:00pm Bayes 5.10 -- Show/hide abstract
  Abstract: Abstract: Irreducible holomorphic symplectic (IHS) varieties can be thought as a generalization of hyperkähler manifolds allowing singularities. Among them we can find for example moduli spaces of sheaves on K3 and abelian surfaces, which have been recently shown to play a crucial role in non abelian Hodge theory. After recalling the main features of IHS varieties, I will present several results concerning their intersection cohomology and the perverse filtration associated with a Lagrangian fibration on them, establishing a compact analogue of the celebrated P=W conjecture by de Cataldo, Hausel and Migliorini for varieties which admit a symplectic resolution. The talk is based on joint works with Mirko Mauri, Junliang Shen and Qizheng Yin. 
  
  
  Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/P7kiHbK3ULU53tYHmf3aRRxJu4Byll-2o2jYiA7GYfZ7Cu94e7du36xd7v78q8R4._ZE94RwkNh_Nab36
• EDGE: Nicola Pagani (Liverpool) - A wall-crossing formula for universal Brill-Noether classes
  18th January 2023, 11:00am to 12:00pm JCMB 6206 -- Show/hide abstract
  Abstract: Abstract: We will discuss an explicit graph formula, in terms of boundary strata classes, for the wall-crossing of universal (=over the moduli space of stable curves) Brill-Noether classes. More precisely, fix two stability conditions for universal compactified Jacobians that are on different sides of a wall in the stability space. Then we can compare the two universal Brill-Noether classes on the two compactified Jacobians by pulling one of them back along the (rational) identity map. The calculation involves constructing a resolution by means of subsequent blow-ups. If time permits, we will discuss the significance of our formula and potential applications. This is joint with Alex Abreu.
  
  
  Zoom Recording: https://ed-ac-uk.zoom.us/rec/play/ZawwlRNgi9QYVYB0OJUg0uarr-scC2YKAlwkQEBEITgpIfmLeiXDSxgiYUF_F3P2J97R5lPESzcRYZz1.I9ujkpq_o7PBKT70?autoplay=true&startTime=1674039789000
• EDGE: Andrew Dancer (Oxford) - Implosion,contraction and Moore-Tachikawa
  25th January 2023, 11:00am to 12:00pm JCMB 5323 -- Show/hide abstract
  Abstract: Abstract: We describe how the Moore-Tachikawa category provides a useful framework for interpreting constructions in symplectic and hyperkahler geometry such as implosion and contraction.This is joint work with Frances Kirwan and Johan Martens.
• EDGE: Jenia Tevelev (UMass Amherst) - Semi-orthogonal decompositions of moduli spaces
  1st February 2023, 11:00am to 12:00pm Bayes 5.10 -- Show/hide abstract
  Abstract: Abstract: Let C be a smooth projective curve of genus g at least 2 and let N be the moduli space of stable rank 2 vector bundles on C with fixed odd determinant. It is a smooth Fano variety of dimension 3g-3, Picard number 1 and index 2. We construct a semi-orthogonal decomposition of the bounded derived category of N conjectured by Narasimhan and by Belmans, Galkin and Mukhopadhyay. It has two blocks for each i-th symmetric power of C for i = 0,...,g−2 and one block for the (g − 1)-st symmetric power. Our proof is based on an analysis of wall-crossing between moduli spaces of stable pairs, combining classical vector bundles techniques with the method of windows. Joint work with Sebastian Torres.
  
  Zoom Recording :https://ed-ac-uk.zoom.us/rec/share/t6NnoQx4ojTY_EPOwJ4YvaR2Lsb5ZWwVfjEU0TjMxfroU9UQn0KPS907G9GoMwNR.W7LHDoMB_q8cTy6u
• EDGE: Dimitri Wyss (EPFL) - BPS-functions from non-archimedean integrals
  8th February 2023, 11:00am to 12:00pm JCMB 5323 -- Show/hide abstract
  Abstract: Abstract: In previous work with F. Carocci and G. Orecchia we discovered, that certain invariants appearing in Donaldson-Thomas theory for moduli of sheaves on del Pezzo surfaces admit a natural interpretation as non-archimedean integrals. Motivated by this, we develop an integration theory for smooth Artin stacks and obtain as application a new expression of the BPS-function on the coarse moduli space. Our results also imply a curious connection between BPS-cohomology and Batyrev’s conjectural stringy cohomology of varieties with Gorenstein canonical singularities. This is joint work in progress with Michael Groechenig and Paul Ziegler. (No previous knowledge of non-archimedean integration is assumed)
  
  
  Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/Gdp5eqRhM1y9Fko7DodFdLOUkfAo7O0KKkmKtVaLF4N2o09P2Ft5o6wfMI1PTDQn.ZOWZ7Cx8oMpkhkAO?startTime=1675854344000
• EDGE Seminar: Alexey Elagin (University of Edinburgh) - Dimensions of Triangulated Categories
  1st March 2023, 11:00am to 12:00pm Bayes 5.10 -- Show/hide abstract
  Abstract: I will talk about two notions of dimension of a triangulated category. The first one is more classical Rouquier dimension, based on generation time with respect to a generator, while the second one is the more recent concept of Serre dimension, based on behavior of iterations of the Serre functor. I will discuss their general properties, both known and conjectural, and provide some examples. This is a joint work with Valery Lunts.
  
  Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/tpN6EQhm8I4c5O8gi_0Uo5Vo38KMKBCVdAql1-9fPWhQ-2zCZ6OwZNNBZM4Mks9d.i9d4rsBbOrV9yn7J
• EDGE: Xiaolei Zhao (UCSB) - The Tate conjecture for h^{2, 0} = 1 varieties over finite fields
  8th March 2023, 11:00am to 12:00pm JCMB 5323 -- Show/hide abstract
  Abstract:

  Abstract: The past decade has witnessed a great advancement on the Tate conjecture for varieties with Hodge number h^{2, 0} = 1. Charles, Madapusi-Pera and Maulik completely settled the conjecture for K3 surfaces over finite fields, and Moonen proved the Mumford-Tate (and hence also Tate) conjecture for more or less arbitrary h^{2, 0} = 1 varieties in characteristic 0.
  
  In this talk, I will explain that the Tate conjecture is true for mod p reductions of complex projective h^{2, 0} = 1 varieties when p is big enough, under a mild assumption on moduli. By refining this general result, we prove that in characteristic p at least 5 the BSD conjecture holds for a height 1 elliptic curve E over a function field of genus 1, as long as E is subject to the generic condition that all singular fibers in its minimal compacification are irreducible. We also prove the Tate conjecture over finite fields for a class of surfaces of general type and Gushel-Mukai varieties. The overall philosophy is that the connection between the Tate conjecture over finite fields and the Lefschetz (1, 1)-theorem over the complex numbers is very robust for h^{2, 0} = 1 varieties, and works well beyond the hyper-Kähler world.
  
  This is based on joint work with Paul Hamacher and Ziquan Yang.
  
  Zoom Recording:
  https://ed-ac-uk.zoom.us/rec/share/Gj2Iok3_stBbvRyw6TyJhnE3CisxI28r4Z5rGzogrkgmkhaqcoBq4CCi4a5dSK6M.mouio52qqDc2-_Hf
  

• EDGE: Tudor Pădurariu (Columbia) - cancelled due to the strike
  15th March 2023, 11:00am to 12:00pm JCMB 5323
• EDGE: Ivan Smith (Cambridge) - Morava K-theory and Hamiltonian loops
  23rd March 2023, 10:00am to 11:00am JCMB 5323 -- Show/hide abstract
  Abstract: I will discuss constraints on the symplectic topology of Hamiltonian fibrations over the two-sphere, arising from the existence of virtual fundamental classes of moduli spaces of sections of these fibrations in Morava K-theory. This talk reports on joint work with Mohammed Abouzaid and Mark McLean.
• EDGE: Kirill Shakhmatov (Higher School of Economics) - Radiant toric varieties
  29th March 2023, 11:00am to 12:00pm Bayes 5.46 -- Show/hide abstract
  Abstract: Abstract: We call a complete toric variety X radiant, if a maximal unipotent subgroup of the automorphism group Aut(X) of X acts on X with an open orbit. Given a radiant toric variety X, we study the set of Demazure roots of X. We describe the structure of a maximal unipotent subgroup of Aut(X) and all its unipotent subgroups, which act on X with an open orbit and are normalized by the acting torus of X. We introduce a graph, that describes the central series of such subgroups of Aut(X). The talk is based on a joint work with Ivan Arzhantsev and Alexander Perepechko (arXiv:2209.04021).
• EDGE Seminar: Noah Porcelli (Imperial College London) - Floer theory and framed cobordisms between exact Lagrangian submanifolds
  5th April 2023, 11:00am to 12:00pm JCMB 5323 -- Show/hide abstract
  Abstract: Lagrangian Floer theory is a useful tool for studying the structure of the homology of Lagrangian submanifolds. In some cases, it can be used to detect more- we show it can detect the framed bordism class of certain Lagrangians and in particular recover a result of Abouzaid-Alvarez-Gavela-Courte-Kragh about smooth structures on Lagrangians in cotangent bundles of spheres. The main technical tool we use is Large's recent construction of a stable-homotopical enrichment of Lagrangian Floer theory.
  This is based on joint work-in-progress with Ivan Smith.
  
  Zoom Link: https://ed-ac-uk.zoom.us/rec/share/4hV-YR2BnVBx0nQr6xQZY4EdkwtZ734XGdetOzGMZEZ4vBnl1M7Q0Z6uqTbqkaTh.iDh-sISqkPoK4sTd
• EDGE: Aleksander Doan (UCL/Cambridge) - Holomorphic Floer theory and the Fueter equation
  12th April 2023, 11:00am to 12:00pm Bayes 5.46 -- Show/hide abstract
  Abstract:

  Lagrangian Floer homology is a powerful invariant associated with a pair of Lagrangian submanifolds in a symplectic manifold. I will discuss a conjectural refinement of this invariant for a pair of complex Lagrangian submanifolds in a complex symplectic manifold. The refined invariant should no longer be a homology group but a category, mimicking the well-known Fukaya-Seidel category, an invariant associated with a holomorphic function on a complex manifold. This proposal leads to many interesting problems in geometric analysis which so far remain largely unexplored. I will talk about some of these problems and discuss the special case of cotangent bundles. This talk is based on joint work with Semon Rezchikov.  

  
  

  Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/Ae-Z4DjJjJ-imeTsCUIqiwaKx6t-1w8WEV9pe1OXCJYfGAfsP2V7mL8kc1eFqVdP.u_OBaGgYZ7FKyPp3?startTime=1681293727000

• EDGE: Yash Deshmukh (Columbia) - Moduli spaces of nodal curves from homotopical algebra
  19th April 2023, 11:00am to 12:00pm JCMB 5323 -- Show/hide abstract
  Abstract:  I will explain how the Deligne-Mumford compactifications of moduli spaces of curves (of all genera) arise from the moduli spaces of framed curves by homotopically trivializing certain circle actions in an appropriate sense. I will sketch how this is relevant to the problem of relating GW invariants (in all genera) with Fukaya categories. A variation of this description gives rise to a partial compactification of moduli spaces of curves. Time permitting, I will talk about some work in progress on using it to define new operations on Rabinowitz Floer cohomology and positive symplectic cohomology.
• EDGE: Amanda Hirschi (Cambridge) - Global Kuranishi charts and a product formula in GW theory
  3rd May 2023, 11:00am to 12:00pm JCMB 5323 -- Show/hide abstract
  Abstract: I will sketch and motivate the construction of a global Kuranishi chart for the moduli space of stable pseudoholomorphic maps in a closed symplectic manifold. This allows for a straightforward definition of symplectic GW invariants in arbitrary genus. I will explain some basic properties and the proof of a formula for the GW invariants of a product of symplectic manifolds. This is joint work with Mohan Swaminathan.
  
  Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/KRpO02Wi9ofR1CgXMAjlTqmUYh9c3Ij-V7RH9GocHvAC1vwC0g1Tyiq5NyCcx2Ir.zfRB7vAbiqGWcJNl
• EDGE: Shaoyun Bai (Simons Center) - Twisted open-closed maps and Shklyarov pairing
  10th May 2023, 11:00am to 12:00pm JCMB 5323 -- Show/hide abstract
  Abstract:

  The Shklyarov pairing defines a non-degenerate bilinear form on the Hochschild homology of a proper and smooth A-infinity category. For the Fukaya category of a closed symplectic manifold, such a pairing can be identified with the Poincare pairing on the quantum cohomology. In this talk, I will present an analogue of the Shklyarov pairing on the Hochschild homology of an A-infinity automorphism, and explain how it is related to the Poincare-type pairing on fixed point Floer cohomology. Applications to Lefschetz fibrations, symplectic automorphisms, and singularity theories will also be discussed. This is joint work with Paul Seidel.

  
  

  Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/bDFKd8d4abmET_e-oh8Toz_FDz_m_zQLP96j6uje_ay6c2pF8GvxzH-fLsp-A2mD.g6SHCXunSUPm5nhz

• EDGE: Umut Varolgunes (Bogazici University)- Involutive covers of symplectic manifolds and closed string mirror symmetry
  17th May 2023, 10:00am to 11:00am Bayes 5.46 -- Show/hide abstract
  Abstract: Consider a closed graded symplectic manifold M with a finite involutive cover (notion will be reviewed). This gives a canonical spectral sequence that starts from the relative SH of the cover and converges to the quantum cohomology of M. I will discuss the compatibility of this SS with various algebraic structures, the consequences of degeneration at the earliest reasonable page and what it all means in the mirror symmetry context. If time permits I will outline a local to global computation of the A-side Yukawa coupling that is a reinterpretation of its equivalence to the B-side Yukawa coupling in mirror symmetry to illustrate the technique.
  
  Zoom Recording: https://ed-ac-uk.zoom.us/rec/play/vqzfyVqGK60DqKCoWHws95KUUoYFweJLuUZNWkknziZrocnY-H3YgmEeJwa4UZ-De54T3dk7goyQ2D8D.tn3giAO_RsyAp2-r?canPlayFromShare=true&from=share_recording_detail&continueMode=true&componentName=rec-play&originRequestUrl=https%3A%2F%2Fed-ac-uk.zoom.us%2Frec%2Fshare%2FlFeIxEMIDj3uY0Pc5ExW2w_MxBgRt1nc1mw6RtiwHXJHvB6jP5HjnoSaTjGMcXA0.hBculIl7TvDg6OZW
• EDGE: Emma Brakkee (Leiden) - Singular symplectic varieties via Prym fibrations
  17th May 2023, 11:00am to 12:00pm Bayes 5.46 -- Show/hide abstract
  Abstract: Abstract: We construct new examples of singular symplectic varieties, as relativePrym varieties associated to linear systems on surfaces with a double cover of a K3 surface. This construction has been studied before for the anti-canonical linear system on low degree del Pezzo surfaces, and for Enriques surfaces. We expand on this by considering arbitrary surfaces with a K3 double cover. I will explain the construction and discuss criteria for when the resulting symplectic variety is primitive or irreducible. This is joint work in progress with C. Camere, A. Grossi, L. Pertusi, G. Saccà and A. Viktorova.
• EDGE: Valentin Bosshard (ETH Zurich) - Lagrangian cobordisms in Liouville manifolds and Lagrangian cobordism groups
  24th May 2023, 10:00am to 11:00am JCMB 5323 -- Show/hide abstract
  Abstract:

  Lagrangian cobordisms induce cone decompositions in the Fukaya category of a symplectic manifold. In this talk, we will look at the wrapped Fukaya category and Lagrangian cobordisms between exact (possibly non-compact) Lagrangian submanifolds in Liouville manifolds, focusing on examples.
  

  How many exact triangles in the Fukaya category do we recover by Lagrangian cobordisms? This can be studied by comparing the Grothendieck group to the Lagrangian cobordism group. We will further discuss several functorial properties of these two groups which lead to explicit computations of Lagrangian cobordism groups.
  

  Link to Zoom Recording:https://ed-ac-uk.zoom.us/rec/share/Dp8ZyX7wxDqED8JQr0DqaCTKSf0D5Hgqaphq8Ul2fIVrtaoxF9j8mtgAIbiS_0iP.lc0SiGb8AvLatVja?startTime=1684918902000

• EDGE: Rob Silversmith (Warwick) - Cross-ratios and perfect matchings
  24th May 2023, 11:00am to 12:00pm JCMB 5323 -- Show/hide abstract
  Abstract: Given a (certain type of) bipartite graph G, the "cross-ratio degree" of G is a non-negative integer invariant of G, defined via a simple counting problem in algebraic geometry. I will discuss some natural contexts in which cross-ratio degrees arise, and will present a perhaps-surprising upper bound on cross-ratio degrees in terms of counting perfect matchings — whose proof involves Gromov-Witten theory. Time permitting, I will discuss the tropical side of the story.
  
  Link to Zoom Recording: https://ed-ac-uk.zoom.us/rec/share/Dp8ZyX7wxDqED8JQr0DqaCTKSf0D5Hgqaphq8Ul2fIVrtaoxF9j8mtgAIbiS_0iP.lc0SiGb8AvLatVja?startTime=1684922935000
• EDGE: Yalong Cao (RIKEN) - Quasimaps to quivers with potentials
  20th September 2023, 10:50am to 11:50am Bayes 5.46 -- Show/hide abstract
  Abstract: Quivers with potentials are fundamental objects in geometric representation theory and important also in Donaldson-Thomas theory of Calabi-Yau 3-categories. In this talk, we will introduce quantum corrections to such objects by counting quasimaps from curves to the critical locus of the potential. Our construction is based on the theory of gauged linear sigma model (GLSM) and uses recent development of DT theory of CY 4-folds. Joint work with Gufang Zhao.
  
  Zoom Recording: https://ed-ac-uk.zoom.us/rec/play/6WnKMiajfGt-pZCjUnPemrLCSe10-fHZXB9X_dAsxEMf-SiTLzhyWFhcHha_Rc8LqkjHtZFOIozqdURl.h2t6ceEV_9pu4OFi?autoplay=true&startTime=1695203552000
• EDGE: Conan Leung (Chinese University of Hong Kong) - Equivariant Floer theory and Teleman’s conjecture
  27th September 2023, 10:50am to 11:50am Bayes 5.46
• EDGE: Brent Pym (McGill) -- Hodge theory for Poisson varieties and nonperturbative quantization
  11th October 2023, 10:45am to 11:45am Bayes 5.46 -- Show/hide abstract
  Abstract: Many of the famous examples of deformation quantizations, such as quantum tori, quantum groups, elliptic Feigin--Odesskii--Sklyanin algebras, etc. have a common feature: when written in the "right" coordinates, the multiplication in the noncommutative algebra looks like a sort of exponentiation of the corresponding Poisson bracket.  Following a suggestion of Kontsevich, I will describe a conceptual explanation for this phenomenon, which combines his canonical quantization formula with Hodge theory.  The result is a natural system of "period coordinates" on the moduli space of Poisson varieties, which in many cases transforms the intractable quantization formula into a linear flow on a complex torus.  This talk is based on forthcoming joint work with A. Lindberg, and on work in progress with T. Raedschelders and S. Sierra.