Forthcoming events in this series


Tue, 28 Oct 2025
15:30
L4

Nearly G2-structures and G2-Laplacian co-flows

Jakob Stein
(State University of Campinas and University of Oxford)
Abstract

Nearly $G_2$-structures in dimension seven are, up to scaling, critical points of a geometric flow called (modified) Laplacian co-flow. Moreover, since nearly $G_2$-structures define Einstein metrics, they can also be associated to critical points of the volume-normalised Ricci flow. In this talk, we will discuss a recent joint work with Jason Lotay, showing that many of these nearly $G_2$ critical points are unstable for the modified co-flow, and giving a lower bound on the index.

Mon, 27 Oct 2025
14:15
L4

Hurwitz-Brill-Noether Theory via K3 Surfaces

Sohelya Feyzbakhsh
(Imperial College London)
Abstract

I will discuss the Brill-Noether theory of a general elliptic š¾3 surface using wall-crossing with respect to Bridgeland stability conditions. As an application, I will provide an example of a general š‘˜-gonal curve from the perspective of Hurwitz-Brill-Noether theory. This is joint work with Gavril Farkas and AndrĆ©s Rojas.

Tue, 21 Oct 2025
15:30
L4

Vector fields on intrinsic mirrors

Mark Gross
(Cambridge)
Abstract
Siebert and I gave a general construction of mirror partners to log
Calabi-Yau pairs, we called these mirror partners "intrinsic mirrors". This talk
is about a small part of a larger project with Pomerleano and Siebert aimed
at understanding this construction at a deeper level. I will explain how to
construct vector fields on the mirror using enumerative geometry of the original
log Calabi-Yau pair.
Tue, 14 Oct 2025
15:30
L4

Vafa-Witten invariants from modular anomaly

Sergey Alexandrov
(Montpelier)
Abstract
I'll present a modular anomaly equation satisfied by generating functions of refined Vafa-Witten invariants 
for the gauge group $U(N)$ on complex surfaces with $b_1=0$ and $b_2^+=1$, 
which has been derived from S-duality of string theory.
I'll show how this equation can used to find explicit expressions for these generating functions
(and their modular completions) on $\mathbb{CP}^2$, Hirzebruch and del Pezzo surfaces.
The construction for $\mathbb{CP}^2$ suggests also a new form of blow-up identities.
Tue, 17 Jun 2025
15:30
L4

Quivers and curves in higher dimensions

Hulya Arguz
(University of Georgia)
Abstract

Quiver Donaldson-Thomas invariants are integers determined by the geometry of moduli spaces of quiver representations. I will describe a correspondence between quiver Donaldson-Thomas invariants and Gromov-Witten counts of rational curves in toric and cluster varieties. This is joint work with Pierrick Bousseau.

Tue, 10 Jun 2025
15:30
L4

Cohomological Donaldson—Thomas invariants for 3-manifolds

Pavel Safronov
(Edinburgh University)
Abstract
Cohomological Donaldson—Thomas theory associates cohomology groups to various moduli spaces in algebraic geometry, such as the moduli space of coherent sheaves on a Calabi—Yau 3-fold. In this talk I will explain some recent results on cohomological DT invariants in the setting of a real 3-manifold $M$. In terms of string theory it corresponds to counting D3 branes in the compactification of a type IIB string theory on $T^* M$. This setting of DT theory is particularly interesting due to its connections to topology (via skein modules), geometric representation theory (geometric Langlands program), and mathematical physics (analytic continuation of Chern—Simons theory). This talk is based on papers joint with Gunningham, Kinjo, Naef, and Park.



 

Tue, 03 Jun 2025
15:30
L4

Bordism categories and orientations of moduli spaces

Dominic Joyce
(Oxford)
Abstract
In many situations in Differential or Algebraic Geometry, one forms moduli spaces $\cal M$ of geometric objects, such that $\cal M$ is a manifold, or something close to a manifold (a derived manifold, Kuranishi space, …). Then we can ask whether $\cal M$ is orientable, and if so, whether there is a natural choice of orientation.
  This is important in the definition of enumerative invariants: we arrange that the moduli space $\cal M$ is a compact oriented manifold (or derived manifold), so it has a fundamental class in homology, and the invariants are the integrals of natural cohomology classes over this fundamental class.
  For example, if $X$ is a compact oriented Riemannian 4-manifold, we can form moduli spaces $\cal M$ of instanton connections on some principal $G$-bundle $P$ over $X$, and the Donaldson invariants of $X$ are integrals over $\cal M$.
  In the paper arXiv:2503.20456, Markus Upmeier and I develop a theory of "bordism categoriesā€, which are a new tool for studying orientability and canonical orientations of moduli spaces. It uses a lot of Algebraic Topology, and computation of bordism groups of classifying spaces. We apply it to study orientability and canonical orientations of moduli spaces of $G_2$ instantons and associative 3-folds on $G_2$ manifolds, and of Spin(7) instantons and Cayley 4-folds on Spin(7) manifolds, and of coherent sheaves on Calabi-Yau 4-folds. These have applications to enumerative invariants, in particular, to Donaldson-Thomas type invariants of Calabi-Yau 4-folds.
   All this is joint work with Markus Upmeier.
Tue, 27 May 2025
15:30
L4

Cored perverse sheaves

Vidit Nanda
(Oxford)
Abstract

I will describe some recent efforts to recreate the miraculous properties of perverse sheaves on complex analytic spaces in the setting of real stratified spaces.

Tue, 20 May 2025
15:30
L4

Relative orientations and the cyclic Deligne conjecture

Nick Rozenblyum
(University of Toronto)
Abstract

A consequence of the works of Costello and Lurie is that the Hochschild chain complex of a Calabi-Yau category admits the structure of a framed E_2 algebra (the genus zero operations). I will describe a new algebraic point of view on these operations which admits generalizations to the setting of relative
Calabi-Yau structures, which do not seem to fit into the framework of TQFTs. In particular, we obtain a generalization of string topology to manifolds with boundary, as well as interesting operations on Hochschild homology of Fano varieties. This is joint work with Chris Brav.

Tue, 13 May 2025
15:30
L4

Parametrising complete intersections

Jakub Wiaterek
(Oxford)
Abstract

We use Non-Reductive GIT to construct compactifications of Hilbert schemes of complete intersections. We then study ample line bundles on these compactifications in order to construct moduli spaces of complete intersections for certain degree types.

Tue, 06 May 2025
15:30
L4

Fukaya categories at singular values of the moment map

Ed Segal
(University College London)
Abstract

Given a Hamiltonian circle action on a symplectic manifold, Fukaya and Teleman tell us that we can relate the equivariant Fukaya category to the Fukaya category of a symplectic reduction.  Yanki Lekili and I have some conjectures that extend this story - in certain special examples - to singular values of the moment map. I'll also explain the mirror symmetry picture that we use to support our conjectures, and how we interpret our claims in Teleman's framework of `topological group actions' on categories.



 

Tue, 29 Apr 2025
15:30
L4

On the birational geometry of algebraically integrable foliations

Paolo Cascini
(Imperial College London)
Abstract

I will review recent progress on extending the Minimal Model Program to algebraically integrable foliations, focusing on applications such as the canonical bundle formula and recent results toward the boundedness of Fano foliations.

Tue, 11 Mar 2025
15:30
L4

Quiver with potential and attractor invariants

Pierre Descombes
(Imperial College London)
Abstract
Given a quiver (a directed graph) with a potential (a linear combination of cycles), one can study moduli spaces of the associated noncommutative algebra and associate so-called BPS invariants to them. These are interesting because they have a deep link with cluster algebras and provide some kind of noncommutative analogue of DT theory, the study of sheaves on Calabi-Yau 3-folds.
The generating series of BPS invariants for interesting quivers with potentials are in general very wild. However, using the Kontsevich-Soibelman wall-crossing formula, a recursive formula expresses the BPS invariants in terms of so-called attractor invariants, which are expected to be simple in interesting situations. We will discuss them for quivers with potential associated to triangulations of surfaces and quivers with potential giving noncommutative resolutions of CY3 singularities.
Tue, 04 Mar 2025
15:30
L4

Mixed characteristic analogues of Du Bois and log canonical singularities

Joe Waldron
(Michigan State University)
Abstract

Singularities are measured in different ways in characteristic zero, positive characteristic, and mixed characteristic. However, classes of singularities usually form analogous groups with similar properties, with an example of such a group being klt, strongly F-regular and BCM-regular.  In this talk we shall focus on newly introduced mixed characteristic counterparts of Du Bois and log canonical singularities and discuss their properties. 

This is joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker and Jakub Witaszek. 

Tue, 25 Feb 2025
15:30
L4

The Logarithmic Hilbert Scheme

Patrick Kennedy-Hunt
(Cambridge)
Abstract

I am interested in studying moduli spaces and associated enumerative invariants via degeneration techniques. Logarithmic geometry is a natural language for constructing and studying relevant moduli spaces. In this talk I  will explain the logarithmic Hilbert (or more generally Quot) scheme and outline how the construction helps study enumerative invariants associated to Hilbert/Quot schemes- a story we now understand well. Time permitting, I will discuss some challenges and key insights for studying moduli of stable vector bundles/ sheaves via similar techniques - a theory whose details are still being worked out. 

Tue, 18 Feb 2025
15:30
L4

Invariance of elliptic genus under wall-crossing

Henry Liu
(IPMU Tokyo)
Abstract

Elliptic genus, and its various generalizations, is one of the simplest numerical invariants of a scheme that one can consider in elliptic cohomology. I will present a topological condition which implies that elliptic genus is invariant under wall-crossing. It is related to Krichever-Hƶhn’s elliptic rigidity. Many applications are possible: to GIT quotients, moduli of sheaves, Donaldson-Thomas invariants, etc.

Tue, 11 Feb 2025
15:30
L4

Equivariant Floer theory for symplectic C*-manifolds

Alexander Ritter
(Oxford)
Abstract
The talk will be on recent progress in a series of joint papers with Filip Živanović, about a large class of non-compact symplectic manifolds, which includes semiprojective toric varieties, quiver varieties, and conical symplectic resolutions of singularities. These manifolds admit a Hamiltonian circle action which is part of a pseudo-holomorphic action of a complex torus. The symplectic form on these spaces is highly non-exact, yet we can make sense of Hamiltonian Floer cohomology for functions of the moment map of the circle action. We showed that Floer theory induces a filtration by ideals on quantum cohomology. I will explain recent progress on equivariant Floer cohomology for these spaces, in which case we obtain a filtration on equivariant quantum cohomology. If time permits, I will also mention a presentation of symplectic cohomology and quantum cohomology for semiprojective toric varities.
Tue, 04 Feb 2025
15:30
L4

Global logarithmic deformation theory

Simon Felten
(Oxford)
Abstract

A well-known problem in algebraic geometry is to construct smooth projective Calabi-Yau varieties $Y$. In the smoothing approach, we construct first a degenerate (reducible) Calabi-Yau scheme $V$ by gluing pieces. Then we aim to find a family $f\colon X \to C$ with special fiber $X_0 = f^{-1}(0) \cong V$ and smooth general fiber $X_t = f^{-1}(t)$. In this talk, we see how infinitesimal logarithmic deformation theory solves the second step of this approach: the construction of a family out of a degenerate fiber $V$. This is achieved via the logarithmic Bogomolov-Tian-Todorov theorem as well as its variant for pairs of a log Calabi-Yau space $f_0\colon X_0 \to S_0$ and a line bundle $\mathcal{L}_0$ on $X_0$.

Tue, 21 Jan 2025
15:30
L4

Deformations and lifts of Calabi-Yau varieties in characteristic p

Lukas Brantner
(Oxford)
Abstract

Derived algebraic geometry allows us to study formal moduli problems via their tangent Lie algebras. After briefly reviewing this general paradigm, I will explain how it sheds light on deformations of Calabi-Yau varieties. 
In joint work with Taelman, we prove a mixed characteristic analogue of the Bogomolov–Tian–Todorov theorem, which asserts that Calabi-Yau varieties in characteristic $0$ are unobstructed. Moreover, we show that ordinary Calabi–Yau varieties in characteristic $p$ admit canonical (and algebraisable) lifts to characteristic $0$, generalising results of Serre-Tate for abelian varieties and Deligne-Nygaard for K3 surfaces. 
If time permits, I will conclude by discussing some intriguing questions related to our canonical lifts.  
 

Tue, 30 May 2017

15:45 - 16:45
L4

Symmetries in monotone Lagrangian Floer theory

Jack Smith
(Cambridge)
Abstract

Lagrangian Floer cohomology groups are extremely hard compute in most situations. In this talk I’ll describe two ways to extract information about the self-Floer cohomology of a monotone Lagrangian possessing certain kinds of symmetry, based on the closed-open string map and the Oh spectral sequence. The focus will be on a particular family of examples, where the techniques can be combined to deduce some unusual properties.

Tue, 23 May 2017

15:45 - 16:45
L4

On Short Time Existence of Lagrangian Mean Curvature Flow

Tom Begley
(Cambridge)
Abstract

The goal of this talk will be to give an overview of recent work, joint with Kim Moore, on a short time existence problem in Lagrangian mean curvature flow. More specifically, we consider a compact initial Lagrangian submanifold with a finite number of singularities, each asymptotic to a pair of transversely intersecting planes. We show it is possible to construct a smooth Lagrangian mean curvature flow, existing for positive times, that attains the singular Lagrangian as its initial condition in a suitable weak sense.  The construction uses a family of smooth solutions whose initial conditions approximate the singular Lagrangian. In order to appeal to compactness theorems and produce the desired solution, it is necessary to first establish uniform curvature estimates on the approximating family. As time allows I hope to focus in particular on the proof of these estimates, and their role in the proof of the main theorem.

Tue, 16 May 2017

15:45 - 16:45
L4

Uniruling of symplectic quotients of coisotropic submanifolds

Tobias Sodoge
(UCL)
Abstract


Coisotropic submanifolds arise naturally in symplectic geometry as level sets of moment maps and in algebraic geometry in the context of normal crossing divisors. In examples, the Marsden-Weinstein quotient or (Fano) complete intersections are often uniruled. 
We show that under natural geometric assumptions on a coisotropic submanifold, the symplectic quotient of the coisotropic is always geometrically uniruled. 
I will explain how to assign a Lagrangian and a hypersurface to a fibered, stable coisotropic C. The Lagrangian inherits a fibre bundle structure from C, the hypersurface captures the generalised Reeb dynamics on C. To derive the result, we then adapt and apply techniques from Lagrangian Floer theory and symplectic field theory.
This is joint work with Jonny Evans.
 

Tue, 09 May 2017

15:45 - 16:45
L4

Limits of Yang-Mills alpha-connections

Casey Lynn Kelleher
(UC Irvine)
Abstract
In the spirit of recent work of Lamm, Malchiodi and Micallef in the setting of harmonic maps, we identify Yang-Mills connections obtained by approximations with respect to the Yang-Mills alpha-energy. More specifically, we show that for the SU(2) Hopf fibration over the four sphere, for sufficiently small alpha values the rotation invariant ADHM connection is the unique alpha-critical point which has Yang-Mills alpha-energy lower than a specific threshold.
Tue, 02 May 2017

15:45 - 16:45
L4

Gopakumar-Vafa type invariants for Calabi-Yau 4-folds

Yalong Cao
(Oxford)
Abstract
As an analogy of Gopakumar-Vafa conjecture for CY 3-folds, Klemm-Pandharipande proposed GV type invariants on CY 4-folds using GW theory and conjectured their integrality. In this talk, we propose a sheaf theoretical interpretation to these invariants using Donaldson-Thomas theory on CY 4-folds. This is a joint work with Davesh Maulik and Yukinobu Toda.
Thu, 09 Mar 2017

16:00 - 17:00
L2

(COW seminar) Gopakumar-Vafa invariants via vanishing cycles

Davesh Maulik
(MIT)
Abstract

Given a Calabi-Yau threefold X, one can count curves on X using various approaches, for example using stable maps or ideal sheaves; for any curve class on X, this produces an infinite sequence of invariants, indexed by extra discrete data (e.g. by the domain genus of a stable map).  Conjecturally, however, this sequence is determined by only a finite number of integer invariants, known as Gopakumar-Vafa invariants.  In this talk, I will propose a direct definition of these invariants via sheaves of vanishing cycles, building on earlier approaches of Kiem-Li and Hosono-Saito-Takahashi.  Conjecturally, these should agree with the invariants as defined by stable maps.  I will also explain how to prove the conjectural correspondence for irreducible curves on local surfaces.  This is joint work with Yukinobu Toda.

Thu, 09 Mar 2017

14:30 - 15:30
L4

(COW seminar) Strange duality on abelian surfaces

Barbara Bolognese
Abstract

With the purpose of examining some relevant geometric properties of the moduli space of sheaves over an algebraic surface, Le Potier conjectured some unexpected duality between the complete linear series of certain natural divisors, called Theta divisors, on the moduli space. Such conjecture is widely known as Strange Duality conjecture. After having motivated the problem by looking at certain instances of quantization in physics, we will work in the setting of surfaces. We will then sketch the proof in the case of abelian surfaces, giving an idea of the techniques that are used. In particular, we will show how the theory of discrete Heisenberg groups and fiber wise Fourier-Mukai transforms, which might be applied to other cases of interest, enter the picture. This is joint work with Alina Marian, Dragos Opera and Kota Yoshioka.

Tue, 07 Mar 2017
15:45
L4

Local cohomology and canonical stratification

Vidit Nanda
(Oxford)
Abstract

Every finite regular CW complex is, ipso facto, a cohomologically stratified space when filtered by skeleta. We outline a method to recover the canonical (i.e., coarsest possible) stratification of such a complex that is compatible with its underlying cell structure. Our construction proceeds by first localizing and then resolving a complex of cosheaves which capture local cohomology at every cell. The result is a sequence of categories whose limit recovers the desired strata via its (isomorphism classes of) objects. As a bonus, we observe that the entire process is algorithmic and amenable to efficient computations!

Tue, 28 Feb 2017

15:45 - 16:45

Tropical compactifications, Mori Dream Spaces and Minkowski bases

Elisa Postinghel
(Loughborough University)
Abstract

Given a Mori Dream Space X, we construct via tropicalisation a model dominating all the small Q-factorial modifications of X. Via this construction we recover a Minkowski basis for the Newton-Okounkov bodies of Cartier divisors on X and hence generators of the movable cone of X. 
This is joint work with Stefano Urbinati.
 

Tue, 21 Feb 2017

15:45 - 16:45
L4

Group actions on quiver moduli spaces

Vicky Hoskins
(Freie UniversitƤt Berlin)
Abstract

We consider two types of actions on moduli spaces of quiver representations over a field k and we decompose their fixed loci using group cohomology. First, for a perfect field k, we study the action of the absolute Galois group of k on the points of this quiver moduli space valued in an algebraic closure of k; the fixed locus is the set of k-rational points and we obtain a decomposition of this fixed locus indexed by the Brauer group of k. Second, we study algebraic actions of finite groups of quiver automorphisms on these moduli spaces; the fixed locus is decomposed using group cohomology and each component has a modular interpretation. If time permits, we will describe the symplectic and holomorphic geometry of these fixed loci in hyperkaehler quiver varieties. This is joint work with Florent Schaffhauser.

Tue, 07 Feb 2017

15:45 - 16:45
L4

Quantum character varieties and the double affine Hecke algebra

David Jordan
(Edinburgh)
Abstract

The character variety of a manifold is a moduli space of representations of its fundamental group into some fixed gauge group.  In this talk I will outline the construction of a fully extended topological field theory in dimension 4, which gives a uniform functorial quantization of the character varieties of low-dimensional manifolds, when the gauge group is reductive algebraic (e.g. $GL_N$).

I'll focus on important examples in representation theory arising from the construction, in genus 1:  spherical double affine Hecke algebras (DAHA), difference-operator q-deformations of the Grothendieck-Springer sheaf, and the construction of irreducible DAHA modules mimicking techniques in classical geometric representation theory.  The general constructions are joint with David Ben-Zvi, Adrien Brochier, and Noah Snyder, and applications to representation theory of DAHA are joint with Martina Balagovic and Monica Vazirani.

Tue, 31 Jan 2017

15:45 - 16:45
L4

Universal flops and noncommutative algebras

Joe Karmazyn
(Sheffield)
Abstract

A classification of simple flops on smooth threefolds in terms of the length invariant was given by Katz and Morrison, who showed that the length must take the value 1,2,3,4,5, or 6. This classification was produced by understanding simultaneous (partial) resolutions that occur in the deformation theory of A, D, E Kleinian surface singularities. An outcome of this construction is that all simple threefold flops of length l occur by pullback from a "universal flop" of length l. Curto and Morrison understood the universal flops of length 1 and 2 using matrix factorisations. I aim to describe how these universal flops can understood for lengths >2 via noncommutative algebra.

Tue, 17 Jan 2017

15:45 - 16:45
L4

The universal property of derived geometry

Andrew MacPherson
(London)
Abstract

I'll show how a simple universal property attaches a category of derived manifolds to any category with finite products and some suitable notion of "topology". Starting with the category of real Euclidean spaces and infinitely differentiable maps yields the category of derived smooth manifolds studied by D. Spivak and others, while starting with affine spaces over some ring and polynomial maps produces a flavour of the derived algebraic geometry of Lurie and Toen-Vezzosi.

I'll motivate this from the differentiable setting by showing that the universal property easily implies all of D. Spivak's axioms for being "good for intersection theory on manifolds".

Tue, 29 Nov 2016

15:45 - 16:45
L4

On short time existence of Lagrangian mean curvature flow

Kim Moore
(Cambridge)
Abstract

One of the simplest, and yet largely still open, questions that one can ask about special Lagrangian submanifolds is whether they exist in a given homology class. One possible approach to this problem is to evolve a given Lagrangian submanifold under mean curvature flow in the hope of reaching a special Lagrangian submanifold in the same homology class. It is known, however, that even for 'nice' initial conditions the flow will develop singularities in finite time. 

I will talk about a joint work with Tom Begley, in which we prove a short time existence result for Lagrangian mean curvature flow, where the initial condition is a Lagrangian submanifold of complex Euclidean space with a certain type of singularity. This is a first step to proving, as conjectured by Joyce, that one may 'continue' Lagrangian mean curvature flow after the occurrence of singularities.

Tue, 22 Nov 2016

15:45 - 16:45
L4

The Cohomological McKay Correspondence and Symplectic Cohomology

Mark McLean
(Stony Brook)
Abstract

Suppose that we have a finite quotient singularity $\mathbb C^n/G$ admitting a crepant resolution $Y$ (i.e. a resolution with $c_1 = 0$). The cohomological McKay correspondence says that the cohomology of $Y$ has a basis given by irreducible representations of $G$ (or conjugacy classes of $G$). Such a result was proven by Batyrev when the coefficient field $\mathbb F$ of the cohomology group is $\mathbb Q$. We give an alternative proof of the cohomological McKay correspondence in some cases by computing symplectic cohomology+ of $Y$ in two different ways. This proof also extends the result to all fields $\mathbb F$ whose characteristic does not divide $|G|$ and it gives us the corresponding basis of conjugacy classes in $H^*(Y)$. We conjecture that there is an extension to certain non-crepant resolutions. This is joint work with Alex Ritter.

Tue, 15 Nov 2016

15:45 - 16:45
L4

The Bayer-Macri map for compact support

Alastair Craw
(Bath)
Abstract

I'll discuss recent joint work with Arend Bayer and Ziyu Zhang in which we define a nef divisor class on moduli spaces of Bridgeland-stable objects in the derived category of coherent sheaves with compact support, generalising earlier work of Bayer and Macri for smooth projective varieties. This work forms part of a programme to study the birational geometry of moduli spaces of Bridgeland-stable objects in the derived category of varieties that need not be smooth and projective.

Tue, 08 Nov 2016

15:45 - 16:45
L4

Towards a compactification of the moduli space of K3 surfaces of degree 2

Alan Thompson
(Warwick)
Abstract

Ever since moduli spaces of polarised K3 surfaces were constructed in the 1980's, people have wondered about the question of compactification: can one make the moduli space of K3 surfaces compact by adding in some boundary components in a "nice" way? Ideally, one hopes to find a compactification that is both explicit and geometric (in the sense that the boundary components provide moduli for degenerate K3's). I will present on joint work in progress with V. Alexeev, which aims to solve the compactification problem for the moduli space of K3 surfaces of degree 2.

Tue, 01 Nov 2016

15:45 - 16:45
L4

A geometric approach to Hall algebras

Adam Gal
(Oxford)
Abstract

The Hall algebra can be constructed using the Waldhausen S-construction. We will give a systematic recipe for this and show how it extends naturally to give a bi-algebraic structure. As a result we obtain a more transparent proof of Green's theorem about the bi-algebra structure on the Hall algebra.

Tue, 18 Oct 2016
15:45
L4

Separating invariants and local cohomology

Emilie DuFresne
(Oxford)
Abstract

The study of separating invariants is a new trend in Invariant Theory and a return to its roots: invariants as a classification tool. For a finite group acting linearly on a vector space, a separating set is simply a set of invariants whose elements separate the orbits o the action. Such a set need not generate the ring of invariants. In this talk, we give lower bounds on the size of separating sets based on the geometry of the action. These results are obtained via the study of the local cohomology with support at an arrangement of linear subspaces naturally arising from the action.

(Joint with Jack Jeffries)

Mon, 11 Jul 2016

16:30 - 17:30
L3

(COW SEMINAR) Monodromy and derived equivalences

Andrei Okounkov
(Columbia)
Abstract

This is will be a progress report on our long-ongoing joint work with Bezrukavnikov on lifting the monodromy of the quantum differential equation for symplectic resolutions to automorphisms of their derived categories of coherent sheaves. I will attempt to define the ingredient that go both into the problem and into its solution.
 

Mon, 11 Jul 2016

14:45 - 15:45
L3

(COW SEMINAR) Higgs bundles and determinant divisors

Nigel Hitchin
(Oxford)
Abstract

Following an idea of Gaiotto, a symplectic representation of a complex Lie group G defines a complex Lagrangian subvariety inside the moduli space of G-Higgs bundles. The talk will discuss the case of G=SL(2) and its link with determinant  divisors, or equivalently Brill-Noether loci, in the moduli space of semistable SL(2)-bundles. 

Mon, 11 Jul 2016

13:30 - 14:30
L3

(COW SEMINAR) Categorification of shifted symplectic geometry using perverse sheaves

Dominic Joyce
(Oxford)
Abstract

Given a (-1)-shifted symplectic derived scheme or stack (X,w) over C equipped with an orientation, we explain how to construct a perverse sheaf P on the classical truncation of X so that its hypercohomology H*(P) can be regarded as a categorification of (or linearisation of) X. Given also a Lagrangian morphism L -> X equipped with a relative orientation, we outline a programme in progress to construct a natural morphism of constructible complexes on the truncation of L from the (shifted) constant complex on L to a suitable pullback of P to L. The morphisms and resulting hypercohomology classes are expected to satisfy various identities under products, composition of Lagrangian correspondences, etc. This programme will have interesting applications, such as proving associativity of a Kontsevich-Soibelman type COHA multiplication on H*(P) when X is the derived moduli stack of coherent sheaves on a Calabi-Yau 3-fold Y, and defining Lagrangian Floer cohomology and the Fukaya cat!
 egory of an algebraic or complex symplectic manifold S.

Tue, 14 Jun 2016

15:45 - 16:45
L4

Symplectic homology for cobordisms

Alexandru Oancea
(Jussieu)
Abstract

I will present a definition of symplectic homology groups for pairs of Liouville cobordisms with fillings, and explain how these fit into a formalism of homology theory similar to that of Eilenberg and Steenrod. This construction allows to understand form a unified point of view many structural results involving Floer homology groups, and yields new applications. Joint work with Kai Cieliebak.

Tue, 07 Jun 2016

15:45 - 16:45
L4

Matrix factorisation of Morse-Bott functions

Constantin Teleman
(Oxford)
Abstract

For a holomorphic function (ā€œsuperpotentialā€)  W: X —> C on a complex manifold X, one defines the (2-periodic) matrix factorisation category MF(X;W), which is supported on the critical locus Crit(W) of W. At a Morse singularity, MF(X;W) is equivalent to the category of modules over the Clifford algebra on the tangent space TX. It had been suggested by Kapustin and Rozansky that, for Morse-Bott W, MF(X;W) should be equivalent to the (2-periodicised) derived category of Crit(W), twisted by the Clifford algebra of the normal bundle. I will discuss why this holds when the first neighbourhood of Crit(W) splits, why it fails in general, and will explain the correct general statement.

Tue, 31 May 2016

15:45 - 16:45
L4

Non-reductive GIT for graded groups and curve counting

Greg Berczi
(Oxford)
Abstract
I will start with a short report on recent progress in constructing quotients by actions of non-reductive algebraic groups and extending Mumford's geometric invariant theory to a wide class of non-reductive linear algebraic groups which we call graded groups. I will then explain how certain components of the Hilbert scheme of points on smooth varieties can be described as non-reductive quotients and why this description is especially efficient to study the topology of Hilbert schemes. In particular I will explain how equivariant localisation can be used to develop iterated residue formulae for tautological integrals on geometric subsets of Hilbert schemes and I present new formulae counting curves on surfaces (and more generally hypersurfaces in smooth varieties) with given singularity classes. This talk is based on joint works with Frances Kirwan, Thomas Hawes, Brent Doran and Andras Szenes. 
Tue, 10 May 2016

15:30 - 17:00
L4

Cohomological DT theory beyond the integrality conjecture

Ben Davison
(EPFL)
Abstract
The integrality conjecture is one of the central conjectures of the DT theory of quivers with potential, which itself is a key tool in understanding the local calculation of DT invariants on moduli spaces of coherent sheaves, as well as having deep links to geometric representation theory, noncommutative geometry and algebraic combinatorics.  I will explain some of the ingredients of the proof of this conjecture by myself and Sven Meinhardt.  In fact the proof gives much more than the original conjecture, which ultimately concerns identities in a Grothendieck ring of mixed Hodge structures associated to moduli spaces of representations, and proves that these equalities categorify to isomorphisms in the category of mixed Hodge structures.  I'll explain what this all means, as well as giving some applications of the categorified version of the theory.