Forthcoming events in this series


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-Höhn’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.