Forthcoming events in this series


Tue, 14 Jun 2022
15:30
Virtual

Co-associative fibrations of $G_{2}$ manifolds: foundations and speculations.

Simon Donaldson
(Imperial College London and SCGP)
Further Information

The talk will be online (Zoom). People who would like to attend the seminar can also meet in person in L3.

Abstract

The introduction to the talk will review basics of $G_{2}$ geometry in seven dimensions, and associative and co-associative submanifolds. In one part of the talk we will explain how fibrations with co-associative fibres, near the “adiabatic limit” when the fibres are very small,  give insights into various questions about moduli spaces of $G_{2}$ structures and singularity formation. This part is mostly speculative. In the other part of the talk we discuss some analysis questions which enter when setting up the foundations of this adiabatic theory. These can be seen as codimension 2 analogues of free boundary problems and related questions have arisen in a number of areas of differential geometry recently.

Mon, 13 Jun 2022
14:15
L5

Open FJRW theory

Mark Gross
(Cambridge)
Abstract

I will describe joint work with Tyler Kelly and Ran Tessler. FJRW (Fan-Jarvis-Ruan-Witten) theory is an enumerative theory of quasi-homogeneous singularities, or alternatively, of Landau-Ginzburg models. It associates to a potential W:C^n -> C given by a quasi-homogeneous polynomial moduli spaces of (orbi-)curves of some genus and marked points along with some extra structure, and these moduli spaces carry virtual fundamental classes as constructed by Fan-Jarvis-Ruan. Here we specialize to the case W=x^r+y^s and construct an analogous enumerative theory for disks. We show that these open invariants provide perturbations of the potential W in such a way that mirror symmetry becomes manifest. Further, these invariants are dependent on certain choices of boundary conditions, but satisfy a beautiful wall-crossing formalism.

Mon, 06 Jun 2022
14:15
L5

Symplectic cohomology of compound Du Val singularities

Jonny Evans
(University of Lancaster)
Abstract

(Joint with Y. Lekili) If someone gives you a variety with a singular point, you can try and get some understanding of what the singularity looks like by taking its “link”, that is you take the boundary of a neighbourhood of the singular point. For example, the link of the complex plane curve with a cusp $y^2 = x^3$ is a trefoil knot in the 3-sphere. I want to talk about the links of a class of 3-fold singularities which come up in Mori theory: the compound Du Val (cDV) singularities. These links are 5-dimensional manifolds. It turns out that many cDV singularities have the same 5-manifold as their link, and to tell them apart you need to keep track of some extra structure (a contact structure). We use symplectic cohomology to distinguish the contact structures on many of these links.

Mon, 30 May 2022
14:15
L5

Drinfeld's conjecture and generalisations

Ana Peón-Nieto
(University of Birmingham)
Abstract

The so called Drinfeld conjecture states that the complement to very stable bundles has pure codimension one in the moduli space of vector bundles. In this talk I will explain a constructive proof in rank three, and discuss if/how it generalises to wobbly fixed points of the nilpotent cone as defined by Hausel and Hitchin. This is joint work with Pauly (Nice).

Mon, 23 May 2022
14:15
L5

Ancient solutions and translators in Lagrangian mean curvature flow

Felix Schulze
(University of Warwick)
Abstract

For almost calibrated Lagrangian mean curvature flow it is known that all singularities are of Type II. To understand the finer structure of the singularities forming, it is thus necessary to understand the structure of general ancient solutions arising as potential limit flows at such singularities. We will discuss recent progress showing that ancient solutions with a blow-down a pair of static planes meeting along a 1-dimensional line are translators. This is joint work with J. Lotay and G. Szekelyhidi.

Mon, 16 May 2022
14:15
L5

Morava K-theory and Hamiltonian loops

Ivan Smith
(Cambridge)
Abstract

A loop of Hamiltonian diffeomorphisms of a symplectic manifold $X$ defines, by clutching, a symplectic fibration over the two-sphere with fibre $X$.  We prove that the integral cohomology of the total space splits additively, answering a question of McDuff, and extending the rational cohomology analogue proved by Lalonde-McDuff-Polterovich in the late 1990’s. The proof uses a virtual fundamental class of moduli spaces of sections of the fibration in Morava K-theory. This talk reports on joint work with Mohammed Abouzaid and Mark McLean.

Mon, 09 May 2022
14:15
L5

Conformally Invariant Energies of Curves and Surfaces

Alexis Michelat
(Oxford University)
Abstract

The integral of mean curvature squared is a conformal invariant of surfaces reintroduced by Willmore in 1965 whose study exercised a tremendous influence on geometric analysis and most notably on minimal surfaces in the last years.


On the other hand, the Loewner energy is a conformal invariant of planar curves introduced by Yilin Wang in 2015 which is notably linked to SLE processes and the Weil-Petersson class of (universal) Teichmüller theory.


In this presentation, after a brief historical introduction, we will discuss some recent developments linking the Willmore energy to the Loewner energy and mention several open problems.


Joint work with Yilin Wang (MIT/MSRI)

Mon, 02 May 2022
14:15
L5

Hypersurfaces with prescribed-mean-curvature: existence and properties

Costante Bellettini
(University College London)
Abstract

Let $N$ be a compact Riemannian manifold of dimension 3 or higher, and $g$ a Lipschitz non-negative (or non-positive) function on $N$. In joint works with Neshan Wickramasekera we prove that there exists a closed hypersurface $M$ whose mean curvature attains the values prescribed by $g$. Except possibly for a small singular set (of codimension 7 or higher), the hypersurface $M$ is $C^2$ immersed and two-sided (it admits a global unit normal); the scalar mean curvature at $x$ is $g(x)$ with respect to a global choice of unit normal. More precisely, the immersion is a quasi-embedding, namely the only non-embedded points are caused by tangential self-intersections: around such a non-embedded point, the local structure is given by two disks, lying on one side of each other, and intersecting tangentially (as in the case of two spherical caps touching at a point). A special case of PMC (prescribed-mean-curvature) hypersurfaces is obtained when $g$ is a constant, in which the above result gives a CMC (constant-mean-curvature) hypersurface for any prescribed value of the mean curvature.

Mon, 25 Apr 2022
14:15
L5

Ricci flows with nonstandard initial data

Peter Topping
(University of Warwick)
Abstract

Most Ricci flow theory takes the short-time existence of solutions as a starting point and ends up concerned with understanding the long-time limiting behaviour and the structure of any finite-time singularities that may develop along the way. In this talk I will look at what you can think of as singularities at time zero. I will describe some of the situations in which one would like to start a  Ricci flow with a space that is rougher than a smooth bounded curvature Riemannian manifold, and some of the situations in which one considers smooth initial data that is only achieved in a non-smooth way. A particularly interesting and useful case is the problem of starting a Ricci flow on a Riemann surface equipped with a measure. I will not be assuming expertise in Ricci flow theory. Parts of the talk are joint with either Hao Yin (USTC) or ManChun Lee (CUHK).

Mon, 07 Mar 2022
14:15
L5

Brakke Regularity for the Allen--Cahn Flow

Huy The Nguyen
(Queen Mary University, London)
Further Information

The talk will be both online (Teams) and in person (L5)

Abstract

In this talk we prove an analogue of the Brakke's $\epsilon$-regularity theorem for the parabolic Allen--Cahn equation. In particular, we show uniform $C^{2,\alpha}$ regularity for the transition layers converging to smooth mean curvature flows as $\epsilon\rightarrow 0$. A corresponding gap theorem for entire eternal solutions of the parabolic Allen--Cahn is also obtained. As an application of the regularity theorem, we give an affirmative answer to a question of Ilmanen that there is no cancellation in BV convergence in the mean convex setting.

Mon, 28 Feb 2022
14:15
L5

Chow quotients and geometric invariant theoretic quotients for group actions on complex projective varieties

Frances Kirwan
(University of Oxford)
Further Information

The talk will be both online (Teams) and in person (L5)

Abstract

When a reductive group G acts on a complex projective variety
X, there exist different methods for finding an open G-invariant subset
of X with a geometric quotient (the 'stable locus'), which is a
quasi-projective variety and has a projective completion X//G. Mumford's
geometric invariant theory (GIT) developed in the 1960s provides one way
to do this, given a lift of the action to an ample line bundle on X,
though with no guarantee that the stable locus is not empty. An
alternative approach due to Kapranov and others in the 1990s is to use
Chow varieties to define a 'Chow quotient' X//G. The aim of this talk is
to review the relationship between these constructions for reductive
groups, and to discuss the situation when G is not reductive.

Mon, 21 Feb 2022
14:15
L5

Anti-self-dual instantons and codimension-1 collapse

Lorenzo Foscolo
(University College London)
Further Information

The talk will be both online (Teams) and in person (L5)

Abstract

We study the behaviour of anti-self-dual instantons on $\mathbb{R}^3 \times S^1$ (also known as calorons) under codimension-1 collapse, i.e. when the circle factor shrinks to zero length. In this limit, the instanton equation reduces to the well-known Bogomolny equation of magnetic monopoles on $\mathbb{R}^3 $. However, inspired by work of Kraan and van Baal in the mathematical physics literature, we show how $SU(2)$ instantons can be realised as superpositions of monopoles and "rotated monopoles" glued into a singular background abelian configuration consisting of Dirac monopoles of positive and negative charges. I will also discuss generalisations of the construction to calorons with arbitrary structure group and potential applications to the hyperkähler geometry of moduli spaces of calorons. This is joint work with Calum Ross.

Mon, 14 Feb 2022
14:15
L5

Quiver varieties and moduli spaces attached to Kleinian singularities

Søren Gammelgaard
(University of Oxford)
Further Information

The talk will be both online (Teams) and in person (L5)

Abstract

Let $\Gamma$ be a finite subgroup of $SL(2, \mathbb{C})$. We can attach several different moduli spaces to the action of $\Gamma$ on $\mathbb{C}^2$, and we show how Nakajima's quiver varieties provide constructions of them. The definition of such a quiver variety depends on a stability parameter, and we are especially interested in what happens when this parameter moves into a specific ray in its associated wall-and-chamber structure. Some of the resulting quiver varieties can be understood as moduli spaces of certain framed sheaves on an appropriate stacky compactification of the Kleinian singularity $\mathbb{C}^2/\Gamma$. As a special case, this includes the punctual Hilbert schemes of $\mathbb{C}^2/\Gamma$.

Much of this is joint work with A. Craw, Á. Gyenge, and B. Szendrői.

Mon, 07 Feb 2022
14:15
L5

Nonabelian Hodge theory and the decomposition theorem for 2-CY categories

Ben Davison
(Edinburgh)
Further Information

The talk will be both online (Teams) and in person (L5)

Abstract

Examples of 2CY categories include the category of coherent sheaves on a K3 surface, the category of Higgs bundles, and the category of modules over preprojective algebras or fundamental group algebras of compact Riemann surfaces.  Let p:M->N be the morphism from the stack of semistable objects in a 2CY category to the coarse moduli space.  I'll explain, using cohomological DT theory, formality in 2CY categories, and structure theorems for good moduli stacks, how to prove a version of the BBDG decomposition theorem for the exceptional direct image of the constant sheaf along p, even though none of the usual conditions for the decomposition theorem apply: p isn't projective or representable, M isn't smooth, the constant mixed Hodge module complex Q_M isn't pure...  As an application, I'll explain how this allows us to extend nonabelian Hodge theory to Betti/Dolbeault stacks.

Mon, 31 Jan 2022
14:15
Virtual

D-critical locus structure for local toric Calabi-Yau 3-folds

Yun Shi
(Harvard University)
Abstract

Donaldson-Thomas (DT) theory is an enumerative theory which produces a virtual count of stable coherent sheaves on a Calabi-Yau 3-fold. Motivic Donaldson-Thomas theory, originally introduced by Kontsevich-Soibelman, is a categorification of the DT theory. This categorification contains more refined information of the moduli space. In this talk, I will explain the role of d-critical locus structure in the definition of motivic DT invariant, following the definition by Bussi-Joyce-Meinhardt. I will also discuss results on this structure on the Hilbert schemes of zero dimensional subschemes on local toric Calabi-Yau threefolds. This is based on joint works with Sheldon Katz. The results have substantial overlap with recent work by Ricolfi-Savvas, but techniques used here are different. 

Mon, 24 Jan 2022
14:15
Virtual

Cayley fibrations in the Bryant-Salamon manifolds

Federico Trinca
(University of Oxford)
Abstract

In 1989, Bryant and Salamon constructed the first Riemannian manifolds with holonomy group $\Spin(7)$. Since a crucial aspect in the study of manifolds with exceptional holonomy regards fibrations through calibrated submanifolds, it is natural to consider such objects on the Bryant-Salamon manifolds.

In this talk, I will describe the construction and the geometry of (possibly singular) Cayley fibrations on each Bryant-Salamon manifold. These will arise from a natural family of structure-preserving $\SU(2)$ actions. The fibres will provide new examples of Cayley submanifolds.

Tue, 18 Jan 2022
15:30
Virtual

TBA

Stephan Stadler
(Max Planck Institute Bonn)
Abstract

TBA

Mon, 17 Jan 2022
14:15
Virtual

Brane quantization of Toric Poisson varieties

Francis Bischoff
((Oxford University))
Abstract

The homogeneous coordinate ring of a projective variety may be constructed by geometrically quantizing the multiples of a symplectic form, using the complex structure as a polarization. In this talk, I will explain how a holomorphic Poisson structure allows us to deform the complex polarization into a generalized complex structure, leading to a non-commutative deformation of the homogeneous coordinate ring. The main tool is a conjectural construction of a category of generalized complex branes, which makes use of the A-model of an associated symplectic groupoid. I will explain this in the example of toric Poisson varieties. This is joint work with Marco Gualtieri (arXiv:2108.01658).

Mon, 06 Dec 2021
14:15
L4

A non-existence result for balanced SU(3)-structures on cohomogeneity one manifolds

Izar Alonso Lorenzo
((Oxford University))
Abstract

The Hull--Strominger system is a system of non-linear PDEs on heterotic string theory involving a pair of Hermitian metrics $(g,h)$ on a six dimensional manifold $M$. One of these equations dictates the metric $g$ on $M$ to be conformally balanced. We will begin the talk by giving a description of the geometry of cohomogeneity one manifolds and SU(3)-structures. Then, we will look for solutions to the Hull--Strominger system in the cohomogeneity one setting. We show that a six-dimensional simply connected cohomogeneity one manifold under the almost effective action of a connected Lie group $G$ admits no $G$-invariant balanced non-Kähler SU(3)-structures. This is a joint work with F. Salvatore.

Mon, 29 Nov 2021
14:15
L4

CoHAs, vertex algebras and torus localisation

Alyosha Latyntsev
((Oxford University))
Abstract

Cohomological Hall algebras and vertex algebras are two structures whose origins are (at least in part) from physics. I will explain what these objects are, how the latter was related to moduli stacks by Joyce, and a theorem relating these two structures. The main tool is torus localisation, a method for "turning geometry into combinatorics", or rather a new formulation of it which works in the singular setting.

Tue, 23 Nov 2021
09:00
Virtual

Deletion and contraction for Hausel-Proudfoot spaces

Michael McBreen
(Hong Kong)
Abstract

Dolbeault hypertoric manifolds are hyperkahler integrable systems generalizing the Ooguri-Vafa space. They approximate the Hitchin fibration near a totally degenerate nodal spectral curve. On the other hand, Betti hypertoric varieties are smooth affine varieties parametrizing microlocal sheaves on the same nodal spectral curve. I will review joint work with Zsuzsanna Dansco and Vivek Shende (arXiv:1910.00979) which constructs a diffeomorphism between the Dolbeault and Betti hypertorics, and proves that it intertwines the perverse and weight filtrations on their cohomologies. I will describe our main tool : deletion-contraction sequences arising from either smoothing a node of the spectral curve or separating its branches. I will also discuss some more recent developments and open questions.

Mon, 22 Nov 2021
14:15
L4

Purely inseparable Galois theory

Lukas Brantner
((Oxford University))
Abstract

A field extension $F/K$ in characteristic $p$ is purely inseparable if for each $x$ in $F$, some power $x^{p^n}$ belongs to $K$. Using methods from homotopy theory, we construct a Galois correspondence for finite purely inseparable field extensions $F/K$, generalising a classical result of Jacobson for extensions of exponent one (where $x^p$ belongs to $K$ for all $x$ in $F$). This is joint work with Waldron.

Mon, 15 Nov 2021
14:15
L4

TBA

Huaxin (Henry) Liu
((Oxford University))
Abstract

TBA

Mon, 08 Nov 2021
14:15
L4

Deformation uniqueness of Calabi-Yau metrics with maximal volume growth

Shih-Kai Chiu
((Oxford University))
Abstract

It is expected that complete noncompact Calabi-Yau manifolds are in some sense governed by their asymptotics at infinity. In the maximal volume growth case, the asymptotics at infinity are given by Calabi-Yau cones. We are interested in deformations of such metrics that fix the asymptotic cones at infinity. In the asymptotically conical case, Conlon-Hein proved uniqueness under such deformations. Their method is based on the corresponding linearized problem, namely the study of subquadratic harmonic functions. We generalize their work to the maximal volume growth case, allowing the tangent cones at infinity to have non-isolated singularities. Part of the talk is based on work in progress joint with Gabor Szekelyhidi.

Mon, 01 Nov 2021
14:15
L4

Stability conditions for polarised varieties

Ruadhaí Dervan
(Cambridge)
Abstract

A central theme of complex geometry is the relationship between differential-geometric PDEs and algebro-geometric notions of stability. Examples include Hermitian Yang-Mills connections and Kähler-Einstein metrics on the PDE side, and slope stability and K-stability on the algebro-geometric side. I will describe a general framework associating geometric PDEs on complex manifolds to notions of stability, and will sketch a proof showing that existence of solutions is equivalent to stability in a model case. The framework can be seen as an analogue in the setting of varieties of Bridgeland's stability conditions on triangulated categories.