University of York Geometry, Analysis and Mathematical Physics seminar 2024-25

All are welcome to attend the seminar, which usually takes place at 2pm on Tuesdays in the Topos during term time. (Directions to the Topos)

Autumn Semester 2024

Date	Speaker	Title
15 October	Benoît Vicedo (York)	(Higher) operations in topological and conformal QFTs from factorisation
22 October	Graeme Wilkin (York)	The Morse complex for semiprojective varieties
5 November	Ian McIntosh (York)	The geometric Toda equations for noncompact symmetric spaces
12 November	Ian McIntosh (York)	Equivariant minimal surfaces in the complex hyperbolic plane
19 November 3-4pm in the Topos Note the time!	Inder Kaur (Glasgow)	A Lefschetz (1,1) theorem for singular varieties
26 November	Martin Kerin (Durham)	Isometric rigidity of Wasserstein spaces
3 December	Ben Lambert (Leeds)	Nonlocal estimates for volume preserving mean curvature flow

Spring Semester 2025

Date	Speaker	Title
4 February	Kasia Wyczesany (Leeds)	Zoo of dualities and related inequalities
11 February
18 February	Francesca Tripaldi (Leeds)	Extracting subcomplexes in the subRiemannian setting
25 February	Stefano Negro (York)	Surfaces in higher-dimensional spaces and quantum integrable systems
4 March	Josh Cork (Leicester)	On the instanton approximation of skyrmions
11 March	Gautam Chaudhuri (Leeds)	The Dynamics of 2-Vortices on Flat Tori
26 March (2pm)	Daniel Platt (Imperial)	Numerics for 1-forms and Calabi-Yau manifolds via neural networks
26 March (3:10pm)	Luca Seemungal (Leeds)	The index of constant mean curvature surfaces in 3-manifolds
26 March (3:40pm)	Sam Engleman (York)	A Morse-Bott cohomology for the moduli space of Higgs bundles
26 March (4:40pm)	Martin Speight (Leeds)	\(L^2\) geometry of vortices
1 April
22 April
29 April	Denis Dobkowski-Ryłko	Isolated horizons – a quasi-local approach to black holes
20 May 4pm in the Topos Note the time!	Alexander Früh (Birmingham)	Maximal abelianisation for non-quasi-split real Higgs bundles

Previous years' seminars

2023-24
2022-23
2021-22
2020-21
2019-20

Abstracts

15 October. Benoît Vicedo (York)
Title. (Higher) operations in topological and conformal QFTs from factorisation
Abstract. Prefactorisation algebras (PFAs) axiomatise the algebraic structure of observables in QFTs. In the case of locally constant PFAs, which correspond to topological QFTs, I will describe how the PFA induces a (higher) algebraic structure on the gauge invariant observables of the TQFT. This involves an ordinary associative, unital product but typically also "higher" operations known as Massey products. I will also discuss the case of holomorphic PFAs in one complex dimension, which correspond to 2d CFTs, and explain how in this setting the PFA encodes the vertex algebra structure of the CFT.

22 October. Graeme Wilkin (York)
Title. The Morse complex for semiprojective varieties
Abstract. Moduli spaces of Higgs bundles admit a natural \(\mathbb{C}^*\) action, which makes them examples of semiprojective varieties. In the first paper on the subject, Hitchin introduced the moment map associated to the action of \(S^1 \hookrightarrow \mathbb{C}^*\) and showed that this is a perfect Morse-Bott function. In this talk I will describe the cup product on the Morse complex for both this moment map as well as minus the moment map. This setup applies more generally to semiprojective varieties satisfying an additional transversality condition. I will spend some time explaining the case of rank \(2\) twisted Higgs bundles, where these constructions relate to questions from classical geometry.

5 November. Ian McIntosh (York)
Title. The geometric Toda equations for noncompact symmetric spaces
Abstract. Different forms of the Toda equations keep turning up both in surface theory and gauge theory. Everyone knows that they are really equations on something like the Cartan subgroup of a simple Lie group. While these have been well studied when the group is compact, the noncompact case has (surprisingly) not been given a general treatment. The aim of the talk is to explain a classification of these equations which fit the geometrical setting of minimal surfaces in noncompact symmetric spaces, and also discuss when methods from gauge theory (especially stability ideas) can be applied to provide spaces of solutions.

12 November. Ian McIntosh (York)
Title. Equivariant minimal surfaces in the complex hyperbolic plane
Abstract. Over a series of three papers, two with John Loftin (Rutgers-Newark), a clear picture has emerged for the structure of the moduli space of equivariant minimal surfaces in the complex hyperbolic plane through the link with PU(2,1)-Higgs bundles. In this talk I'll give a survey of the interesting features, focussing on what we can learn about the minimal surfaces from their Higgs bundles, and some remaining open questions.

19 November. Inder Kaur (Glasgow)
Title. A Lefschetz (1,1) theorem for singular varieties
Abstract. The Lefschetz (1, 1) theorem is a classical result that tells us that for any smooth projective variety, a rational (1, 1) Hodge class comes from a algebraic cycle of codimension 1. In 1994, Barbieri-Viale and Srinivas gave a counter-example to the obvious generalization of this result to singular varieties. Inspired by Totaro, in this talk, I will give a modification of the statement of Lefschetz (1, 1) and show that it is satisfied by several singular varieties such as those with ADE-singularities and rational singularities. In particular, this Lefschetz (1,1) statement is satisfied by the varieties considered by Barbieri-Viale and Srinivas.
This is joint work with A. Dan.

26 November. Martin Kerin (Durham)
Title. Isometric rigidity of Wasserstein spaces
Abstract. The Wasserstein spaces over a metric space X are sets of Borel probability measures on X which are equipped with metrics derived from optimal transport. In general, the Wasserstein spaces reflect properties of the underlying metric space X. In this context, it is natural to ask whether the isometries of the Wasserstein space are related to those of X. In this talk, I will discuss some work in this direction conducted together with Mauricio Che, Fernando Galaz-García and Jaime Santos-Rodríguez.

3 December. Ben Lambert (Leeds)
Title. Nonlocal estimates for volume preserving mean curvature flow
Abstract. The volume preserving mean curvature flow (VPMCF) is the gradient flow of the area functional under the fixed enclosed volume constraint, meaning that stationary solutions of this flow are hypersurfaces of constant mean curvature. However, the non-local term in this evolution means that the flow satisfies significantly different properties to mean curvature flow and many "standard" mean curvature flow methods simply don't work. In this talk I will discuss some progress on nonlocal estimates which, amongst other things, imply that the singularities of VPMCF are in fact modelled by the (better understood) ancient solutions of mean curvature flow.
This work is joint with Elena Maeder-Baumdicker.

4 February. Kasia Wyczesany (Leeds)
Title. Zoo of dualities and related inequalities
Abstract. In this talk, we will discuss order-reversing quasi-involutions, which are dualities on their image, and their properties. We will relate this concept to the classical example of "polarity", which corresponds to the duality between finite-dimensional normed spaces. Already this classical example incites many easy-to-formulate but difficult-to-proof questions, such as the Mahler conjecture ('39).
We show that any order-reversing quasi-involution is of a special form and discuss how this unified point of view helps to deepen the understanding of the underlying structures and principles.
This talk is based on joint work with Shiri Artstein-Avidan and Shay Sadovsky.

18 February. Francesca Tripaldi (Leeds)
Title. Extracting subcomplexes in the subRiemannian setting
Abstract. On subRiemannian manifolds, the de Rham complex is not the ideal candidate to use to carry out geometric analysis. However, special subcomplexes have successfully been applied in very specific settings, such as Heisenberg groups and the Cartan group. I will give an overview of different techniques used to obtain such subcomplexes, as well as point out their limitations when used on arbitrary Carnot groups, and a possible way to overcome them.

25 February. Stefano Negro (York)
Title. Surfaces in higher-dimensional spaces and quantum integrable systems
Abstract. Integrable systems are deeply intertwined with the geometry of surfaces. It is in this context that many of the renowned classical integrable equations - such as the sine-Gordon and the Tzizéica-Bullough-Dodd equations - first arose in the mathematical literature. A perhaps less widely known fact is the existence of a direct relation between classical and quantum integrable systems known as ODE/IM correspondence. After reviewing how the construction of minimal surfaces embedded in AdS produce classical integrable systems I will concentrate on a simple example to explicitly show the relation to quantum integrable models.

4 March. Josh Cork (Leicester)
Title. On the instanton approximation of skyrmions
Abstract. Skyrmions are critical points of a generalised harmonic map energy for maps between Riemannian 3-manifolds. In the case where the target is a 3-sphere, they play a key role in nuclear physics, with skyrmions interpreted as baryons in a limit of quantum chromodynamics. To extract physical properties of nuclei from skyrmions, a semi-classical approach is taken involving studying quantum mechanics on a finite-dimensional configuration space; this is necessary due to the non-integrable nature of the Skyrme field equations. Over the years, many different configuration spaces have been used, but each typically fails to describe all physical configurations of interest. One of the best candidates uses moduli spaces of instantons (self-dual gauge fields) on \(\mathbb{R}^4\), and recently several powerful tools have been developed which make using this both more explicit and more versatile.
In this talk, after a brief general overview of skyrmions, we shall review this recent progress, and illustrate in detail how these tools may be applied explicitly to studying configurations of skyrmions, and quantum problems, via some examples.
This talk is based on multiple joint works with Linden Disney-Hogg (Leeds), Chris Halcrow (Edinburgh), and Derek Harland (Leeds).

11 March. Gautam Chaudhauri (Leeds)
Title. The Dynamics of 2-Vortices on Flat Tori
Abstract. Vortices are topological solitons on Riemann surfaces arising in the Abelian Higgs model. Low energy dynamics of vortices are well approximated by geodesic motion on an associated moduli space, but in many cases the metric on this moduli space has no explicit global description. In this talk, we will look at how the metric can be approximated in the dissolving limit, i.e. when the volume of the base Riemann surface tends to a critical minimum value. We will then see some new results on how this approximation can be used to model the dynamics of 2-vortices on flat tori, yielding explicit approximations of vortex dynamics and characterising the relative motion of the vortices.
This talk is based on joint work with Derek Harland and Martin Speight.

26 March (2pm). Daniel Platt (Imperial)
Title. Numerics for 1-forms and Calabi-Yau manifolds via neural networks
Abstract. Calabi-Yau manifolds are manifolds admitting a unique Ricci-flat metric. Even though existence is known, no explicit formulae for these metrics are known. That frequently causes problems when one wants to compute things that depend on the metric, in particular in Physics. One example in maths is the following: does there exist a harmonic 1-form on a real locus of a Calabi-Yau manifold that is nowhere vanishing? No example is known. In the talk I will explain a conjectural example and an interesting non-example. To define the manifolds, some real algebraic geometry is used. We then numerically (approximately) solve the Ricci-flat equation and the harmonic 1-form equation. It turns out that a neural network is good at that and the approximate solution is easily interpretable.
This is based on arXiv:2405.19402, which is joint work with Michael Douglas and Yidi Qi.
Time permitting, I will comment on ongoing efforts to turn this into a numerically verified proof that there is a genuine solution near our approximate solution.

26 March (3:10pm). Luca Seemungal (Leeds)
Title. The index of constant mean curvature surfaces in 3-manifolds
Abstract. Constant mean curvature (CMC) surfaces are special geometric variational objects, closely related to minimal surfaces. The key properties of a CMC surface are its area, mean curvature, genus, and index. The index of a CMC surface measures its stability: the index counts how many ways one can perturb the surface to decrease the area while keeping the enclosed volume constant. In this talk we discuss relationships between these key properties. In particular we present recent joint work with Ben Sharp, where we bound the index of CMC surfaces linearly from above by genus and the correct scale-invariant quantity involving mean curvature and area.

26 March (3:40pm). Sam Engleman (York)
Title. A Morse-Bott cohomology for the moduli space of Higgs bundles
Abstract. TBA

26 March (4:40pm). Martin Speight (Leeds)
Title. \(L^2\) geometry of vortices
Abstract. TBA

29 April. Denis Dobkowski-Ryłko (Warsaw)
Title. Isolated horizons – a quasi-local approach to black holes
Abstract. One of the most fascinating predictions of general relativity is the existence of black holes. While the first exact black hole solution was found by Karl Schwarzschild shortly after Einstein introduced his field equations, a complete description of general stationary black holes took significantly more time and effort. It wasn't until the 1960s that new mathematical tools in differential geometry and topology—developed by Penrose, Hawking, Geroch, and others—enabled a deeper understanding of the global structure of spacetimes containing black holes. These global insights led to the discovery of many key properties of black holes, including the uniqueness and rigidity theorems, the no-hair conjecture, area laws, black hole thermodynamics, and Hawking radiation. However, realistic black holes are neither static nor stationary, and their description should ideally not depend on teleological knowledge of the entire spacetime. To address this, a local framework known as isolated horizons was introduced by Pajerski and Newman in 1971, gaining broader recognition in the late 1990s. This approach offers a quasi-local characterization of black holes and reproduces many of the essential features associated with their global counterparts. In this talk, I will outline the theory of isolated horizons and highlight some of the contributions made by the Warsaw relativity group in this area.

20 May. Alexander Früh (Birmingham)
Title. Maximal abelianisation for non-quasi-split real Higgs bundles
Abstract. The Hitchin map is a central tool in the study of the moduli space of \(G\)-Higgs bundles, and if \(G\) is a complex reductive group it equips the space with the structure of an integrable system. A natural question is to ask what can be said about the Hitchin map for \(G_\mathbb{R}\)-Higgs bundles, where \(G_\mathbb{R}\) is a real group. If \(G_\mathbb{R}\) is quasi-split, Peón-Nieto showed that the Hitchin fibres can still be described as semi-abelian varieties. However if \(G_\mathbb{R}\) is not quasi-split, the behaviour of the Hitchin fibres can be quite different. Hitchin and Schaposnik calculated non-abelian spectral data for the cases \(SU^*(2m)\), \(SO^*(4m)\) and \(Sp(2m,2m)\), where Higgs bundles correspond to spaces of rank 2 bundles on a spectral curve. We consider the examples \(SU(p,q)\) and \(SO^*(4m+2)\). In these cases, the Hitchin fibres are described by a combination of both abelian and non-abelian data. Such a phenomenon has already been observed for \(SO(p,q)\) by Baraglia and Schaposnik. We take a global approach, using a rational map which maximally abelianises the moduli spaces. In the \(SU(p,q)\) case, we will also consider this description in terms of spectral data on a non-reduced curve.

Last updated 13 May, 2025