University of York Geometry, Analysis and Mathematical Physics seminar 2022-23

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

Autumn Term 2022

Date	Speaker	Title
4 October	Martin Speight (Leeds)	The \(L^2\) geometry of the moduli space of vortices on the two-sphere in the dissolving limit
11 October	Wilhelm Klingenberg (Durham)	The Toponogov Conjecture on convex proper embedded discs in \(\mathbb{R}^3\)

Spring Term 2023

Date	Speaker	Title
17 January
31 January
14 February
21 February
28 February	Gonçalo Oliveira (IST Lisbon)	From electrostatics to geodesics in K3 surfaces
14 March	Sebastian Schulz (Johns Hopkins)	Nilpotent Higgs bundles and families of flat connections

Summer Term 2023

Date	Speaker	Title
18 April	Aleksander Doan (UCL)	Holomorphic Floer theory and the Fueter equation
25 April 2-3pm in the Topos	Hartmut Weiß (Kiel)	On the modularity of gravitational instantons of type ALG
25 April 3:30-4:30pm in G/N/135	Michael Singer (UCL)	Compactification of Nahm and other moduli spaces
2 May	Alex Waldron (Wisconsin)	Strong gap theorems via Yang-Mills flow
9 May	Claudio Meneses (Kiel)	Variation of Kähler metrics on moduli of parabolic bundles

Previous years' seminars

2021-22
2020-21
2019-20

Abstracts

4 October. Martin Speight (Leeds)
Title. The \(L^2\) geometry of the moduli space of vortices on the two-sphere in the dissolving limit
Abstract. There is an upper bound on how many abelian Higgs vortices a compact Riemann surface can accommodate, proportional to its area. In the limit that the area shrinks to the minimum allowed (for a given vortex number), the vortices spread out and delocalize completely. Hence this is sometimes called the "dissolving limit". For vortices on a two-sphere, the moduli space \(\mathcal{M}_n\) of \(n\)-vortices is biholomorphic to \(\mathbb{C}P^n\). This space carries a natural metric, the \(L^2\) metric, which controls the classical and quantum dynamics of vortices. In 2002, Baptista and Manton conjectured that, in the dissolving limit, this metric converges to the Fubini-Study metric on \(\mathbb{C}P^n\). I will describe how this conjecture can be made precise and proved.
Joint work with René García Lara.

11 October. Wilhelm Klingenberg (Durham)
Title. The Toponogov Conjecture on convex proper embedded discs in \(\mathbb{R}^3\)
Abstract. We prove a conjecture of Toponogov on complete convex surfaces, namely that such surfaces must contain an umbilic point, albeit at infinity. Our proof is indirect. It uses Fredholm regularity of an associated Riemann-Hilbert boundary value problem and an existence result for holomorphic discs with Lagrangian boundary conditions, both of which apply to a putative counterexample.
This is joint work with Brendan Guilfoyle.

28 February. Gonçalo Oliveira (IST Lisbon)
Title. From electrostatics to geodesics in K3 surfaces
Abstract. Motivated by some conjectures originating in the Physics literature, I have recently been looking for closed geodesics in the K3 surfaces constructed by Lorenzo Foscolo. It turns out to be possible to locate several such with high precision and compute their index (their length is also approximately known). Interestingly, in my view, the construction of these geodesics is related to an open problem in electrostatics posed by Maxwell in 1873.

14 March. Sebastian Schulz (Johns Hopkins)
Title. Nilpotent Higgs bundles and families of flat connections
Abstract. In studying Higgs bundles, one is naturally led to consider \(\mathbb{C}^*\)-families of flat connections. One fruitful approach to studying the asymptotic behavior of such a family is a procedure known as the "exact WKB method", at least for sufficiently generic Higgs bundles. I will describe how these results can be generalized to the most degenerate case of nilpotent Higgs bundles. Time permitting, I will explain how this sheds light on a conjecture by Simpson concerning a stratification of the moduli space of flat connections.

18 April. Aleksander Doan (UCL)
Title. Holomorphic Floer theory and the Fueter equation
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.

25 April. Hartmut Weiß (Kiel)
Title. On the modularity of gravitational instantons of type ALG
Abstract. I will report on recent progress towards the modularity conjecture of Phil Boalch in the case of gravitational instantons of type ALG. Slightly rephrased, it claims that all of them should be given by Hitchin moduli spaces.
This is joint work with Laura Fredrickson, Rafe Mazzeo and Jan Swoboda.

25 April. Michael Singer (UCL)
Title. Compactification of Nahm and other moduli spaces
Abstract. If \(G\) is a compact Lie group, then \(T^*G^c\), the complex cotangent bundle of its complexification, has a natural hyperKähler metric. This surprising fact was proved by Kronheimer in 1988 by showing that a moduli space of solutions of Nahm's equations on a finite real interval is diffeomorphic to \(T^*G^c\). (The story was further elucidated by Dancer and Swann). Little is known explicitly about this metric except in the simplest cases. In joint work with Richard Melrose and Raphael Tsiamis we have been studying the asymptotic behaviour of this metric, which turns out to be quite intricate. The goal of my talk is to explain some of this work, setting it in the context of compactification of moduli spaces more generally.

2 May. Alex Waldron (Wisconsin)
Title. Strong gap theorems via Yang-Mills flow
Abstract. Given a principal bundle over a compact Riemannian manifold, it is natural to ask whether a gap exists between the instanton energy and that of any non-minimal Yang-Mills connection. This question is quite open in general, although a fair number of positive results exist in the literature. We'll review several of these gap theorems and strengthen them to statements of the following type: the space of all connections below a certain energy deformation retracts (under Yang-Mills flow) onto the space of instantons. We'll also mention a few planned applications.

9 May. Claudio Meneses (Kiel)
Title. Variation of Kähler metrics on moduli of parabolic bundles
Abstract. Moduli spaces of parabolic bundles on Riemann surfaces possess a peculiar dependence on a set of real parameters, leading to wall-crossing phenomena in the study of their birational geometry. At the same time, they come equipped with natural Kähler metrics, whose parameter-dependence is known to be real-analytic. It is compelling to try to express this dependence as a suitable "Torelli theorem", and to answer the following question: to what extent does the Kähler metric (an analytic invariant) determine the stability conditions that define the moduli problem?
In this talk I will describe how this variation problem can be addressed in terms of analytic invariants associated to the spectral geometry of Fuchsian groups. I will also explain how this problem determines the analogous variations of hyperkähler metrics on moduli of parabolic Higgs bundles, and its relation to the classification of gravitational instantons of ALG type.
This is work in progress, joint with Hartmut Weiß.

Last updated 29 April, 2023