University of York Geometry, Analysis and Mathematical Physics seminar 2023-24

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 2023

Date	Speaker	Title
26 September	Jakob Stein (UCL)	\(G_2\) instantons bubbling at infinity on \(S^3 \times \mathbb{R}^4\)
3 October
10 October 3:30pm in the Topos Note the time!	Fernando Galaz-Garcia (Durham)	Topology and geometry of 3-dimensional Alexandrov spaces
17 October	Ruadhai Dervan (Glasgow)	The universal structure of moment maps in complex geometry
24 October	Tathagata Ghosh (Leeds)	Instantons on Asymptotically Conical Spin(7)-Manifolds
8 November Lewis Fry Richardson lecture 4:30-5:30pm in P/L/002	David Tong (Cambridge)	A Chern-Simons Theory for the North Atlantic Ocean
14 November
21 November
28 November	Ronald Zúñiga-Rojas (Universidad de Costa Rica)	Stratifications on the nilpotent cone of the moduli space of Hitchin pairs
5 December 3pm in the Dusa McDuff room Note the time and location!	Ana Peon-Nieto (Universidad de Santiago de Compostela)	Hodge bundles of length two in the moduli space of Higgs bundles
12 December	Dave Smith (Yale-NUS)	Fokas Diagonalization

Spring Semester 2024

Date	Speaker	Title
16 January
23 January	Samuel Borza (Trieste)	The measure contraction property in the sub-Finsler Heisenberg group
30 January	Jaime Mendizabal (UCL)	Hyper-Kähler moduli spaces of monopoles with arbitrary symmetry breaking
6 February	Georgios Kydonakis (Patras)	Fock bundles and Teichmüller spaces
20 February	Ilyas Khan (Duke)	A compact and non-convex ancient solution to curve shortening flow
19 March
16 April	Andries Salm (Université Libre de Bruxelles)	Construction of gravitational instantons of type \(ALG^\) and \(ALH^\)
30 April 2:00pm in the Topos	Calum Ross (Edge Hill)	Vortices and Cartan Geometry
30 April 3:30pm via zoom (Live audience in the Topos)	Thomas Mark (Virginia)	Constraints on contact type hypersurfaces in symplectic 4-manifolds
14 May
21 May	Marie-Amelie Lawn (Imperial)	On the spinorial characterization of submanifolds
28 May	Ivan Tulli (Sheffield)	Special Joyce structures and hyperkähler metrics

Previous years' seminars

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

Abstracts

26 September Jakob Stein (UCL)
Title. \(G_2\) instantons bubbling at infinity on \(S^3 \times \mathbb{R}^4\)
Abstract. In this talk, we will discuss two sorts of bubbling for Yang-Mills instantons on a \(G_2\)-manifold, with some explicit examples arising from solutions to an ODE. The first describes bubbling off instantons on a fixed asymptotic model space, in this case, the asymptotic cone of the Bryant-Salamon \(S^3 \times \mathbb{R}^4\). The second describes the limiting behaviour of instantons as one varies the asymptotic geometry in family of Taub-NUT-type metrics.
This is joint work with Matt Turner (Bath).

10 October Fernando Galaz-Garcia (Durham)
Title. Topology and geometry of 3-dimensional Alexandrov spaces
Abstract. Alexandrov spaces (with curvature bounded below) are metric generalizations of complete Riemannian manifolds with a uniform lower sectional curvature bound. Instances of Alexandrov spaces include compact Riemannian orbifolds and orbit spaces of isometric compact Lie group actions on compact Riemannian manifolds. In addition to being objects of intrinsic interest, Alexandrov spaces play an important role in Riemannian geometry, for example, in Perelman's proof of the Poincaré Conjecture. In this talk, I will discuss the topology and geometry of 3-dimensional Alexandrov spaces, focusing on their classification and structure under certain geometric constraints (for example, positive or non-negative curvature and collapse).

17 October Ruadhai Dervan (Glasgow)
Title. The universal structure of moment maps in complex geometry
Abstract. Much of complex geometry is motivated by linking the existence of solutions to geometric PDEs (producing "canonical metrics") to stability conditions in algebraic geometry. I will address a more basic question: what is the recipe to produce geometric PDEs in complex geometry? The solution to this will use some old tools from symplectic geometry, namely equivariant differential geometry and the theory of moment maps (though I will not assume any prior knowledge of complex or symplectic geometry).
This is joint work with Michael Hallam.

24 October Tathagata Ghosh (Leeds)
Title. Instantons on Asymptotically Conical Spin(7)-Manifolds
Abstract. In this talk we discuss instantons on asymptotically conical Spin(7)-manifolds where the instanton is asymptotic to a fixed nearly G2-instanton at infinity. In particular, we focus on the deformation theory of these instantons; by relating the deformation complex with spinors, we identify the space of infinitesimal deformations with the kernel of the twisted negative Dirac operator on the asymptotically conical Spin(7)-manifold. Finally, we apply this theory to describe the deformations of Fairlie-Nuyts-Fubini-Nicolai (FNFN) Spin(7)-instantons on \(\mathbb{R}^8\). We calculate the virtual dimension of the moduli space using Atiyah-Patodi-Singer index theorem and the spectrum of the twisted Dirac operator.

8 November David Tong (Cambridge)
Title. A Chern-Simons Theory for the North Atlantic Ocean
Abstract. In 1922, Richardson published a remarkable book in which he documented his failure to predict the weather six hours in advance. He was missing two of the most important breakthroughs of 20th century science: computers, and low-energy effective field theory. In this talk, I’ll revisit the equations that Richardson studied and show that they can be recast as a gauge theory in d=2+1 dimensions. In a particular limit, this gauge theory becomes Maxwell-Chern-Simons theory and describes a class of waves in the ocean and atmosphere. Chern-Simons theory is well known to exhibit edge modes, and these have been observed in quantum Hall states. These same edge modes also manifest themselves as chiral waves in the ocean that run along the coast.

28 November Ronald Zúñiga-Rojas (Universidad de Costa Rica)
Title. Stratifications on the nilpotent cone of the moduli space of Hitchin pairs
Abstract. We consider the problem of finding the limit at infinity (corresponding to the downward Morse flow) of a Higgs bundle in the nilpotent cone under the natural \(\mathbb{C}^*\)-action on the moduli space. For general rank we provide an answer for Higgs bundles with regular nilpotent Higgs field, while in rank three we give the complete answer. Our results show that the limit can be described in terms of data defined by the Higgs field, via a filtration of the underlying vector bundle.

5 December Ana Peon-Nieto (Universidad de Santiago de Compostela)
Title. Hodge bundles of length two in the moduli space of Higgs bundles
Abstract. The geometry of the moduli space of Higgs bundles is governed by the so called global nilpotent cone. This is the fibre over 0 of the Hitchin map, sending a Higgs bundle to its characteristic polynomial. A particularly important subset of the nilpotent cone is given by fixed points of a \(\mathbb{C}^*\)action, also called systems of Hodge bundles. Indeed, these determine important geometric information, such as the irreducible components of the nilpotent cone, and in the cases where very stable points exist, their multiplicity, as proven by Hausel and Hitchin. These two authors identified some potential obstructions to the existence of very stable points, in terms of the polynomiality of some invariants (virtual equivariant multiplicites, and equivariant Euler classes). In this talk, I will analyse the case of nilpotent order two fixed points, showing that, except for one case, they never contain very stable points, and contrast these results with the computation of the afore mentioned invariants.

12 December Dave Smith (Yale-NUS)
Title. Fokas Diagonalization
Abstract. We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed; differential operators with piecewise constant coefficients are thus included.

23 January Samuel Borza (Trieste)
Title. The measure contraction property in the sub-Finsler Heisenberg group
Abstract. The Heisenberg group is a source of inspiration for many fields in mathematics and physics, including quantum theory, metric geometry, and harmonic analysis. I will discuss the sub-Finsler geometry of the Heisenberg group and explain how it is related to the isoperimetric problem in the non-Euclidean (Finsler) plane. We will then explore approaches to studying the curvature of the sub-Finsler Heisenberg group, focusing particularly on the measure contraction property that appears in the analysis of metric measure spaces.
This is a joint work with Kenshiro Tashiro, Mattia Magnabosco, and Tommaso Rossi.

30 January Jaime Mendizabal (UCL)
Title. Hyper-Kähler moduli spaces of monopoles with arbitrary symmetry breaking
Abstract. Monopoles are solutions to the gauge-theoretic Bogomolny equations, a set of PDEs closely related to the anti-self-dual Yang-Mills equations. For the gauge group \(SU(2)\) they have been studied extensively, where they have been shown to form complete hyper-Kähler moduli spaces. For other gauge groups not so much is known, especially in the case of non-maximal symmetry breaking, a condition on their asymptotic behaviour. In this talk we present a construction of the moduli spaces for arbitrary gauge group and symmetry breaking, which furthermore provides them with a hyper-Kähler metric.

6 February Georgios Kydonakis (Patras)
Title. Fock bundles and Teichmüller spaces
Abstract. Higher Teichmüller theory is concerned with the study of special connected components of character varieties sharing analogous properties to the classical Teichmüller space. Fixing a complex structure on the underlying topological surface introduces powerful holomorphic techniques through certain holomorphic pairs called Higgs bundles, which correspond to fundamental group representations via the non-abelian Hodge correspondence. Yet, a rather adverse aspect of the correspondence is that is fails to transfer the action of the mapping class group on character varieties to the moduli space of Higgs bundles.
We will introduce a similar class of augmented bundles over a topological surface that we call Fock bundles which does not require fixing any complex structure on the underlying surface. We conjecture that there is an alternative passage to the one given by the non-abelian Hodge correspondence from such pairs to certain higher rank Teichmüller spaces that is independent of the complex structure on the surface.
This is joint work with Charles Reid (Austin) and Alexander Thomas (Heidelberg).

20 February Ilyas Khan (Duke)
Title. A compact and non-convex ancient solution to curve shortening flow
Abstract. Understanding ancient solutions is a central problem in the study of geometric flows, which in the case of curve shortening flow (CSF) has seen significant recent progress. This includes the classification of compact, convex ancient solutions to CSF by P. Daskalopoulos, R. Hamilton, and N. Sesum. In this talk, we discuss the construction of a non-convex ancient solution to CSF which is at all times compact and embedded, which shows that the convexity assumption cannot be dropped in the D-H-S classification.
This talk is based on joint work with S. Angenent, C. Olson, and Y. Zhang.

16 April Andries Salm (Université Libre de Bruxelles)
Title. Construction of gravitational instantons of type \(ALG^*\) and \(ALH^*\)
Abstract. From the 70’s physicists and mathematicians have been interested in the Yang-Mills analogue for gravity. These gravitational instantons were constructed using various methods and have recently been classified. These spaces do not only show up in physics, but they have also been realized as the bubbles of degenerating Calabi-Yau metrics on the K3 surfaces. In this talk, we explore a gluing construction of gravitational instantons of type \(ALG^*\) and \(ALH^*\). Compared to the other types of gravitational instantons these spaces have a non-standard geometry, and we explain the consequence this has on the functional analysis. Due to the explicit nature of this construction, we will see how the metric bubbles off Atiyah-Hitchin and Taub-NUT spaces as the metric degenerates to a flat 3-manifold.

30 April Calum Ross (Edge Hill)
Title. Vortices and Cartan Geometry
Abstract. Certain abelian vortex equations are known to be integrable on constant curvature Riemann surfaces. These vortex equations can be reinterpreted as flat non-abelian Cartan connections, encoding the geometric structure of the Riemann surface. This Cartan picture gives a link between vortex equations and the geometry of three-dimensional Lie groups. I will outline this construction and discuss some generalisations, such as defects that preserve the integrability of the vortex equations, some generalisations of the vortex equations, and the relationship to zero modes and instantons.

30 April Thomas Mark (Virginia)
Title. Constraints on contact type hypersurfaces in symplectic 4-manifolds
Abstract. In joint work with Bulent Tosun, it was shown that Heegaard Floer theory provides an obstruction for a contact 3-manifold to embed as a contact type hypersurface in standard symplectic 4-space. As one consequence, no Brieskorn homology sphere admits such an embedding (regardless of the contact structure). I will review the ideas that lead to these results, and discuss recent extensions that can obstruct suitably convex embeddings in closed symplectic 4-manifolds, particularly rational complex surfaces.

21 May Marie-Amelie Lawn (Imperial)
Title. On the spinorial characterization of submanifolds
Abstract. In the last few decades, characterizations of submanifolds in terms of the existence of spinors satisfying particular equations have become a subject of interest. We will give such a representation for submanifolds of any dimension and codimension into Riemannian space forms and explain how this can be generalized to the case of other ambient spaces, in particular when the ambient manifold is a (reductive) homogeneous space.

28 May Ivan Tulli (Sheffield)
Title. Special Joyce structures and hyperkähler metrics
Abstract. Joyce structures were introduced by T. Bridgeland in the context of the space of stability conditions of a 3-Calabi-Yau category and its associated Donaldson-Thomas invariants. In subsequent work, T. Bridgeland and I. Strachan showed that Joyce structures satisfying a certain non-degeneracy condition encode a complex hyperkähler structure on the tangent bundle of the base of the Joyce structure. The goal of this talk will be to define an analogous structure over an affine special Kähler (ASK) manifold, which we call a special Joyce structure, and show that it encodes a real hyperkähler (HK) structure on the tangent bundle of the ASK manifold. Particular examples include the rigid c-map metric, and the HK metrics associated with certain uncoupled variations of BPS structures over the ASK manifold.
This talk is based on arXiv:2403.00548.

Last updated 18 May, 2024