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# University of York Geometry, Analysis and Mathematical Physics seminar 2020-21

All are welcome to attend the seminar, which usually takes place at 2pm on Tuesdays during term time.
For the near future, the seminar will be held via Zoom. If you would like to receive the regular weekly announcements then please e-mail me and I can add you to the e-mail list.

## Autumn Term 2020

 Date Speaker Title 6 October 13 October Sebastian Heller (Hannover) The energy of holomorphic sections of the Deligne-Hitchin moduli space 20 October Georgios Kydonakis (MPIM Bonn) Poisson structures on moduli spaces of Higgs bundles over stacky curves 27 October Ben Sibley (Université libre de Bruxelles) Compactifcations of Hermitian-Yang-Mills moduli space and the Yang-Mills flow on projective manifold 3 November 10 November4pm this week only Paul Feehan (Rutgers) Virtual Morse-Bott theory on analytic spaces, moduli spaces of SO(3) monopoles, and applications to four-manifolds (slides) 17 November Lynn Heller (Hannover) Area Estimates for High genus Lawson surfaces via DPW 24 November Jason Lotay (Oxford) Deformed $$G_2$$-instantons (slides) 1 December Eloise Hamilton (IMJ-PRG) Moduli spaces for unstable Higgs bundles of rank 2 and their geometry

## Spring Term 2021

 Date Speaker Title 19 January Chikako Mese (Johns Hopkins) Holomorphic Rigidity of Teichmuller space 26 January 2 February 9 February 16 February Mark Haskins (Duke) Solitons in Bryant's G_2-Laplacian flow (slides) 23 February 2 March Sushmita Venugopalan (IMSc Chennai) Tropical Fukaya Algebras 9 March 16 March Chris Woodward (Rutgers) Morse flow trees and deformations of exact Lagrangians

## Summer Term 2021

 Date Speaker Title 20 April 27 April 4 May 11 May Alastair King (Bath) Categorification of perfect matchings 15 June Zeno Huang (CUNY) TBA

2019-20

## Abstracts

13 October. Sebastian Heller (Hannover)
Title. The energy of holomorphic sections of the Deligne-Hitchin moduli space
Abstract. The aim of the talk is to introduce a natural functional on the space of holomorphic sections of the twistor spaces of a class of hyper-Kähler manifolds, including the Hitchin moduli spaces.
In the first part, we recall basic concepts of hyper-Kähler manifolds and their twistor spaces. For hyper-Kähler spaces admitting a circle action (and some additional conditions) we explain the construction (by Haydys and Hitchin) of a holomorphic line bundle with a meromorphic connection on its twistor space. The energy of a section is given by the residue of the pull-back of the meromorphic connection. It is shown that the energy on the spaces of twistor lines coincides with the moment map for the circle action. In the second part of the talk we consider the particular case of the Higgs bundle moduli space. Its twistor space can be identified with the Deligne-Hitchin moduli space of $$\lambda$$-connections. We prove a natural formula for the energy of a section of the Deligne-Hitchin moduli spaces in terms of a Serre-type pairing. Finally, we compute the energy for a class of examples given by Willmore surface.
The talk is based on joint work with F. Beck and M. Röser, and on joint work with L. Heller.

20 October. Georgios Kydonakis (MPIM Bonn)
Title. Poisson structures on moduli spaces of Higgs bundles over stacky curves
Abstract. For a connected complex reductive group $$G$$, the $$G$$-character varieties over a complex algebraic curve consist a much studied class of Poisson manifolds. Under the non-abelian Hodge correspondence such character varieties correspond to appropriate algebro-geometric quotients, over which one can study the construction of certain Poisson structures from a rather algebraic point of view. In this talk, we shall discuss a certain construction via Atiyah algebroids for the moduli spaces of Higgs bundles over stacky curves.
This is joint work with Hao Sun and Lutian Zhao.

27 October. Ben Sibley (Université libre de Bruxelles)
Title. Compactifcations of Hermitian-Yang-Mills moduli space and the Yang-Mills flow on projective manifold
Abstract. One of the cornerstones of gauge theory and complex geometry in the late 20th century was the so-called "Kobayashi-Hitchin correspondence", which provides a link between Hermitian-Yang-Mills connections (gauge theory) and stable holomorphic structures (complex geometry) on a vector bundle over projective (or merely Kähler) manifold. On the one hand, this gives an identification of (non-compact) moduli spaces. On the other, one proof of the correspondence goes through a natural parabolic (up to gauge) flow called Yang-Mills flow. Namely, Donaldson proved the convergence of this flow to an Hermitian-Yang-Mills connection in the case that the initial holomorphic structure is stable. Two questions that this leaves open are: 1. Do the moduli spaces admit compactifications, and if so what sort of structure do they have? Are they for example complex spaces? Complex projective? What is the relationship between the compactifications on each side? 2. What is the behaviour of the flow at infinity in the case when the initial holomorphic structure is unstable? I will touch on aspects of my previous work on these problems and explain how they connect up with each other.
This work is spread out over several papers, and is partly joint work with Richard Wentworth, and with Daniel Greb, Matei Toma, and Richard Wentworth.

10 November. Paul Feehan (Rutgers)
Title. Virtual Morse-Bott theory on analytic spaces, moduli spaces of SO(3) monopoles, and applications to four-manifolds
Abstract. We introduce an approach to Morse-Bott theory, called virtual Morse-Bott theory, for Hamiltonian functions of circle actions on closed, real analytic, almost Hermitian spaces. In the case of Hamiltonian functions of circle actions on closed, smooth, almost Kähler (symplectic) manifolds, virtual Morse-Bott theory coincides with classical Morse-Bott theory due to Bott (1954) and Frankel (1959). We prove that positivity of virtual Morse-Bott indices implies downward gradient flow in the top stratum of smooth points in the analytic space. When applied to moduli spaces of Higgs pairs on a smooth Hermitian vector bundle over a Riemann surface of genus two or more, virtual Morse-Bott theory allows one to relax assumptions due to Hitchin (1987) that ensure that those moduli spaces are smooth, for example, that the degree and rank of the vector bundle are coprime and hence that the defining Higgs equations for Higgs pairs are unobstructed in the sense of Kuranishi (1965) and that gauge transformations act freely. We apply our method to the moduli space of SO(3) monopoles over a complex, Kähler surface, use the Atiyah-Singer Index Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces), and prove that these indices are positive in a setting with implications for the geography of closed, smooth four-manifolds of Seiberg-Witten simple type.
Reference: https://arxiv.org/abs/2010.15789

17 November. Lynn Heller (Hannover)
Title. Area Estimates for High genus Lawson surfaces via DPW
Abstract. Starting at a saddle tower surface, we give a new existence proof of the Lawson surfaces $$\xi_{m,k}$$ of high genus by dropping some closing conditions of the surface and then deforming the corresponding DPW potential. As a byproduct, we obtain for fixed m estimates on the area of $$\xi_{m,k}$$ in terms of their genus $$g= mk \gg 1$$
This is joint work with Sebastian Heller and Martin Traizet.

24 November. Jason Lotay (Oxford)
Title. Deformed $$G_2$$-instantons
Abstract. Deformed $$G_2$$-instantons are special connections occurring in $$G_2$$ geometry in 7 dimensions. They arise as “mirrors” to certain calibrated cycles, providing an analogue to deformed Hermitian-Yang-Mills connections, and are critical points of a Chern-Simons-type functional. I will describe an elementary construction of the first non-trivial examples of deformed $$G_2$$-instantons, and their relation to 3-Sasakian geometry, nearly parallel $$G_2$$-structures, isometric $$G_2$$-structures, obstructions in deformation theory, the topology of the moduli space, and the Chern-Simons-type functional.

1 December. Eloise Hamilton (Institut de Mathématiques de Jussieu-Paris Rive Gauche)
Title. Moduli spaces for unstable Higgs bundles of rank 2 and their geometry
Abstract. The moduli space of semistable Higgs bundles of arbitrary rank and degree on a nonsingular projective curve was first constructed by Nitsure in 1990, using Geometric Invariant Theory (GIT). Thanks to its rich geometric structure, this moduli space continues to represent an active area of research. The aim of this talk is to describe how recent results in Non-Reductive GIT can be used to construct moduli spaces for Higgs bundles which are not semistable, and to describe initial steps towards the study of their geometry in the rank 2 case.

19 January. Chikako Mese (Johns Hopkins)
Title. Holomorphic Rigidity of Teichmuller space
Abstract. The holomorphic rigidity of Teichmuller space can be loosely stated as follows: The action of the mapping class group uniquely determines the complex structure of Teichmuller space. The proof applies the theory of finite and infinite energy harmonic maps into NPC (non-positively curved) metric spaces.

Title. Solitons in Bryant's $$G_2$$-Laplacian flow
Abstract. I will introduce some basic features of $$G_2$$-geometry, a geometry peculiar to 7 dimensions defined in terms of the exceptional compact simple Lie group $$G_2$$. I will then describe a geometric flow on 3-forms, due to Robert Bryant, called Laplacian flow that aims to produce Riemannian manifolds with holonomy group $$G_2$$. My talk will concentrate on certain special solutions to Laplacian flow called solitons: in particular I will describe a recent construction of noncompact shrinking, steady and expanding solitons in Laplacian flow all with asymptotically conical geometry. In other better-understood geometric flows (e.g. Ricci flow and mean curvature flow) such solitons have played a key role in understanding singularity formation and hence in understanding the long-time behaviour of these flows.
This is joint work with Johannes Nordström and also in part with Rowan Juneman at Bath.

2 March. Sushmita Venugopalan (IMSc Chennai)
Title. Tropical Fukaya Algebras
Abstract. A multiple cut operation on a symplectic manifold produces a collection of cut spaces, each containing relative normal crossing divisors. We explore what happens to curve count-based invariants when a collection of cuts is applied to a symplectic manifold. The invariant we consider is the Fukaya algebra of a Lagrangian submanifold that is contained in the complement of relative divisors. The ordinary Fukaya algebra in the unbroken manifold is homotopy equivalent to a broken Fukaya algebra' whose structure maps count broken disks' associated to rigid tropical graphs. Via a further degeneration, the broken Fukaya algebra is homotopy equivalent to a `tropical Fukaya algebra' whose structure maps are sums of products over vertices of tropical graphs.
This is joint work with Chris Woodward.

16 March. Chris Woodward (Rutgers)
Title. Morse flow trees and deformations of exact Lagrangians
Abstract. The adiabatic limit of the moduli of Floer trajectories associated to Fukaya-theoretic isomorphisms between exact Lagrangians in cotangent bundles is a union of Morse flow trees. I will discuss some of its properties and explain how in good cases one obtains a canonical deformation of the Lagrangian.

11 May. Alastair King (Bath)
Title. Categorification of perfect matchings
Abstract. I will describe what role dimer models on a disc play in understanding the positroid stratification of Grassmannians. I will then explain how some representation theory sheds light on the associated combinatorics.

Last updated 7 May, 2021