# Spring 2024

The Seminar runs on *Tuesdays* at **STRAND BLDG S4.29**. This term, some talks will run from *15:00 to 16:00*, and others will be in two parts – an introductory talk from *15:00 to 15:40* and a research talk from *15:45 to 16:30*.

23 April, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29

Speaker: **Dan Kaplan** (University of Hasselt)

Title: Quiver varieties for the working geometer

Abstract: This talk is divided into two related, yet self-contained sections. The first section is an elementary introduction to (Nakajima) quiver varieties, beginning with representations of quivers and emphasizing small examples. The second section shifts gears to symplectic resolutions of singularities, including the minimal resolutions of du Val singularities and the Springer resolution of the nilpotent cone of a Lie algebra.

The sections unite as we construct symplectic resolutions for quiver varieties by varying a stability parameter. In joint work with Travis Schedler, we leverage these symplectic resolutions to build resolutions for spaces that are (analytically) locally quiver varieties. The key idea here is to choose local resolutions at the most singular points and then demonstrate that certain compatible, monodromy-free choices extend and glue to a global resolution.

30 April, 15:00-16:00, STRAND BLDG S4.29

Speaker: **Ulrike Tillmann** (University of Oxford)

Title: Homology stability for generalised Hurwitz spaces and asymptotic monopoles

Abstract: Configuration spaces have played an important role in mathematics and its applications. In particular, the question of how their topology changes as the cardinality of the underlying configuration changes has been studied for some fifty years and has attracted renewed attention in the last decade.

While classically additional information is associated “locally” to the points of the configuration, there are interesting examples when this additional information is “non-local”. With Martin Palmer we have studied homology stability in some of these cases, including Hurwitz space and moduli spaces of asymptotic monopoles.

7 May, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29

Speaker: **Ilaria Di Dedda** (King’s College London)

Title: Type A symplectic Auslander correspondence

Abstract: In this talk, we will study invariants of complex isolated hypersurface singularities. In the first half I will review the basics of Floer theory, and I will describe Fukaya-Seidel categories, a powerful and geometric derived invariant of singularities. In the second half, I will describe invariants of a special family of isolated singularities, whose Fukaya-Seidel categories play an important role in bordered Heegaard Floer theory. Motivated by representation theory, I will relate these singularities to abstract objects associated to algebras of type A (named after the quiver of Dynkin type A). I will introduce “type A symplectic Auslander correspondence”, a purely geometrical construction which realises a notable result in representation theory. Most of the talk will be example-based.

14 May, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29

Speaker: **Eloise Hamilton** (University of Cambridge)

Title: Geometric Invariant Theory and moduli of representations of quivers [Part 1] Non-Reductive Geometric Invariant Theory and moduli of representations of quivers with multiplicities [Part 2]

Abstract: Geometric Invariant Theory (GIT) is a powerful theory for constructing quotients in algebraic geometry. An important classical application of GIT is to the construction of moduli spaces of representations of quivers. I will explain the basics of GIT and this particular application in the first part of the talk. In the second part of the talk I will turn to a natural generalisation of representations of quivers, called representations of quivers with multiplicities. These are representations of quivers in the category of modules over a truncated polynomial ring, instead of in the category of vector spaces. Unfortunately, moduli spaces of such objects cannot be constructed using GIT because the group involved is not reductive, yet reductivity plays a key role in GIT. I will explain recent work, joint with Victoria Hoskins and Joshua Jackson, in which we extend existing work on a non-reductive version of GIT to enable the construction of moduli spaces for representations of quivers with multiplicities.

21 May, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29

Speaker: **Anna Felikson** (Durham University)

Title: Polytopal realizations of non-crystallographic associahedra

Abstract: An associahedron is a polytope arising from combinatorics of Catalan-type objects (for example, from a collection of all triangulations of a given polygon). Fomin and Zelevinsky found a way to construct the same combinatorial structure from considering the Coxeter group of type A_n. This allowed them to define a generalized associahedron for every finite reflection group. For generalized associahedra arising from crystallographic reflection groups, it was also shown that they can be realized as polytopes. We use the folding technique to construct polytopal realisations of generalized associahedra for all non-simply-laced root systems, including non-crystallographic ones. This is a joint work with Pavel Tumarkin and Emine Yildrim.

In the first half of the talk, I will sketch the history of the associahedron and introduce generalised associahedra, then in the second half we will discuss how to produce the associahedra in the non-crystallographic case. The talk will not require any special background.

28 May, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29

Speaker: **Nick Lindsay** (University of Cologne)

Title: On a symplectic generalization of a Hirzebruch problem.

Abstract: The talk is based on a recent joint preprint with Leonor Godinho and Silvia Sabatini arxiv number 2403.00949. The main result is a classification of closed, 8-dimensional symplectic manifolds having a Hamiltonian T^2-action with exactly 6 fixed points. In the first part of the talk I will give a brief introduction to Hamiltonian torus actions with focus on material relevant for the paper. In the second part, I will begin by discussing motivation for the main result which comes from algebraic geometry. I will then discuss the main ideas of the proof.

11 June, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29

Speaker: **Filippo Baroni** (University of Oxford)

Title: Using the curve graph and train tracks to classify surface homeomorphisms

Abstract: The Nielsen-Thurston classification theorem states that there are three kinds of surface homeomorphisms up to homotopy: periodic, reducible, and pseudo-Anosov.

In the introductory part of the talk, we will investigate the differences between these three categories, focusing on the wide array of geometric, topological, and dynamical properties that set pseudo-Anosov mapping classes apart from the rest. To this end, we will also introduce the curve graph, a combinatorial object associated with a surface, and describe how the dynamics of the action of a mapping class on the curve graph can be used to detect pseudo-Anosovness.

In the second part of the talk, we will see how to turn this characterisation into an algorithm for deciding whether a given mapping class is pseudo-Anosov. The key tools will be the theory of train tracks and their connection to the curve graph developed by Masur and Minsky.