Wed, 04 Jun 2025
16:00
L6

Even the Loch Ness monster deserves a curve graph

Filippo Baroni
(University of Oxford)
Abstract
Every topologist knows that a mug is a doughnut, but did you know that the Loch Ness monster is a baguette?
 
This talk is meant as a gentle introduction to the theory of big surfaces and their mapping class groups. This is a topic that has gained significant traction in the last few years, and is undergoing an exciting phase of explosive expansion.
 
We will start by giving lots of examples of surfaces of infinite type, working our way towards a general classification theorem. We will then introduce big mapping class groups, and outline some of their topological properties that are reminiscent of classical geometric group theory. Finally, following a programme proposed by Calegari in 2009,  we will investigate to what extent the classical theory of curve and arc graphs of finite-type surfaces generalises to the infinite-type setting. 
 
The level of prior required knowledge on the topic of big mapping class groups will be the same as that of the speaker one week before the talk — that is, none.
Wed, 28 May 2025
16:00
L6

Instanton homology for $\mathfrak{gl}_2$ webs and foams

Alex Epelde Blanco
(Harvard University)
Abstract

In the definition of the skein lasagna module of a $4$-manifold $X$, it is essential that the input TQFT be fully functorial for link cobordisms in $S^3 \times [0, 1]$. I will describe an approach to resolve existing sign ambiguities in Kronheimer and Mrowka's spectral sequence from Khovanov homology to singular instanton link homology. The goal is to obtain a theory that is fully functorial for link cobordisms in $S^3 \times [0,1]$, and where the $E_2$ page carries a canonical isomorphism to Khovanov-Rozansky $\mathfrak{gl}_2$ link homology. Possible applications include non-vanishing theorems for $4$-manifold Khovanov skein lasagna modules à la Ren-Willis.

Wed, 21 May 2025
16:00
L2

Fat minors and where to find them

Joseph MacManus
(University of Oxford)
Abstract

Recently, much attention has been paid to the intersection between coarse geometry and graph theory, giving rise to the fresh, exciting new field aptly known as ‘coarse graph theory’. One aspect of this area is the study of so-called ‘fat minors’, a large-scale analogue of the usual idea of a graph minor.

In this talk, I will introduce this area and motivate some interesting questions and conjectures. I will then sketch a proof that a finitely presented group is either virtually planar or contains arbitrarily ‘fat’ copies of every finite graph.

No prior knowledge or passion for graph theory will be assumed in this talk.

Wed, 14 May 2025
16:00
L6

Coarse cohomology of metric spaces and quasimorphisms

William Thomas
(University of Oxford)
Abstract

In this talk, we give an accessible introduction to the theory of coarse cohomology of metric spaces in the sense of Margolis, which we present in direct analogy with group cohomology for discrete groups. We explain how this yields the robust notion of coarse cohomological dimension (due to Margolis), which is a genuine quasi-isometry invariant of metric spaces generalising the cohomological dimension of groups when the latter is finite. We then give applications to geometric properties of quasimorphisms and motivate how such considerations might be useful in the setting of non-positively curved groups. This is joint reading/work with Paula Heim.

Wed, 30 Apr 2025
16:00
L3

Property (T) via Sum of Squares

Gargi Biswas
(University of Oxford)
Abstract

Property (T) is a rigidity property for group representations. It is generally very difficult to determine whether an infinite group has property (T) or not. It has long been known that a discrete group with a finite symmetric generating set has property (T) if and only if the group Laplacian is a positive element in the maximal group C*-algebra. However, this characterization has not been useful in addressing the question for automorphism groups of (non-abelian) free groups. In his 2016 paper, Ozawa proved that the phenomenon of 'positivity' of the group Laplacian is observed in the real group algebra, meaning that the Laplacian can be decomposed into a 'sum of squares'. This result transformed checking property (T) into a finite-dimensional condition that can be performed with the assistance of computers. In this talk, we will introduce property (T) and discuss Ozawa's result in detail.

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