TTT124

Date Thursday 16th July 2026

Speakers

The speakers will be Brad Ashley (de Montfort), Jan Steinebrunner (Cambridge) and Lukas Waas (Oxford).

Programme

 The talks will be in MATH210 on the second floor of the maths building. 

1:30 - 2:30 Lukas Waas (Oxford) 

2:30 - 3:30 Brad Ashley (de Montfort)

tea/coffee break

4:00 - 5:00 Jan Steinebrunner (Cambridge) 

early dinner 

Titles and abstracts

Lukas Waas - Presenting the stratified homotopy hypothesis

One of the fundamental principles of higher category theory is the homotopy hypothesis - the equivalence between the homotopy theory of spaces - as presented by the Quillen-model structure on topological spaces - and the homotopy theory of infinity groupoids - as presented, for example, by the Kan-Quillen model structure on simplicial sets. Conveniently, we can just present this equivalence in terms of the classical realization and singular simplicial set adjunction - the classical Kan-Quillen equivalence.

A version of this statement concerning stratified topological spaces was conjectured by Ayala-Francis and Rozenblyum and first proven by Peter Haine. Here, I want to explain how to elevate this result to a similarly convenient situation as in the classical unstratified setting. Namely, I will provide an answer to a conjecture of Wolf and Nand-Lal that inquired after the existence of a model structure for stratified spaces in which classical geometrical examples are bifibrant, and explain how Lurie's stratified version of the classical adjunction presents the aforementioned stratified version of the homotopy hypothesis in terms of a stratified Kan-Quillen equivalence.

Brad Ashley -  Asymptotic Topology and Roe-Style Coarse Geometry

In this talk I will briefly introduce some of the key ideas in the study of asymptotic topology (also known as large-scale geometry/topology or coarse geometry). I will discuss some commonly studied asymptotic geometric properties, namely Freudenthal’s ends and Gromov’s asymptotic dimension, and say a few words on their application in geometric group theory. In the second part of the talk, I will introduce Roe’s axiomatic coarse structures that generalise large-scale geometry, briefly discuss some non-metric examples, and if time permits speak a little on the current ideas I am working on.

Jan Steinebrunner -  The prime decomposition fibre sequence for moduli spaces of 3-manifolds

Milnor's prime decomposition theorem states that every oriented 3-manifold M can be written as a connected sum of "prime" manifolds in an essentially unique way: M == P_1 # ... # P_n # (S^1 x S^2)^{#g}. This reduces many questions about 3-manifolds to the prime case, but when studying 3-manifolds in families this reduction is not so straightforward. For example, a diffeomorphism of M need not respect the decomposition into prime factors.

I will explain recent joint work with Boyd and Bregman, in which we use a homotopical version of the prime decomposition theorem to describe the classifying space BDiff(M) (the "moduli space" of M) in terms of moduli spaces of the P_i. More precisely, we establish a "prime decomposition fibre sequence" that describes the moduli space in terms of BDiff(P_1 u ... u P_n) and a space of handle-attachments that is amenable to computations. To illustrate this, I will discuss our calculation of the rational cohomology ring of BDiff((S^1 x S^2)#(S^1 x S^2)).