Topology seminars: 2024-25

Recently (together with Andrew Fisher and James Cranch) I have thinking about cohomology of diagram algebras. These arise in all sorts of areas of maths but at present I don't understand any of this as much as I would like. In this talk, I'll focus on one particular algebra: the Temperley-Lieb algebra. Time and understanding permitting, I hope to say a little about these algebras in the context of algebra, representation theory, combinatorics, topology, homological algebra and (at least very vaguely) statistical mechanics. Some of what I hope to say can be found in the following recently-revised arXiv submission: https://arxiv.org/pdf/2307.11929

Large-scale geometry (or coarse geometry) is the study of spaces, not through local features, but through global or asymptotic ones, for example through the growth function or an asymptotic notion of dimension. These ideas have proven powerful in understanding metric spaces and beyond, but most notably in geometric group theory, where results like the Švarc–Milnor lemma allow us to study the structure of finitely generated groups via the large-scale geometry of their Cayley graphs.

In this talk, I will discuss two coarse-geometric tools that describe how spaces behave “at infinity,” and explore how they relate. The first is the classical notion of ends, introduced by Freudenthal, which records the number of distinct ways an observer may head off to infinity in a given space. The second is the more recent concept of coarse path components, arising from the coarse homotopy theory developed by Mitchener, Norouzizadeh, and Schick. I will outline both notions, and present examples illustrating when they coincide and when they differ.