There are a number of features in statistical mechanics which we believe are shared by a large number of models: universality, scale invariance, etc. These properties are often assumed for a model before we have shown rigorously that they hold, and then used to derive useful information. In this talk, I will use the example of Galton-Watson trees and their scaling limit, the continuum random tree, to explain what we mean by universality and scale invariance, and how studying these properties of models can give us information about combinatorial objects such as trees.

We will introduce the notion of "asymptotic expansion" recently defined by Li, Nowak, Spakula and Zhang and give operator algebraic motivation for its study. We will explore this definition in the familiar setting of Cayley graphs, and show that this leads to a C*-algebraic characterisation of usual expanders for graphs that are sufficiently symmetric. This is joint work with Li, Vigolo and Zhang.

I'll try and explain how to compute the wrapped Fukaya category of a thrice-punctured sphere.

TBA