A graph $X$ is defined inductively to be $(a_0, . . . , a_{n−1})$-regular if $X$ is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius 1 around $v$ is an $(a_1, . . . , a_{n−1})$-regular graph. A family $F$ of graphs is said to be an expander family if there is a uniform lower bound on the Cheeger constant of all the graphs in $F$. 

After briefly (re)introducing Coxeter groups and their geometries, we will describe how they can be used to construct very regular polytopes, which in turn can yield highly regular graphs. We will then use the super-approximation machinery, whenever the Coxeter group is hyperbolic, to obtain the expansion of these families of graphs. As a result, we obtain interesting infinite families of highly regular expander graphs, some of which are related to the exceptional groups. 

The talk is based on work joint with Conder, Lubotzky, and Schillewaert. 

Optimal transport theory is a natural way to define both a distance and a geometry on the space of probability measures. In settings like graphical causal models (also called Bayes networks or belief networks), the space of probability measures is enriched by an information structure modeled by a directed graph. This talk introduces a variant of optimal transport including such a graphical information structure. The goal is to provide a concept of optimal transport whose topological and geometric properties are well suited for structural causal models. In this regard, we show that the resulting concept of Wasserstein distance can be used to control the difference between average treatment effects under different distributions, and is geometrically suitable to interpolate between different structural causal models.