What do fiber polymers and ice sheets have in common? They both flow with a directionally dependent - anisotropic - viscosity. This behaviour occurs in other geophysical flows, such as the Earth's mantle, where a material's microstructure affects its large-scale flow. In ice, the alignment of crystal orientations can cause the viscosity to vary by an order of magnitude, consequently having a strong impact on the flow of ice sheets and glaciers. However, the effect of anisotropy on large-scale flow is not well understood, due to a lack of understanding of a) the best physical approximations to model crystal orientations, and b) how crystal orientations affect rheology. In this work, we aim to address both these questions by linking rheology to crystal orientation predictions, and testing a range of models against observations from the Greenland ice sheet. The results show assuming all grains experience approximately the same stress provides realistic predictions, and we suggest a set of equations and parameters which can be used in large-scale models of ice sheets. 

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.