The Bruhat-Tits building is a crucial combinatorial tool in the study of reductive p-adic groups and their representation theory. Given a p-adic group, its Bruhat-Tits building is a simplicial complex upon which it acts with remarkable properties. In this talk I will give an introduction to the Bruhat-Tits building by sketching its definition and going over some of its basic properties. I will then show the usefulness of the Bruhat-Tits by determining the maximal compact subgroups of a p-adic group up to conjugacy by using the Bruhat-Tits building.

The Brunn-Minkowski inequality gives a lower bound on the volume of the set of midpoints of line segments joining two sets. On a Riemannian manifold, line segments are replaced by geodesic segments, and the Brunn-Minkowski inequality characterizes manifolds with nonnegative Ricci curvature. I will present a generalization of the Riemannian Brunn-Minkowski inequality where geodesics are replaced by magnetic geodesics, which are minimizers of a functional given by length minus the integral of a fixed one-form on the manifold. The Brunn-Minkowski inequality is then equivalent to nonnegativity of a suitably defined magnetic Ricci curvature. More generally, I will present a magnetic version of the Borell-Brascamp-Lieb inequality of Cordero-Erausquin, McCann and Schmuckenschläger. The proof uses the needle decomposition technique.