Last time we defined what it means to be a Minkowski weight over a k-skeleton of a fan (complete or not). To a loopless matroid, there is an associated simplicial fan called Bergman fan. Following de Medrano, Rincon and Shaw we will define the CSM classes of a loopless matroid as Minkowski weights over this fan. We will show that it satisfies the balancing condition. This will involve some combinatorial machinery concerning matroid that we will review.

Topological Data Analysis provides a framework for analyzing data that is robust to perturbations of the data. One way in which it accomplishes this is by introducing stable invariants: invariants of data sets that vary continuously with respect to suitable metrics on the collection of data sets. In this talk I will present several well known invariants of different kinds of data (such as metric spaces, metric measure spaces, dynamic metric spaces, and filtered metric spaces) and a theoretical framework that lets us prove known stability results as well as novel ones.

Regularity structures were introduced by Martin Hairer to deal with the divergences that arise from stochastic partial differential equations which typically involve white noise. Exploring the underlying geometry reveals the role played in this context by direct connections on a vector bundle. Originally introduced by Nikolai Teleman in the context of non commutative geometry, these provide a direct transport of fibres from point to point. We generalise them to groupoids and propose an interpretation of re-expansion maps arising in regularity structures in the language of groupoids. Re-expansion maps were introduced by Hairer to transform a singular stochastic differential equation into a fixed point problem, based on an ad hoc ``Taylor expansion'' of the solutions at any point in space-time and a ``re-expansion map'' which relates the values at different points. For gauge groupoids, namely those built from a principal bundle, a re-expansion map can be viewed as a (local) ``gaugeoid field'', the groupoid counterpart of a (local) gauge field. We investigate the case of jet bundles arising in polynomial regularity structures.