Invariant theory, that is the art of finding polynomials invariant under the action of a given group, has played a major role in the historical development of commutative algebra. In this theory reflection groups are singled out for having rings of invariants that are isomorphic to polynomial rings. From a geometric perspective, reflection groups give rise to beautiful and very symmetric arrangements of hyperplanes termed reflection arrangements.

This talk will take a close look at the ideals defining the singular loci of reflection arrangements, which are in turn symmetric subspace arrangements. We describe their syzygies in terms of invariant polynomials for the relevant reflection groups. We leverage this information to settle many aspects of the containment problem asking for containments between the ordinary and the symbolic powers of the ideals in this family. This talk is based on joint work with Ben Drabkin.

Matroid theory had its origins in linear algebra and graph theory. In recent years, the geometric roots of the field have grown much deeper, bearing many new fruits. The interplay between matroid theory and algebraic geometry has opened up interesting research directions at the intersection of combinatorics, algebra, and geometry, and led to the solution of long-standing questions.

This talk will discuss my recent joint work with Graham Denham and June Huh. We introduce the conormal fan of a matroid M. Inside its Chow ring, we find simple interpretations of the Chern-Schwartz-MacPherson cycle of M (a tropical geometric construction) and of the h-vector of M (a combinatorial invariant). We then use the Hodge-Riemann relations to prove Brylawski's and Dawson's conjectures that the h-vector of a matroid is log-concave.

I will make the talk as self-contained as possible, and assume no previous knowledge of matroid theory.