Lagrangian Geometry of Matroids

Abstract: 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.

commutative algebra

Audience: researchers in the topic

Fellowship of the Ring

Series comments: Description: National Commutative Algebra Seminar

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