Matroid theory had its origins in linear algebra and graph theory, and it turns out to have deep connections with many other fields. With time, the geometric roots of the field have grown much deeper, bearing many new fruits. The interplay between matroid theory and algebraic geometry has recently led to the solution of long-standing combinatorial questions. Perhaps more importantly, it has opened up new and interesting research directions at the intersection of combinatorics, algebra, and geometry.
This talk will discuss my recent joint work with Graham Denham and June Huh, where we use ideas from Lagrangian geometry to prove Brylawski and Dawson’s conjectures on the log-concavity of the h-vector of a matroid. I will gear the talk towards a combinatorial audience, and assume no previous knowledge of algebraic geometry.