Hodge Theory in Combinatorial Geometry

Speaker

I am interested in the topology of algebraic varieties, and its interaction with combinatorics and algebraic statistics. I am currently interested in the following topics.

☑The topological aspects of algebraic optimization problems

☑Fourier-Mukai (or Mellin) transformation and the generic vanishing theorems

☑Kazhdan-Lusztig theory for matroids

☑Schubert varieties of hyperplane arrangements and their generalizations

Course Description

The purpose of these lectures is to give a survey on the recent resolution of two conjectures in matroid theory. The first one is the Heron-Rota-Welsh conjecture by Adiprasito-Huh-Katz, and the second one is the Dowling-Wilson conjecture by Braden-Huh-Matherne-Proudfoot-Wang. The main idea is to attach algebraic geometric invariants to a given matroid, and use ideas from Hodge theory to derive combinatorial properties. If time permits, we will also survey some ongoing developments of intersection cohomology of matroids and some new applications.