Tutte seminar - Eric Katz

Log concavity of characteristic polynomials and tropical intersection theory

Abstract:

In a recent joint work with June Huh, we proved the log concavity of the characteristic polynomial of a realizable matroid by relating its coefficients to intersection numbers on an algebraic variety and applying an algebraic geometric inequality. This extended earlier work of Huh which resolved a conjecture in graph theory.

In this talk, we rephrase the problem in terms of intersection theory in algebraic geometry, outline the proof, and discuss an approach to extending this proof to all matroids. Our proposed extension involves finding an analogue of the Hodge index theorem in a certain variant of algebraic graph theory.