Jonathan Leake, Technische Universität Berlin, Germany

Wednesday, October 6, 2021 - 3:30pm to 5:00pm

PDL C-38 and via Zoom Link: https://washington.zoom.us/j/91547335974

**Abstract:**

About 5 years ago, the Heron-Rota-Welsh conjecture (log-concavity of the coefficients of the reduced characteristic polynomial of a matroid) was proven by Adiprasito, Huh, and Katz via the development of a new combinatorial Hodge theory for matroids. In very recent work with Petter Brändén, we have given a new short "polynomial proof" of the Heron-Rota-Welsh conjecture. Our proof uses an extension of the theory of Lorentzian polynomials to convex cones, and in particular reproves the Hodge-Riemann relations of degree one for the Chow ring of a matroid. In the pre-seminar, I will discuss the basics of Lorentzian (aka completely log-concave) polynomials as developed by Anari-Liu-Oveis Gharan-Vinzant and by Brändén-Huh in recent years. In the remainder of the talk, I will give an overview of our new proof of the Heron-Rota-Welsh conjecture, emphasizing newly developed tools in the theory of Lorentzian polynomials.

**Note: This talk begins with a pre-seminar (aimed at graduate students) at 3:30–4:00. The main talk starts at 4:10.**

**Join Zoom Meeting: https://washington.zoom.us/j/ 91547335974Meeting ID: 915 4733 5974**