Jonathan Leake: Log-concavity and entropic inequalities via stable polynomials
Time: Wed 2019-09-04 10.15 - 11.00
Participating: Jonathan Leake
Abstract: In the past decade or two, the theory of stable polynomials (and subsequently, that of Lorentzian polynomials) has inspired new and combinatorially intuitive ways to prove various inequalities. In this talk, we will discuss two aspects of these developments in an accessible way. First, we will discuss an interesting connection between the log-concavity properties of stable polynomials and representation theory. In particular, we will show how log-concavity properties of a certain invariant linear operator is at the root of the polynomial stability theory. Second, we will discuss some entropic inequalities of stable polynomials. As an important example, we will use these inequalities to sketch a proof of a lower bound (originally due to Csikvari) on the matchings of a regular bipartite graph. We will finally discuss an open problem at the intersection of these two aspects, due to Gurvits.