# Lorentzian polynomials I: Theory

I’m organizing a reading seminar this semester on Lorentzian polynomials, mainly following this paper by Brändén and Huh but also covering some of the work of Anari et. al. In this post, I’d like to give a quick introduction to this active and beautiful subject. I’ll concentrate on the basic theory for now, and in a follow-up post I’ll discuss some of the striking applications of this theory.

One major goal of the theory of Lorentzian polynomials is to provide new techniques for proving that various naturally-occurring sequences of non-negative real numbers $a_k$ are log-concave, meaning that $a_k^2 \geq a_{k-1} a_{k+1}$ for all $k$. More generally, the authors consider homogeneous multivariate polynomials $p(x_1,\ldots,x_n)$ with non-negative coefficients and study certain natural extensions of log-concavity to this setting. (For some background on log-concave sequences, see this survey paper which I wrote for the Bulletin of the AMS.) So let me begin by introducing two “classical” methods for proving (an even sharper version of) log-concavity of the coefficients of certain polynomials.

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# Hodge Theory in Combinatorics

From L to R: Karim Adiprasito, June Huh, Eric Katz

In January 2016, my colleague Josephine Yu and I are organizing a workshop called Hodge Theory in Combinatorics. The goal of the workshop is to present the recent proof of a 50-year-old conjecture of Rota by Karim Adiprasito, June Huh, and Eric Katz. In this post, I want to explain what the conjecture says and give a brief outline of its marvelous proof. I will follow rather closely this paper by Adiprasito-Huh-Katz (henceforth referred to as [AHK]) as well as these slides from a talk by June Huh. Continue reading