CAREER: Determinantal, hyperbolic, and log-concave polynomials in theory and applications

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Determinants and polynomials are fundamental objects used regularly by researchers across the mathematical sciences. The aim of this project is to better understand some classes of polynomials that, like determinants, are ubiquitous in mathematics. Hyperbolic polynomials and, even more generally, log-concave polynomials are real polynomials that share many useful functional properties of determinants. The study of hyperbolic polynomials originated in the context of partial differential equations in the 1950's and has been used to understand problems in convex optimization, operator theory, and combinatorics. There are still many important open questions about such polynomials, and because of their prevalence and concrete nature, structural results often have far-ranging applications. The educational component of this project includes a series of summer workshops for undergraduate and graduate students providing them a background in an important and active area of mathematical research as well as exposure to its applications in other fields. Recently, close connections between log-concave polynomials and matroids, which are combinatorial models of independence structures, were developed. Coefficients of these polynomials give discrete probability distributions that can be efficiently sampled. This research aims to use real algebraic and discrete geometry to further develop our understanding of these classes of polynomials, with a view towards applications in approximation algorithms, combinatorics, convex optimization, and operator theory. This award will also support workshops for undergraduate and graduate students to train them in areas including real algebraic geometry, the geometry of polynomials, tropical geometry, and their applications outside of mathematics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.