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

行列式和多项式是数学科学研究人员经常使用的基本对象。这个项目的目的是更好地理解某些类型的多项式，如行列式，在数学中无处不在。双曲多项式，甚至更一般地，对数凹多项式是真正的多项式，它们具有许多有用的行列式的泛函性质。双曲多项式的研究起源于20世纪50年代的偏微分方程，并被用来理解凸优化、算子理论和组合学中的问题。关于这些多项式仍然有许多重要的开放性问题，并且由于它们的普遍性和具体性质，结构结果通常具有广泛的应用。这个项目的教育部分包括一系列针对本科生和研究生的夏季研讨会，为他们提供一个重要和活跃的数学研究领域的背景知识，以及它在其他领域的应用。近年来，对数凹多项式与拟阵之间的密切联系被发展为独立结构的组合模型。这些多项式的系数给出了可以有效采样的离散概率分布。本研究的目的是利用真实的代数和离散几何来进一步发展我们对这类多项式的理解，以期在近似算法、组合学、凸优化和算子理论中的应用。该奖项还将支持为本科生和研究生举办的讲习班，培训他们的领域包括真正的代数几何、多项式几何、热带几何及其在数学之外的应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.