Stable Polynomials, Rational Singularities, and Operator Theory

这个项目涉及数学的一个经典领域，称为数学分析，特别关注它的两个子领域，复分析和算子理论。复杂分析是一门成熟的学科，具有广泛的适用性——从绘制全球地图到理解算法的运行时间。算子理论最初是为了研究量子力学而创建的，后来发展成为一个同样成熟的领域，适用于工程和优化。分析能力的一部分是将具体任务（如设计恒温器或理解质数的分布）转化为有关称为函数的数学对象的问题的能力。该项目涉及这些学科的现代基础研究，重点关注具有内在多变量性质的问题，这些问题需要对多变量函数（更具体地说是多变量有理函数）有更深入的理解。这项研究将被纳入本科和研究生阶段的教育角色，并通过指导学生对这些现代和重要的分析领域进行研究。本课题主要集中在三个方面：(1)在多变量域上刻画有理函数的有界性；(2)理解有理函数的可积性；(3)系统地确定多变量有理函数的系数渐近性。这项研究也与稳定多项式理论密切相关，这一领域在过去十年中获得了许多令人惊讶的应用。将继续发展稳定多项式的局部理论，在旧的和新的设置，以获得关于一个有理函数在奇点附近的行为的详细信息。此外，某些由自然算子理论结构（如行列式表示）构建的多项式和有理函数在应用和算子理论中都是重要的特殊情况。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。

This project concerns a classical area of mathematics called mathematical analysis with specific attention on two of its subfields, complex analysis and operator theory. Complex analysis is a mature subject with wide applicability – from mapping the globe to understanding the runtime of algorithms. Operator theory was originally created to study quantum mechanics and has since grown into a similarly mature field with applicability to engineering and optimization. Part of the power of analysis is the ability to convert concrete tasks, such as designing a thermostat or understanding the distribution of prime numbers, into questions about mathematical objects called functions. This project concerns modern fundamental research in these subjects with a focus on questions with an inherent multivariable nature necessarily requiring a deeper understanding of multivariable functions (and more specifically multivariable rational functions). The research will be incorporated into educational roles at both the undergraduate and graduate levels and by mentoring of students on these modern and important areas of analysis.This project focuses on three main areas: (1) characterizing the boundedness of rational functions on domains in several variables, (2) understanding the integrability of rational functions, and (3) systematically determining the asymptotics of coefficients of multivariable rational functions. This research is also closely tied to the theory of stable polynomials, an area which has enjoyed numerous surprising applications in the last decade. Development will be continued of the local theory of stable polynomials in old and new settings to get detailed information about the behavior of a rational function near a singularity. In addition, certain polynomials and rational functions built out of natural operator theoretic constructions such as determinantal representations represent important special cases of interest both in applications and to operator theory.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.