Stable Polynomials, Rational Singularities, and Operator Theory

该项目涉及数学的一个经典领域，称为数学分析，特别关注其两个子领域，复分析和算子理论。 复分析是一门成熟的学科，具有广泛的适用性--从绘制地球仪到理解算法的运行时。 算子理论最初是为了研究量子力学而创建的，后来发展成为一个类似的成熟领域，适用于工程和优化。 分析能力的一部分是能够将具体的任务（如设计恒温器或理解素数的分布）转化为关于称为函数的数学对象的问题。 该项目关注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，这些问题必然需要对多变量函数（更具体地说，多变量有理函数）有更深入的理解。 该研究将被纳入本科和研究生两个层次的教育角色，并通过指导学生对这些现代和重要的分析领域。该项目侧重于三个主要领域：（1）刻画多元区域上有理函数的有界性，（2）理解有理函数的可积性，（3）系统地确定了多元有理函数系数的渐近性。 这项研究也与稳定多项式理论密切相关，这是一个在过去十年中获得了许多令人惊讶的应用的领域。 发展将继续在旧的和新的设置的稳定多项式的局部理论，以获得有关的行为的一个合理的功能附近的一个奇点的详细信息。 此外，某些多项式和有理函数建立了自然的运营商理论的建设，如行列式表示，代表了重要的特殊情况下的利益，无论是在应用和运营商theory.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.