Stable Polynomials, Rational Singularities, and Operator Theory

稳定多项式、有理奇点和算子理论

• 批准号: 2247702
• 负责人: Greg Knese
• 金额: $ 22.73万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-15 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247702&HistoricalAwards=false
• 关键词: Stable; Polynomials; Rational; Singularities; Operator

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.

这个项目涉及一个被称为数学分析的经典数学领域，特别关注它的两个子领域--复分析和算子理论。复杂分析是一门具有广泛适用性的成熟学科--从绘制全球地图到理解算法的运行时。算符理论最初是为了研究量子力学而创立的，后来发展成为一个类似成熟的领域，适用于工程和优化。分析的力量之一是能够将具体的任务，如设计恒温器或了解质数的分布，转化为与称为函数的数学对象有关的问题。这个项目涉及这些学科的现代基础研究，重点放在具有内在多变量性质的问题上，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。这项研究将被纳入本科生和研究生的教育角色，并通过指导学生在这些现代和重要的分析领域。本项目集中在三个主要领域：(1)刻画有理函数在多变量区域上的有界性；(2)理解有理函数的可积性；(3)系统地确定多变量有理函数系数的渐近性。这项研究还与稳定多项式理论密切相关，在过去十年中，稳定多项式理论得到了许多令人惊讶的应用。稳定多项式的局部理论将在新旧环境中继续发展，以获得有关有理函数在奇点附近的行为的详细信息。此外，基于自然算符理论构建的某些多项式和有理函数，如行列式表示，代表了应用程序和算符理论感兴趣的重要特例。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。