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.

该项目涉及数学分析的经典领域，特别关注其两个子领域：复数分析和算子理论。复数分析是一门具有广泛适用性的成熟学科——从绘制地球图到理解算法的运行时间。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。分析的部分力量在于转换具体任务的能力，例如设计恒温器或理解素数的分布。 ，进入有关称为函数的数学对象的问题。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，这些问题必然需要对多变量函数（更具体地说是多变量理性函数）有更深入的了解。该研究将纳入本科生和研究生的教育角色中。并指导学生了解这些现代且重要的分析领域。该项目侧重于三个主要领域：(1) 表征多个变量域上有理函数的有界性，(2) 理解有理函数的可积性，以及 (3 ）系统地确定这项研究也与稳定多项式理论密切相关，在过去的十年中，稳定多项式的局部理论在新旧环境中得到了许多令人惊讶的应用。获得有关奇点附近有理函数行为的详细信息此外，由自然算子理论构造（例如行列式表示）构建的某些多项式和有理函数代表了应用和研究中感兴趣的重要特殊情况。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。