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.

该项目涉及一个名为数学分析的经典数学领域，并特别注意其两个子场，复杂的分析和操作者理论。复杂分析是一个具有广泛适用性的成熟主题 - 从映射地球到了解算法的运行时。操作者理论最初是为了研究量子力学而创建的，此后已发展成为一个类似成熟的领域，适用于工程和优化。分析功能的一部分是能够转换具体任务（例如设计恒温器或理解质数的分布）的能力，将其转换为有关称为函数的数学对象的问题。该项目涉及这些主题中的现代基本研究，重点是具有继承多变量性质的问题，必须深入了解多变量功能（更具体地说是多元可变的合理功能）。这项研究将在基础和研究生层面以及学生在这些现代和重要分析领域的心理上纳入教育角色。本项目侧重于三个主要领域：（1）表征多个变量中理性功能在域上的界限，（2）理解正性功能的集成，以及（3）系统性地确定倍增功能的综合量。这项研究也与稳定的多项式理论紧密相关，该理论在过去十年中享有许多惊喜应用。在新旧环境中稳定多项式的局部理论将继续发展，以获取有关奇异性附近理性函数行为的详细信息。此外，由自然运营商理论结构（例如确定的表示）构建的某些多项式和理性功能代表了应用程序和操作者理论中的重要特殊案例。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的影响审查标准来通过评估来获得的支持。