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Operator Theory and Stable Polynomials

算子理论和稳定多项式

基本信息

批准号：
1900816
负责人：
Greg Knese
金额：
$ 19.1万
依托单位：
Washington University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-06-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900816&HistoricalAwards=false
关键词：
Operator Theory Stable Polynomials

项目摘要

Operator theory is a broad and mature area of pure mathematics with close ties to the mathematics of quantum mechanics and control systems engineering. Indeed, von Neumann is often credited with formulating the foundations of operator theory as a language for quantum mechanics, while Nobert Wiener initiated an approach to engineering prediction problems (such as jet tracking) based on ideas from operator theory and harmonic analysis. In more recent decades, operator theory has been used in H-infinity control theory which has applications in automatic pilot design. On the other hand, a stable polynomial is not a field per se but a basic concept in mathematics that has become profoundly useful in the study of a diverse range of problems in mathematics: combinatorics (enumeration problems), graph theory (the study of networks), and operator theory. The purpose of this project is to use operator theory to study stable polynomials and vice versa. This project focuses on three problems: (1) the generalized Lax conjecture, (2) a quantitative understanding of linear preservers of stability, and (3) extensions of the theory of stable polynomials to entire functions. Semi-definite programming is a powerful technique in optimization and the generalized Lax conjecture is a bold assertion about the sets on which this theory can successfully be implemented. The conjecture claims that these feasibility sets have a geometric description in terms of hyperbolic polynomials (a slight generalization of the concept of a stable polynomial). Although evidently important in optimization, this would further establish links between stable polynomials and operator theory as the question is closely related to the concept of representing stable polynomials via operator theoretic determinantal representations. Problem (2) is about taking the highly successful theorems of Borcea-Branden that characterized the linear maps on stable polynomials that preserve stability and making them quantitative. How exactly do stability preservers modify zero sets? One approach to this problem is through analyzing how stability preservers affect various sums-of-squares formulas for stable polynomials. Problem (3) is about a natural extension of stable polynomials to various classes of entire functions (the Polya class, Hermite-Biehler class) and would require developing multivariable theories of Hilbert spaces of entire functions. This latter endeavor is important in its own right as the successes in one variable Hilbert spaces of entire functions have been profound (i.e. the Poltoratski approach to problems of uncertainty type).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.

算子理论是纯数学的一个广泛而成熟的领域，与量子力学和控制系统工程的数学密切相关。事实上，冯·诺依曼经常被认为奠定了算子理论的基础作为量子力学的语言，而诺伯特·维纳则提出了一种基于算子理论和调和分析思想的工程预测问题（例如射流跟踪）的方法。近几十年来，算子理论已被用于 H-无穷大控制理论，该理论在自动驾驶仪设计中得到了应用。另一方面，稳定多项式本身并不是一个领域，而是数学中的一个基本概念，它在研究各种数学问题中变得非常有用：组合学（枚举问题）、图论（网络研究）和算子理论。该项目的目的是使用算子理论来研究稳定多项式，反之亦然。该项目重点关注三个问题：(1) 广义 Lax 猜想，(2) 对线性稳定性保持器的定量理解，以及 (3) 将稳定多项式理论扩展到整个函数。半定规划是一种强大的优化技术，而广义 Lax 猜想是关于可以成功实现该理论的集合的大胆断言。该猜想声称这些可行性集具有双曲多项式的几何描述（稳定多项式概念的轻微概括）。尽管在优化中显然很重要，但这将进一步建立稳定多项式和算子理论之间的联系，因为该问题与通过算子理论行列式表示来表示稳定多项式的概念密切相关。问题（2）是关于采用非常成功的 Borcea-Branden 定理，该定理描述了稳定多项式上保持稳定性的线性映射，并将其定量化。稳定保持器究竟如何修改零集？解决此问题的一种方法是分析稳定性保持器如何影响稳定多项式的各种平方和公式。问题（3）是关于稳定多项式到各种类型的整个函数（Polya 类、Hermite-Biehler 类）的自然扩展，并且需要开发整个函数的希尔伯特空间的多变量理论。后一项努力本身就很重要，因为整个函数的一个变量希尔伯特空间取得了深远的成功（即解决不确定性类型问题的 Poltoratski 方法）。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

期刊论文数量（4）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

An [O iii] search for extended emission around AGN with H i mapping: a distant cloud ionized by Mkn 1

[O-iii] 使用 H-i 映射搜索 AGN 周围的扩展发射：由 Mkn 1 电离的遥远云

DOI：
10.1093/mnras/staa1510
发表时间：
2020
期刊：
Monthly Notices of the Royal Astronomical Society
影响因子：
4.8
作者：
Knese, Erin Darnell;Keel, William C;Knese, Greg;Bennert, Vardha N;Moiseev, Alexei;Grokhovskaya, Aleksandra;Dodonov, Sergei N
通讯作者：
Dodonov, Sergei N

Cyclicity Preserving Operators on Spaces of Analytic Functions in $${\mathbb {C}}^n$$

$${mathbb {C}}^n$$ 解析函数空间上的循环保留算子