Operator Theory and Stable Polynomials

算符理论是一个广泛而成熟的纯数学领域，与量子力学和控制系统工程的数学有着密切的联系。事实上，冯·诺伊曼经常被认为是将算子理论作为量子力学语言的基础，而诺伯特·维纳则基于算子理论和谐波分析的思想，提出了一种解决工程预测问题（如射流跟踪）的方法。近几十年来，算子理论在h -∞控制理论中得到了应用，在自动驾驶仪的设计中得到了应用。另一方面，稳定多项式本身不是一个领域，而是数学中的一个基本概念，它在数学中各种各样的问题的研究中变得非常有用：组合学（枚举问题）、图论（网络的研究）和算子理论。本课题的目的是利用算子理论来研究稳定多项式，反之亦然。本课题主要研究三个问题：(1)广义Lax猜想；(2)稳定性线性保持器的定量认识；(3)稳定多项式理论在整个函数中的推广。半定规划是一种强大的优化技术，广义Lax猜想是关于该理论可以成功实现的集合的大胆断言。该猜想声称这些可行性集具有双曲多项式（稳定多项式概念的稍微推广）的几何描述。虽然在优化中很重要，但这将进一步建立稳定多项式和算子理论之间的联系，因为这个问题与通过算子理论行列式表示稳定多项式的概念密切相关。问题(2)是关于采用Borcea-Branden非常成功的定理，该定理描述了稳定多项式上保持稳定性的线性映射，并使其量化。稳定保持器究竟是如何修改零集的？解决这个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)是关于稳定多项式的自然扩展到整个函数的各种类（Polya类，Hermite-Biehler类），并且需要发展整个函数的希尔伯特空间的多变量理论。后一种努力就其本身而言是重要的，因为在整个函数的单变量希尔伯特空间中的成功是深刻的（即Poltoratski解决不确定性类型问题的方法）。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.