RUI: Representations of Analytic Functions in Several Variables and Applications

RUI：多个变量和应用中解析函数的表示

• 批准号: 2055098
• 负责人: Ryan Tully-Doyle
• 金额: $ 11.23万
• 依托单位: California Polytechnic State University Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-15 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2055098&HistoricalAwards=false
• 关键词: RUI; Representations; Analytic; Functions; Several

This project sits at the intersection of operator theory and complex analysis, two areas of mathematics. Complex analysis is concerned with functions involving complex numbers, which have broad use in mathematics, physics, and engineering applications. Complex analysis is often used when systems (such as a circuit) are encoded into a function which is studied as a means of understanding the system. A common approach in the analysis of functions is to transform them into more readily accessible objects. In classical analysis, this transformation is often an integral transform, such as the Fourier transform, where members of the same family of functions are transformed into a form with a common structure (called a kernel) and a geometric object that encodes individual behavior (called a measure). Another approach to studying families of functions representing systems common in engineering is called realization, which turns a system into a simple form involving matrices that represent properties of the system being studied. For functions with more than one input, both approaches are enveloped by operator theory, which arose from the mathematical foundation of quantum physics. This project will use operators to study representations of more general families of functions and apply them to questions in several complex variables. There are many questions in this project that will be explored by groups of student researchers.Specifically, this project is concerned with representations of functions that are bounded on certain domains as expressions of operators acting on associated Hilbert spaces and how the behavior of these functions at the boundary of their domains is encoded in the representations. Examples include the Pick functions, which play an important role in interpolation, probability, and engineering, and which possess well-known and useful representations in one variable (such as the Nevanlinna representation). One goal of the project is to extend methods of operator realizations into more general settings, including understanding concrete formulas for the operators in realizations and detailing the interplay between operator components and function regularity (in parallel with boundary measures in the classical theory) for functions that take matrix inputs. A second topic of the project is boundary problems in two or more variables, such as analogues of the Denjoy-Wolff problem on iteration of bounded functions on the complex disk. The connected goals of this project are motivated by the important historical and continuing interplay between the representation of functions bounded on special domains and the study of boundary properties. This research has potential applications in several complex variables and free analysis.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.

这个项目位于算符理论和复数分析这两个数学领域的交叉点上。复数分析涉及涉及复数的函数，在数学、物理和工程应用中有着广泛的应用。当系统(如电路)被编码成作为理解系统的一种手段而被研究的函数时，通常使用复杂分析。分析函数的一种常见方法是将它们转换为更容易访问的对象。在经典分析中，这种变换通常是积分变换，例如傅里叶变换，在这种变换中，同一函数族的成员被变换成具有共同结构(称为核)和编码个人行为的几何对象(称为度量)的形式。另一种研究工程中常见的表示系统的函数族的方法称为实现，它将系统变成一种简单的形式，其中包含表示所研究系统的性质的矩阵。对于有多个输入的函数，这两种方法都受到算符理论的影响，算符理论起源于量子物理的数学基础。这个项目将使用运算符来研究更一般的函数族的表示，并将它们应用于几个复变量中的问题。在这个项目中有许多问题将由学生研究小组来探索。具体地说，这个项目涉及到在某些区域上有界的函数的表示，作为作用在相关Hilbert空间上的算子的表示，以及这些函数在其区域边界上的行为是如何在表示中编码的。例如Pick函数，它在插值法、概率法和工程学中扮演着重要的角色，并且在一个变量中拥有众所周知和有用的表示法(如Nevanlinna表示法)。该项目的一个目标是将运算符实现方法扩展到更一般的环境，包括了解实现中运算符的具体公式，以及详细说明运算符组件与接受矩阵输入的函数的函数正则性(与经典理论中的边界度量并行)之间的相互作用。该项目的第二个主题是两个或多个变量的边值问题，例如关于复圆盘上有界函数迭代的Denjoy-Wolff问题的类似。这个项目的相关目标是由特殊区域上有界函数的表示和边界性质的研究之间的重要的历史和持续的相互作用所推动的。这项研究在几个复杂变量和自由分析中具有潜在的应用。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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

The royal road to automatic noncommutative real analyticity, monotonicity, and convexity

通往自动非交换实解析性、单调性和凸性的康庄大道

• DOI: 10.1016/j.aim.2022.108548
• 发表时间: 2022
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Pascoe, J.E.;Tully-Doyle, Ryan
• 通讯作者: Tully-Doyle, Ryan

Dynamics of low-degree rational inner skew-products on $\mathbb{T}^2$

$mathbb{T}^2$ 上低度有理内偏积的动力学

• DOI: 10.4064/ap211108-28-2
• 发表时间: 2022
• 期刊: Annales Polonici Mathematici
• 影响因子: 0.5
• 作者: Sola, Alan;Tully-Doyle, Ryan
• 通讯作者: Tully-Doyle, Ryan

Induced Stinespring Factorization and the Wittstock Support Theorem

诱导 Stinespring 分解和 Wittstock 支持定理

• DOI: 10.1007/s00025-023-01908-4
• 发表时间: 2023
• 期刊: Results in Mathematics
• 影响因子: 2.2
• 作者: Pascoe, J. E.;Tully-Doyle, Ryan
• 通讯作者: Tully-Doyle, Ryan

Monotonicity of the principal pivot transform

主枢轴变换的单调性

• DOI: 10.1016/j.laa.2022.02.016
• 发表时间: 2022
• 期刊: Linear Algebra and its Applications
• 影响因子: 1.1
• 作者: Pascoe, J.E.;Tully-Doyle, Ryan
• 通讯作者: Tully-Doyle, Ryan