Operator related function theory and algebraic varieties

算子相关函数论和代数簇

• 批准号: 1048775
• 负责人: Greg Knese
• 金额: $ 9.3万
• 依托单位: University of Alabama Tuscaloosa
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-11 至 2014-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1048775&HistoricalAwards=false
• 关键词: Operator; related; function; theory; algebraic

The PI will develop operator related function theory and its interaction with algebraic varieties. The main theme is that multi-variable polynomials whose zero sets have a natural relationship to the torus or the polydisk in complex euclidean space offer an interesting setting for the study of complex analysis and operator theory. The goal is to better understand "stable" polynomials (those without zeros on a specified domain), because of their frequent appearance in function theory (as in rational inner functions and interpolation problems), mathematical physics, and engineering, as well as to view algebraic varieties as domains on which to study function theory and operator theory. Function theory on varieties can enrich one variable function theory and while shining light on difficult problems in function theory in several variables.Much of the work has its intellectual roots in the works of Norbert Wiener, Andrey Kolmogorov, and Arne Beurling (to name just a few) on areas of probability theory and mathematical analysis that formed the mathematical underpinnings of signals analysis (or communications), control theory (as in automatic pilots), and time series analysis (the study of sequential data like stock prices). This project will continue in this long and fruitful tradition by developing and generalizing the underlying mathematics further (most notably by emphasizing relations to algebraic topics). The project should have connections to areas of scientific endeavor with multidimensional data (e.g. an image) as opposed to the previous examples that feature primarily one dimensional data (the one dimension being time).

PI 将开发算子相关函数理论及其与代数簇的相互作用。 主题是零集与复欧几里德空间中的圆环或多圆盘具有天然关系的多变量多项式为复分析和算子理论的研究提供了有趣的环境。 目标是更好地理解“稳定”多项式（在指定域上没有零的多项式），因为它们经常出现在函数论（如有理内函数和插值问题）、数学物理和工程学中，以及查看代数簇作为研究函数论和算子理论的领域。 簇函数理论可以丰富一个变量函数理论，同时阐明多个变量函数理论中的难题。该工作的大部分内容都源于 Norbert Wiener、Andrey Kolmogorov 和 Arne Beurling（仅举一例）的著作。概率论和数学分析领域构成了信号分析（或通信）、控制理论（如自动驾驶仪）和时间序列分析（对股票价格等序列数据的研究）的数学基础。 该项目将通过进一步发展和推广基础数学（最显着的是通过强调与代数主题的关系）来延续这一悠久而富有成果的传统。 该项目应该与具有多维数据（例如图像）的科学努力领域建立联系，而不是之前主要以一维数据（一维是时间）为特征的示例。