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将发展算子相关函数理论及其与代数簇的相互作用。 主要的主题是，多元多项式的零集有一个自然的关系，环面或复欧氏空间中的多圆盘复分析和算子理论的研究提供了一个有趣的设置。 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），数学物理和工程中，以及将代数簇视为研究函数论和算子论的域。 簇的函数论可以丰富一元函数论，同时也可以揭示多元函数论中的一些难题，这些工作的思想根源在于Norbert Wiener、Andrey Kolmogorov和阿恩Beurling的著作（仅举几例）概率论和数学分析领域，形成了信号分析的数学基础（或通信），控制理论（如自动驾驶仪）和时间序列分析（研究股票价格等序列数据）。 这个项目将继续在这一长期和富有成效的传统，进一步发展和推广基础数学（最显着的是通过强调关系代数主题）。 该项目应该与多维数据（例如图像）的科学奋进领域建立联系，而不是以前的主要是一维数据（一维是时间）的例子。