Mathematical Sciences: Connections of Modern Analysis with Geometry and Topology

数学科学：现代分析与几何和拓扑的联系

• 批准号: 9304283
• 负责人: Ronald Douglas
• 金额: $ 17.63万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-06-01 至 1997-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9304283&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Connections; Modern; Analysis

Douglas will study the connection between problems in operator theory on one hand and algebraic and analytic geometry in the presence of a hermitian metric. The goal is to understand multi- variate phenomena and to provide techniques to study more general cases. He will also investigate various invariants for elliptic operators on manifolds. Starting from the case of elliptic operators on compact, smooth manifolds without boundary which is now classical, one is seeking to understand what happens when these hypothesis are relaxed. The general area of mathematics of this project has its basis in the theory of algebras of Hilbert space operators. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These abstract objects have a surprising variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA, and they are of fundamental importance in noncommutative geometry, which is becoming increasingly important in physics.

道格拉斯将研究算子理论问题与厄米度规存在下的代数和解析几何问题之间的联系。目标是了解多变量现象，并提供研究更一般案例的技术。他还将研究流形上椭圆算子的各种不变量。从紧致光滑无边界流形上的椭圆算子开始，这是一个经典的例子，人们试图理解当这些假设被放宽时会发生什么。这个项目的一般数学领域在希尔伯特空间算子的代数理论中有其基础。运算符可以被认为是有限或无限的复数矩阵。特殊类型的运算符通常放在一个代数中，自然地称为运算符代数。这些抽象对象有各种各样的应用。例如，它们在结理论中发挥着关键作用，而结理论目前正被用于研究DNA的结构，它们在非交换几何中具有重要的基础意义，而非交换几何在物理学中正变得越来越重要。