Topology, Noncommutative Geometry, and Applications

拓扑、非交换几何及其应用

• 批准号: 0805003
• 负责人: Jonathan Rosenberg
• 金额: $ 37.91万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805003&HistoricalAwards=false
• 关键词: Topology; Noncommutative; Geometry; Applications

Professor Jonathan Rosenberg will study problems in classical and noncommutative topology, as well as various applications, especially to differential geometry and mathematical physics. One main focus of the proposal will be the application of noncommutative geometry to mathematical physics, especially string theory. Noncommutative sigma-models will be analyzed and compared with their classical counterparts. This will require development of some aspects of a new theory of noncommutative nonlinear elliptic partial differential equations. Other topics will include the classification of metrics of positive scalar curvature on manifolds, the purely algebraic K-theory of operator algebras, and the use of invariants coming from C*-algebras (especially Kasparov?s KK-theory) to study the geometry and topology of manifolds. For example, the K-homology classes coming from the classical elliptic operators, such as the signature operator and the Dolbeault operator, will be computed, and their invariance and rigidity properties will be determined. As a result, we expect to understand better the links between topological and analytic approaches to geometry of manifolds and singular spaces.Many physical theories, such as general relativity, are formulated in terms of geometry and partial differential equations. However, the principles of quantum mechanics require studying space-time ``geometries''in which the coordinate functions do not commute with one another. One main focus of this project will be the reformulation of some of the partial differential equations of mathematical physics in the setting of such noncommutative geometries. We will develop tools for studying these noncommutative equations and will compare the noncommutative geometries with their classical counterparts. This should advance the language for formulating quantum theories of gravity.Professor Rosenberg will also train graduate students and advanced undergraduate students in algebra, analysis, geometry, topology, and mathematical physics, and will also work toward integration of mathematical software into the undergraduate mathematics curriculum.

乔纳森·罗森伯格教授将研究经典拓扑学和非交换拓扑学中的问题，以及各种应用，特别是在微分几何和数学物理中的应用。该提案的一个主要焦点将是将非对易几何应用于数学物理，特别是弦理论。我们将分析非对易的西格玛模型，并将其与经典模型进行比较。这需要发展一种新的非对易非线性椭圆型偏微分方程理论的某些方面。其他主题将包括流形上正标量曲率度量的分类，算子代数的纯代数K-理论，以及使用来自C*-代数(特别是Kasparov？S KK-理论)的不变量来研究流形的几何和拓扑。例如，计算来自经典椭圆算子(如签名算子和Dolbeault算子)的K-同调类，并确定它们的不变性和刚性性质。因此，我们希望更好地理解流形和奇异空间几何的拓扑学和解析法之间的联系。许多物理理论，如广义相对论，都是根据几何和偏微分方程来表述的。然而，量子力学原理要求研究坐标函数不相互交换的时空‘几何’。这个项目的一个主要焦点将是在这种非对易几何的背景下重新表述数学物理中的一些偏微分方程式。我们将开发研究这些非对易方程的工具，并将非对易几何与它们的经典对应几何进行比较。这将推动形成量子引力理论的语言。罗森伯格教授还将培训研究生和高级本科生在代数、分析、几何、拓扑和数学物理方面，并将致力于将数学软件整合到本科数学课程中。