Noncommutative Geometry: Its Applications to Geometry and Analysis

非交换几何：其在几何和分析中的应用

• 批准号: 0900985
• 负责人: Xiang Tang
• 金额: $ 10.91万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900985&HistoricalAwards=false
• 关键词: Noncommutative; Geometry; Its; Applications; Analysis

This project will apply noncommutative geometry to study several problems in geometry and analysis. The project will study index theory on various singular spaces. Using the recent developments in Molino's theory, Hopf cyclic theory, and algebraic local index theory, PI aims to obtain an explicit description of the topological index of a transverse elliptic differential operator of a riemannian foliation. PI will also study a covering index theorem on orbifolds and investigate homotopy invariance properties of signature numbers on orbifolds. The project will develop some noncommutative geometry tools, especially groupoid Mackey machine, to study a physics conjecture about duality between gerbes on orbifolds. Finally, several interesting applications of Connes-Moscovici's Hopf algebras will be made to geometry, analysis, and number theory.The project is an interplay between analysis and geometry.Noncommutative geometry tools will be used to study geometry and topology of spaces with singularities; and ideas and motivations in geometry will lead to new structures and developments in operator algebras and harmonic analysis.

这个项目将应用非对易几何来研究几何和分析中的几个问题。该项目将研究各种奇异空间上的指数理论。利用Molino理论、Hopf循环理论和代数局部指数理论的最新发展，PI的目的是得到黎曼叶分的横向椭圆微分算子的拓扑指数的显式刻画。PI还将研究orborold上的复盖指数定理，并研究orbioles上签名数的同伦不变性。该项目将开发一些非对易的几何工具，特别是群核Mackey机器，来研究关于奥比诺德上细菌之间的对偶的一个物理猜想。最后，Connes-Moscovici的Hopf代数将在几何、分析和数论中得到一些有趣的应用。这个项目是分析和几何之间的相互作用。非交换几何工具将被用来研究具有奇点的空间的几何和拓扑；几何中的思想和动机将导致算子代数和调和分析的新的结构和发展。