Noncommutative Geometry: Its Applications to Geometry and Analysis

本计画将运用非交换几何来研究几何与分析中的几个问题。本课题将研究各种奇异空间上的指标理论。利用Molino理论、Hopf循环理论和代数局部指标理论的最新进展，PI旨在获得黎曼叶理的横椭圆微分算子的拓扑指标的显式描述。此外，还将研究轨道上的一个覆盖指数定理，并研究轨道上签名数的同伦不变性。本项目将开发一些非交换几何工具，特别是群样麦基机，来研究关于轨道上格布之间对偶性的物理猜想。最后，将介绍Connes-Moscovici的Hopf代数在几何、分析和数论中的几个有趣的应用。该项目是分析和几何之间的相互作用。非交换几何工具将用于研究奇异空间的几何和拓扑；几何中的思想和动机将导致算子代数和调和分析的新结构和发展。

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.