Noncommutative Geometry and Index Theory

在生活中，做事的顺序极其重要。在工作B之前做工作A和在工作B之后做工作A可能会产生完全不同的结果。非交换几何用几何学的思想研究这种现象。经典几何中的坐标函数都是交换的，在AB等于BA的意义上，它们在非交换几何中不交换，在非交换几何中AB可以不同于BA。在过去的三十年里，非交换几何迅速发展成为数学中最活跃的研究领域之一，在数学和物理的许多分支中都有应用。在这个项目中，首席研究员将研究非交换几何和指数理论中几个有趣而重要的问题，其中一些问题是由物理学中的量子场论激发的。首席研究员将在非交换几何和指数理论方面进行三个研究项目。(1)他将研究由Alain Connes提出的一个长期存在的指标问题，即群似椭圆微分算子在一般李群似上的指标问题。(2)他将继续他的计划，利用非交换几何和辛拓扑的联合力来研究一个对偶猜想，灵感来自物理学中的弦理论，关于轨道上的gerbes。(3)他将研究一个有趣的指标问题，由多元算子理论驱动，利用复分析和非交换几何的最新发展，研究一些本质上正常的希尔伯特模块。

In life, the order of doing things is extremely important. Doing job A before B may produce a completely different result from doing A after B. Noncommutative geometry studies this phenomena using ideas from geometry. While coordinate functions in classical geometry all commute, in the sense that the product AB equals BA, they do not commute in noncommutative geometry, where AB can be different from BA. In the last thirty years, noncommutative geometry has quickly grown into one of most active research areas in mathematics, with applications in many branches of mathematics and physics. In this project the principal investigator will study several interesting and important problems in noncommutative geometry and index theory, some of which are motivated by quantum field theory in physics.The principal investigator will pursue three research projects in noncommutative geometry and index theory. (1) He will study a long-standing index problem, raised by Alain Connes, about the index of a groupoid elliptic differential operator on a general Lie groupoid. (2) He will continue his program of using the joint force of noncommutative geometry and symplectic topology to study a duality conjecture, inspired from string theory in physics, about gerbes on orbifolds. (3) He will investigate an interesting index problem, motivated by multivariate operator theory, about some essentially normal Hilbert modules using the recent developments in complex analysis and noncommutative geometry.