Mathematical Sciences: Non-Commutative Harmonic Analysis

本项目是关于光滑流形和与之相关的某些微分算子的数学研究。流形是一个曲面（或高维类似物），它具有一个常识性的性质，即在任何点的附近，可以引入一个坐标系，其行为与二维（或n维）空间中的普通系统相似。微分算子可以用这些局部坐标来定义。可以从算子中提取的某些数字告诉我们关于底层流形的非常有趣的事情。当流形是一个对称空间时，群表示论就出现了。算子代数理论的最新发展所提供的非交换观点在流形带有某些附加结构时是非常有用的，例如带有病态叶空间的叶化。该项目的一个具体目标是将半单李群的调和分析方法应用于研究具有非正截面曲率的局部对称流形的eta不变量和解析挠率，旨在用测地线流来表达这些不变量。另一个方面的工作涉及陈德铭字符和指数可求和Fredholm模块在非交换微分几何，以期对应用程序的运营商代数与某些群体。

This project is mathematical research concerning smooth manifolds and certain differential operators associated with them. A manifold is a surface (or higher-dimensional analog) with the commonsensical property that in the near vicinity of any point, a coordinate system can be introduced that behaves like the ordinary system in two- (or n-) dimensional space. Differential operators can be defined in terms of these local coordinates. Certain numbers that can be extracted from the operators tell us very interesting things about the underlying manifold. Group representation theory comes into the picture when the manifold is a symmetric space. The noncommutative perspective afforded by recent developments in the theory of operator algebras is very helpful when the manifold carries some additional structure, such as a foliation with pathological leaf space. One specific objective of the project is to apply methods of harmonic analysis on semisimple Lie groups to the study of eta invariants and analytic torsion of locally symmetric manifolds with non-positive sectional curvature, aiming to express these invariants in terms of the geodesic flow. Another aspect of the work involves the Chern character and exponentially summable Fredholm modules in noncommutative differential geometry, with a view towards applications to operator algebras associated with certain groups.