Mathematical Sciences: Non-Commutative Harmonic Analysis

本课题是关于光滑流形及其相关微分算子的数学研究。流形是一个曲面（或高维模拟曲面），它具有一个常识性，即在任意点的附近，可以引入一个与二维（或n维）空间中的普通系统行为相似的坐标系。微分算子可以用这些局部坐标来定义。从运算符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。当流形是对称空间时，群表示理论就出现了。算子代数理论的最新发展所提供的非交换视角在流形带有一些附加结构时非常有用，例如带有病态叶空间的叶。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非正截面曲率局部对称流形的不变量和解析扭转，旨在用测地线流来表示这些不变量。本研究的另一个方面涉及非交换微分几何中的chen特征和指数可和Fredholm模块，并将其应用于与某些群相关的算子代数。

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.