Methods of operator K-theory in geometry, topology, analysis

这项研究是上一个题为“几何、拓扑、调和分析中的算子K-理论方法”的继续。本项目的主要目的是将算子K-理论应用于椭圆算子的几何、拓扑学、表示理论和指数理论中的一些著名问题。这些问题是群C*-代数K-理论中的Baum-Connes猜想，环空间上关于Dirac算子的正Ricci曲率猜想，以及构造无限维流形上椭圆算子的有用指标理论的问题。该项目的Baum-Connes猜想部分由两个不同的问题组成：针对Lp空间上适当作用且仿射等距的离散群的Baum-Connes猜想和针对离散算术群的Baum-Connes猜想。关于无限维流形上的指数理论，理论物理中已经存在一些不严格的雏形，但为这一理论提供坚实的基础将是本课题的另一个问题。正Ricci曲率猜想是关于所谓Witten亏格消失的一个著名的拓扑猜想。它的主要方法是利用无穷维流形上的指数理论。20世纪量子力学的发展导致了希尔伯特空间中算符理论的产生，后来又产生了C*-代数理论。C~*-代数理论与当代物理学场论、几何学、拓扑学、群表示论有着密切的联系。算子K-理论从拓扑学发展到C*-代数理论，并发展成为一个非常强大的工具。目前，算子K-理论是非对易几何的一部分，它是分析和几何的综合，在数学和理论物理的许多领域都有广泛的应用。本项目的重点是算子K-理论在拓扑学、几何学和无限维分析中的几个著名问题上的应用。例如，构建无限维的指数理论，这是我们项目的一部分，将为数学和理论物理中的许多应用提供基础。算子K-理论方法将是本研究的主要工具。

This research is a continuation of the previous project titled ``Operator K-theory methods in geometry, topology, harmonic analysis''. The main objectives of the present project are applications of operator K-theory to a number of well known problems in geometry, topology, representation theory and index theory of elliptic operators. These problems are the Baum-Connes conjecture in K-theory of group C*-algebras, the positive Ricci curvature conjecture in relation to the Dirac operator on a loop space and the problem of constructing a useful index theory of elliptic operators on infinite-dimensional manifolds. The Baum-Connes conjecture part of the project consists of two different problems: the Baum-Connes conjecture for discrete groups which act properly and affine-isometrically on Lp spaces and the Baum-Connes conjecture for discrete arithmetic groups. Concerning index theory on infinite-dimensional manifolds, some non-rigorous rudiments of it already exist in theoretical physics, but providing a solid basis for this theory will be another problem of the present project. The positive Ricci curvature conjecture is a well-known topological conjecture on the vanishing of the so called Witten genus. The main approach to it is by the use of index theory on infinite-dimensional manifolds. The development of quantum mechanics in the 20th century has led to the creation of the theory of operators in Hilbert space and later to the creation of the theory of C*-algebras. The theory of C*-algebras is closely related with the contemporary field theory in physics, as well as with geometry, topology, group representation theory. Operator K-theory came to the C*-algebra theory from topology and grew out into a highly powerful tool. Nowadays, operator K-theory is part of the non-commutative geometry which is a synthesis of analysis and geometry with wide applications in many areas of mathematics and theoretical physics. The emphasis of the present project is on the applications of operator K-theory to several well known problems in topology, geometry, and infinite-dimensional analysis. For instance, constructing index theory in infinite dimensions, which is part of our project, will provide the basis for numerous applications in mathematics and theoretical physics. Operator K-theory methods will be the main tool for this research.