Operator K-theory methods in geometry, topology, harmonic analysis

这个项目的主要目标是将算子K-理论应用于拓扑学、调和分析和指数理论中的一些著名问题。这些问题是关于光滑流形高签名的Novikov猜想，群C*-代数K-理论中的Baum-Connes猜想，自同构形式的重数问题，横截椭圆算子的指标理论。算子K-理论起源于椭圆算子的指标理论，利用算子K-理论研究包括横切椭圆算子在内的椭圆算子是最自然的。此外，算子K-理论在与几何群作用有关的应用中是一个非常合适的工具，这使得它对于像Novikov猜想和Baum-Connes猜想这样的问题特别有用。自同构形式的多重性是算子K理论为老问题提供新途径的另一个主题。20世纪量子力学的发展导致了希尔伯特空间中算子理论的产生，后来又产生了C*-代数理论。C*-代数理论与当代物理学中的场论密切相关，也与几何学、拓扑学、李群表示理论等密切相关。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-理论在分析和拓扑学中几个著名问题上的应用。

Abstract KasparovThe main objective of this project are applications of operator K-theory to a number of well known problems in topology, harmonic analysis and index theory. These problems are the Novikov conjecture on higher signatures of smooth manifolds, the Baum-Connes conjecture in K-theory of group C*-algebras, the problem of multiplicities of automorphic forms, index theory for transversally elliptic operators. Operator K-theory has its origins in the index theory of elliptic operators, and it is most natural to study elliptic operators, including transversally elliptic operators, using operator K-theory. Also operator K-theory is a very suitable tool in applications related to geometric group actions, which makes it exceptionally useful for problems like the Novikov and the Baum-Connes conjectures. Multiplicities of automorphic forms is another topic where operator K-theory gives new approaches to old problems.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, Lie group representation theory, etc. 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 analysis and topology.