The 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.
