Mathematics

Kasparov's $KK$-theory is a bivariant homology theory for $C^*$-algebras providing a general framework for index theory. Cycles for the theory come from the abstract analogue of a first-order differential operator. After an exposition of the theory for non-specialists, I will discuss some recent results. These include analytic properties of self-adjoint operators, applications of $KK$-theory in the theory of arithmetic manifolds and automorphic forms and the physics of topological insulators.