Conformal geometry, dynamics, and unbounded KK-theory

I will briefly introduce noncommutative geometry and its main technical device, unbounded KK-theory, and then discuss how dynamical systems can be represented therein. One important concept for this purpose is equivariant KK-theory. In the development of unbounded KK-theory, an aspect which has been left unresolved is the definition of equivariance. One reason for this is that the natural definition fails to capture all the degrees of freedom available in the usual, bounded equivariant KK-theory. This is apparent already in the classical, commutative, case, as we shall see. In light of the forgoing, conformal geometry will be briefly introduced and I will discuss the definition of conformally equivariant unbounded KK-theory. I will also touch on ongoing work which uses these tools to analyse more general dynamical systems.