This talk is a continuation from the Departmental Seminar on Monday 6th April. At the start of the talk we will briefly review the contents of the first part.
The first part introduced a generalisation of the Riemann zeta function using the compact operators on a separable Hilbert space, and linked the residue of the zeta function of a compact operator to the Dixmier trace, a non-normal trace used as the foundation for the "noncommutative integral" in Alain Connes' theory of Noncommutative Geometry.
In this second part we consider specific zeta functions associated to Laplacians on compact Riemannian manifolds. For example, the zeta function associated to the Laplacian on the circle is just a multiple of the Riemann Zeta Function. We introduce zeta functions weighted by bounded operators and show how, in recent work, we solved a problem concerning the "noncommutative integral" that has been open for 20 years.
Namely, we recover the Lebesgue integral of any bounded (and then any) integrable function as the residue of a zeta function. If time permits, we will introduce the integral on the "noncommutative torus" and show, using the same technique, that it can be recovered from zeta functions associated to the "noncommutative Laplacian".