Title: "Atiyah's L2-Betti numbers: properties and applications"

Abstract: 1976, Atiyah defined the L2-Betti numbers of a compact manifold M. (This was done in terms of the eigenvalues and eigenspaces of the Laplace Beltrami operaotrs on hte universla covering of M). A priori, these are sone non-negative real numbers.

Using these L2-Betti numbers, one can compute e.g. the Euler characteristic of M. On the other hand, strong vanishing theorems for L2-Betti numbers exist. Putting this together, one can obtain new information about the Euler characteristic for interesting classes of spaces.

An important question about L2-Betti numbers are their possible values. More precisely, are they always integers (if the fundamental group of M is torsion-free). This has far reaching algebraic implications. In particular, the Kaplanski conjecture follows, which says that the rational group ring of the fundamental group does not have non-trivial zero divisors.

The talk tries to give a survey over the theory and the results obtained so far.