Abstract: By an old result of Raoul Bott, the index of homogeneous Dirac operators can be computed using Weyl character formula. Thus at least in this case one can in principle bypass a substantial amount of analysis and reduce the Atiyah-Singer index theorem to representation theory of compact Lie groups. In my talk I shall recall these results and discuss their impact on equivariant index theorems.

Abstract: In the 1970's Lichtenbaum conjectured a formula for special values of zeta-functions of number fields at negative integers in terms of algebraic K-groups. I will give an overview of the results on this and related conjectures and show how Voevodsky's proof of the Bloch-Kato Conjecture not only allowed to prove the Lichtenbaum conjecture for abelian number fields, but suggests a more complete motivic cohomology version, which includes the prime 2.