Given a compact Riemannian manifold $M$ and $D^{2}$ the Dirac Laplacian on the Clifford bundle, Bochner discovered the existence of a self-adjoint, non-negative Laplacian $\Delta$ such that the difference $D^{2} - \Delta$ is a zero-order operator that can be expressed in terms of the curvature tensor of $M$. In fact, combined with some harmonic theory, these operators allowed Bochner to obtain fundamental vanishing theorems involving the Betti numbers of $M$. In this talk, I will recall the Dirac and connection Laplacian operators and prove the general Bochner identity. Using this, I will prove the Weitzenbock formula and a vanishing theorem of the first Betti number $b_{1}(M) = dim (H^{1}(M, \mathbb{R}))$.

Various results by Tannaka, Krein, Deligne, Lurie, Hall, Schaeppi, Chirvasitu and B. show that a scheme / algebraic stack can be recovered from its tensor category of quasi-coherent sheaves. This motivates to generalize several constructions from algebraic geometry to tensor category theory. I would like to illustrate this process for affine and projective morphisms, tangent bundles, and fiber products.

A holomorphic function on a planar domain $\Omega$ is said to possess a universal Taylor series about a point in $\Omega$ if the partial sums of the Taylor series have the following surprising property: they can approximate arbitrary polynomials on arbitrary compact sets $K$ outside $\Omega$ (provided only that $K$ has connected complement). In the last few years, central questions about universal Taylor series have been addressed using potential theory. In this talk we will discuss some of these results and in particular we will focus on the boundary behaviour of such functions.

This is the last part of the syzygy saga (but independent of the previous talks).

I will recall how syzygies interpolate between torsion-free and free modules and why certain syzygies are ruled out in the torus-equivariant cohomology of compact orientable manifolds. More precisely, by a result of Allday, Puppe and myself there is a bound on the syzygy order unless the equivariant cohomology is free. The main point of this talk is to show that this bound is sharp. I will do so by exhibiting a new class compact orientable manifolds with torus action. These manifolds are related to polygon spaces as studied by Hausmann, Farber and many others.

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.

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.