Veranstaltungen am Mathematischen Institut

The idea of six functors formalisms originates in Grothendieck's work on duality for étale cohomology of schemes. Much more recently, a simple and powerful definition of this structure was given using the theory of higher categories. This has greatly improved our ability to work with such structures. However it does not simplify our task of constructing six functor formalisms, and in fact apriori it makes it much harder. Nevertheless, work of Liu-Zheng formalized the most important construction principle, which goes back to the original work of Artin, Grothendieck and Verdier on the six functor formalism on étale cohomology. I will explain a new approach to this construction principle which is joint work with Bastiaan Cnossen and Tobias Lenz. To do this we recast the problem as that of computing a certain universal (infinity,2)-category, which we then do by combining methods from parametrized and (infinity,2)-category theory.

By showing that there is a topologically free boundary action, we prove that the outer automorphism groups of the topological full groups of the Cuntz?Krieger groupoids are C*-simple, meaning that their reduced C*-algebras are simple (this is an extreme case of non-amenability). These groups generalise the outer automorphism groups of the Higman?Thomson groups, so our result applies, in particular to, Out(V_n), where V_n is a Higman?Thompson group. This is joint work with Xin Li and Takuya Takeishi.

In this talk we compute the index homomorphism of even K-groups arising from a class in even KK-theory via the Kasparov product. Due to the seminal work of Baaj and Julg, under mild conditions on the C*-algebras in question, every class in KK-theory can be represented by an unbounded Kasparov module. We then describe the corresponding index homomorphism of even K-groups in terms of spectral localizers. This means that our explicit formula for the index homomorphism does not depend on the full spectrum of the abstract Dirac operator D, but rather on the intersection between this spectrum and a compact interval. The size of this compact interval does however reflect the interplay between the K-theoretic input and the abstract Dirac operator. Since the spectral projections for D are not available in the general context of Hilbert C*-modules we instead rely on certain continuous compactly supported functions applied to D to construct the spectral localizer. In the special case where even KK-theory coincides with even K-homology, our work recovers the pioneering work of Loring and Schulz-Baldes on the index pairing.