Hamburg

 For a semisimple modular tensor category the Reshetikhin-Turaev construction yields an extended three-dimensional topological field theory and hence by restriction a modular functor. By work of Lyubachenko-Majid the construction of a modular functor from a modular tensor category remains possible in the non-semisimple case. We explain that the latter construction is the shadow of a derived modular functor featuring homotopy coherent mapping class group actions on chain complex valued conformal blocks and a version of factorization and self-sewing via homotopy coends. On the torus we find a derived version of the Verlinde algebra, an algebra over the little disk operad (or more generally a little bundles algebra in the case of equivariant field theories). The concepts will be illustrated for modules over the Drinfeld double of a finite group in finite characteristic. This is joint work with Christoph Schweigert (Hamburg).

The interplay of minimum degree and 'structural properties' of large graphs with a given forbidden subgraph is a central topic in extremal graph theory. For a given graph $F$ we define the homomorphism threshold as the infimum $\alpha$ such that every $n$-vertex $F$-free graph $G$ with minimum degree $>\alpha n$ has a homomorphic image $H$ of bounded size (independent of $n$), which is $F$-free as well. Without the restriction of $H$ being $F$-free we recover the definition of the chromatic threshold, which was determined for every graph $F$ by Allen et al. The homomorphism threshold is less understood and we present recent joint work with O. Ebsen on the homomorphism threshold for odd cycles.

"In a paper from 2001, Diestel and Leader characterised uncountable graphs with normal spanning trees through a class of forbidden minors. In this talk we investigate under which circumstances this class of forbidden minors can be made nice. In particular, we will see that there is a nice solution to this problem under Martin’s Axiom. Also, some connections to the Stone-Chech remainder of the integers, and almost disjoint families are uncovered.”