Abstract: To any object in a braided tensor category we can associate a Nichols algebra, either through braid group combinatorics or through a universal property. The main motivation for Nichols algebras is to systematically construct the quantum group from its Cartan part. This idea goes back to Lusztig, but I will present it in modern categorical language. Then I will sketch how Nichols algebras can be used to construct solutions of the Knishnik Zomolochikov differential equation from its respective abelian version, and to construct a conformal field theory associated to the (small) quantum group from a free field theory. This parallelism is a main ingredient in our recent proof of the logarithmic Kazhdan Lusztig correspondence, which is the nonsemisimple small-quantum-group-analog of the more familiar Kazhdan Lusztig correspondence. At the same time, the construction produces many new nonsemisimple braided tensor categories, topological field theories and conformal field theories, correspondingly.

Del Pezzo surfaces and their automorphism groups play a key role in the classification up to conjugacy of finite subgroups of the Cremona group of the plane. Over an algebraically closed field, they are completely classified together with their automorphism groups. In this talk, we will focus on real rational Del Pezzo surfaces of degree 4. Unlike larger degrees, the…

I will recall a class of action of inverse semigroups, how they form a topos and their connections with presheaves over a semilattice. Then, I will present some new work showing which representations these actions correspond to and how this result can be used to relate two inverse semigroups, one coming from automorphisms of a sheaf and one coming from automorphisms of a bundle.