The rigidity theorem for motives of non-archimedean analytic spaces
Friday, 6.7.18, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
I will give a quick introduction to the notions of motive and of motivic sheaves. Then, after recalling the main ideas of non-Archimedean analytic geometry I will define the category of motivic sheaves of non-Archimedean analytic spaces. Finally, I will state the Rigidity Theorem in this context and if time permits I will briefly sketch the main ideas about its proof and mention some applications.
Global Serre dualities
Monday, 9.7.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Serre equivalences are important autoequivalences of k-linear categories appearing in different fields of mathematics. In this talk we will ask the following question. In which way are Serre equivalences compatible with k-linear functors? For this we first review the situation in the case of algebraic geometry, where some compatibilty results are known. This motivates us to introduce the notion of a "global Serre duality", which is an abstract framework encoding the naturality of Serre equivalences. Afterwards we show the existence of global Serre dualities in the case of (abstract) representation theory. In interesting special cases, we obtain explicit descriptions of Serre equivalences. This last step will require some techniques from abstract cubical homotopy theory. This is part of an on-going project with Moritz Groth.\n
Sheaves on the alcoves and modular representations
Friday, 13.7.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
I will give an overview of some recent results obtained jointly with Martina Lanini. While trying to understand the intricacies of the combinatorial category of Andersen, Jantzen and Soergel we came up with a new category that consists of ordinary sheaves on the space of alcoves of an affine Weyl group. I will show how this category provides new methods and tools for the problem of determining rational characters of algebraic groups in positive characteristics.
Variations on the theme of moment graphs
Friday, 20.7.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
Naturally arising as the 1-skeletons of torus actions on\n(nice) complex projective algebraic varieties, moment graphs were\noriginally introduced by Goresky, Kottwitz and MacPherson to compute\nequivariant cohomology of such varieties. In this talk, I will review\nsome applications of moment graph theory, starting from the equivariant\ncohomology of the flag variety, and the representation theory of a\ncomplex finite dimensional simple Lie algebra. Time permitting, I will\nalso discuss some ongoing joint work with Tomoyuki Arakawa on a certain\nclass of modules ("admitting a Wakimoto flag") for an affine Kac-Moody\nalgebra at a negative level.