Marc Levine (Duisburg-Essen), 14:00-15:00
Motivic, quadratic and real D-T invariants
The Déglise-Jin-Khan machinery of cohomology/Borel-Moore homology coming out of the six-functor formalism for the motivic stable homotopy category has allowed us to give refinements of the usual virtual fundamental classes valued in the Chow groups to classes in the Borel-Moore homology of an arbitrary motivic cohomology theory. We have applied this to yield invariants in the cohomology of the sheaf of Witt rings of quadratic forms. For the dimension 0 Hilbert schemes of a smooth project spin threefold, this yields versions of Donaldson-Thomas invariants with values in the Witt ring of the base-field. Using a quadratic refinement of classical torus localization, Anneloes Viergever has computed some of these invariants for P^3 over the reals, and together, we have computed all these invariants for (P^1)^3 over the reals. We will give an overview of these methods and results and point to some possible new developments. 

Niklas Kipp (Regensburg), 15:30-16:30
Constructing six-sunctors for syntomic cohomology of p-adic formal schemes
In this talk, we explain a way to obtain a six-functor formalism for syntomic cohomology developed by Bhatt-Morrow-Scholze and Bhatt-Scholze. The starting point is the fact that Bhatt-Lurie and Drinfeld found a way to interpret this cohomology theory as the quasi-coherent cohomology of a p-adic formal stack. On the other hand, constructing a well-behaved quasi-coherent cohomology with compact support is a subtle question. One world where this is achievable is the world of analytic stacks developed by Clausen-Scholze. So the task is to understand these p-adic formal stacks as analytic stacks, such that one obtains a well-behaved six-functor formalism on the quasi-coherent sheaves of the latter. Well-behaved here, for example, means that this six-functor formalism recovers Poincaré duality for syntomic cohomology. In the talk, we elaborate on some subtleties one encounters in achieving this idea.

Page web du séminaire : https://www.imo.universite-paris-saclay.fr/~matthew.morrow/MotivesSeminar.html