|Speaker:||Anthony Blanc (SISSA)|
|Title:||Motivic realizations of dg-categories, matrix factorizations and vanishing cycles|
|Date:||Thu, Dec 07, 2017, 13:30 - 15:00|
|Place:||Seminar Room B|
(joint work with M. Robalo, B. Toen, G. Vezzosi).
Given a cohomology theory for schemes, it is often a hard task to generalize this notion to dg-categories (aka noncommutative spaces). One method consists in approximating a noncommutative space by the geometric stack of objects inside it, or more accurately by a certain motive depending functorially on the former. In this talk we will explain how to use Morel--Voevodsky's homotopy theory of schemes and realization functors in order to define some cohomology theories for noncommutative spaces (Betti, l-adic). Given a LG model over a discrete valuation ring with perfect residue field, with potential induced by a uniformizer, we will see how the l-adic cohomology of the associated category of matrix factorizations is given by the inertia invariant part of vanishing cohomology.