Motivic realizations of dg-categories, matrix factorizations and vanishing cycles

Given a cohomology theory for schemes, it is often hard 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, which gives rise to a motive. 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. (Joint work with M. Robalo, B. Toën, G. Vezzosi.)