Posted in

Speaker:

Anthony Blanc
Affiliation:

MPIM
Date:

Tue, 2017-02-28 14:00 - 15:00
Location:

MPIM Lecture Hall
Parent event:

Seminar on Algebra, Geometry and Physics Given a Landau--Ginzburg model over the complex numbers, one can associate to it a dg-category of

matrix factorizations. By a result of Efimov, the periodic cyclic homology of this dg-category identifies

with the vanishing cohomology of the singular fiber. In this situation, certain Landau-Ginzburg models

appears as mirrors of certain symplectic manifolds, and this result allows an interpretation of homological

mirror symmetry via the associated noncommutative Hodge structures.

matrix factorizations. By a result of Efimov, the periodic cyclic homology of this dg-category identifies

with the vanishing cohomology of the singular fiber. In this situation, certain Landau-Ginzburg models

appears as mirrors of certain symplectic manifolds, and this result allows an interpretation of homological

mirror symmetry via the associated noncommutative Hodge structures.

In these two talks, I want to report on a joint work with Robalo, Toën and Vezzosi, which adresses a

similar situation above, but where we replace the complex field by a complete discrete valuation ring

(e.g. the ring of p-adic integers). This involves constructing a nice cohomology theory of noncommutative

spaces over such a ring, using the theory of noncommutative motives.

similar situation above, but where we replace the complex field by a complete discrete valuation ring

(e.g. the ring of p-adic integers). This involves constructing a nice cohomology theory of noncommutative

spaces over such a ring, using the theory of noncommutative motives.

I: In the first part we will recall the analog of our result over the complex numbers by Efimov. This will

involve preliminaries on dg-categories and cyclic homology. From there we will set up more preliminaries

for the sequel, on motives and noncommutative motives, and the method of taking motivic realizations.

involve preliminaries on dg-categories and cyclic homology. From there we will set up more preliminaries

for the sequel, on motives and noncommutative motives, and the method of taking motivic realizations.

II: In the second part we will review Orlov comparison theorem with the singularity category, the sheaf

of vanishing cycles over a dvr and present some elements of the proof that the

matrix factorization is given by vanishing cohomology. Finally we hope to put the result in perspective

of vanishing cycles over a dvr and present some elements of the proof that the

*l*-adic realization ofmatrix factorization is given by vanishing cohomology. Finally we hope to put the result in perspective

regarding Toën--Vezzosi approach to the conductor formula.