# Derived Noncommutative Geometry

## Monday, May 29

 09:00 - 09:30 Self-Registration 09:30 - 10:30 Dmitry Kaledin: Topological field theory in dimension one 10:30 - 11:00 Coffee break 11:00 - 12:00 Dmitry Kaledin: Topological field theory in dimension one 12:00 - 14:00 Lunch break 14:00 - 15:00 Julia Ramos González: On the tensor product of large categories 15:00 - 15:30 Coffee break 15:30 - 16:30 Emile Bouaziz: Chiral cohomology and Landau-Ginzburg models

## Tuesday, May 30

 09:00 - 10:00 Andrei Konovalov: Dg-categories and infinity-categories 10:00 - 10:30 Coffee break and Group Photo 10:30 - 12:00 Bertrand Toën: Matrix factorizations, motives and conductor formula 12:00 - 14:00 Lunch break 14:00 - 15:30 Dmitry Kaledin: Topological field theory in dimension one 15:30 - 16:00 Coffee break 16:00 - 17:00 Thi Quynh Trang Nguyen: The Geometric Langlands correspondance

## Wednesday, May 31

 09:00 - 10:30 Bertrand Toën: Matrix factorizations, motives and conductor formula 10:30 - 11:00 Coffee break 11:00 - 12:30 Tony Pantev: Non-abelian Hodge theory and mirror symmetry for character varieties 12:30 - 13:30 Lunch break 13:30 - 16:00 Excursion at Drachenfels (boat leaving at 14:00)

## Thursday, June 1

 09:00 - 10:30 Dmitry Kaledin: Topological field theory in dimension one 10:30 - 11:00 Coffee break 11:00 - 12:30 Bertrand Toën: Matrix factorizations, motives and conductor formula 12:30 - 14:00 Lunch break 14:00 - 15:30 Tony Pantev: Non-abelian Hodge theory and mirror symmetry for character varieties 15:30 - 16:00 Coffee break 16:00 - 17:00 Discussion Session 19:00 - Conference Dinner at Salvators

## Friday, June 2

 09:00 - 10:00 Pieter Belmans: Fully faithful functors versus Hochschild-Kostant-Rosenberg 10:00 - 10:30 Coffee break 10:30 - 12:00 Tony Pantev: Non-abelian Hodge theory and mirror symmetry for character varieties 12:00 - 14:00 Lunch break 14:00 - 15:00 Tasos Moulinos: On the Topological K-theory of derived Azumaya algebras

## Pieter Belmans: Fully faithful functors versus Hochschild-Kostant-Rosenberg

Given a fully faithful functor between the derived categories of smooth projective varieties, we can obtain a morphism between their Hochschild cohomologies. On the other hand, we can compute the Hochschild cohomology of a smooth projective variety using the Hochschild-Kostant-Rosenberg decomposition. In several instances there is an interesting (numerical) relationship between certain components of the Hochschild cohomology of a variety and the Hochschild cohomology of a moduli space associated to this variety, related to each other via a fully faithful functor. This suggests an interaction between the (noncommutative) deformations of the variety and the moduli space, which is not understood yet. I will explain some of the ingredients required for this observation, discuss various instances of this phenomenon, and its relation to noncommutative algebraic geometry.

## Emile Bouaziz: Chiral cohomology and Landau-Ginzburg models

I will introduce some ongoing work with I. Grojnowski dealing with a certain infinite dimensional analogue of vanishing cycles cohomology. It generalises both the twisted de Rham complex and the chiral de Rham complex in the appropriate degenerate cases.

## Dmitry Kaledin: Topological field theory in dimension one

At a first level approximation, TQFT in dimension one sounds like a completely vacuous topic: after all, the only smooth compact manifold of dimension one is the circle, so what structure could there possibly be. If you look closer, cellular decompositions of the circle come into play; this brings about A. Connes' cyclic category $\Lambda$ and the whole story of cyclic homology, a non-commutative generalization of de Rham cohomology. But if you look even closer, you notice that the circle can cover itself. It turns out that if one takes account of these covering maps in a proper way, then the resulting extra structures control at least the Frobenius action on the cristalline cohomology of algebraic varieties in positive characteristic, and possibly much more. I am going to give a general introduction to this circle of ideas.

## Tasos Moulinos: On the Topological K-theory of derived Azumaya algebras

Recent work of Anthony Blanc constucts a localizing invariant of dg categories over the complex numbers, called topological K-theory. The name arises from the fact that for $X\rightarrow Spec(C)$, a sufficiently nice scheme, $K^{top}(Perf(X)) =KU*(X(C))$, the topological K-theory of the associated space of complex points. In this talk I describe an ongoing project extending this idea to Azumaya algebras over such schemes. Namely, if T is a derived Azumaya algebra over $X$, we can identify $K^{top}(Perf(T))$ with a form of twisted topological K-theory of $X(C)$.

## Tony Pantev: Non-abelian Hodge theory and mirror symmetry for character varieties

I will discuss a construction of the homological mirror correspondence on algebraic integrable systems arising as moduli of flat bundles on curves. The focus will be on non-abelian Hodge theory as a tool for implementing hyper Kaehler rotations of objects in the Fukaya category. I will discuss in detail a specific example of the construction building automorphic sheaves on the moduli space of rank two bundles on the projective line with parabolic structure at five points. This is a joint work with Ron Donagi.

## Julia Ramos González: On the tensor product of large categories

The content of this talk is joint work with Wendy Lowen and Boris Shoikhet.

Grothendieck abelian categories and enhanced triangulated categories play an important role in noncommutative algebraic geometry, where they are used as models for noncommutative spaces. The analysis of a suitable tensor product construction in both settings is hence of interest in order to generalize the tensor product of schemes to noncommutative settings.

In first place, we define a tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. As an example of our construction, we describe the tensor product of noncommutative projective schemes in terms of Z-algebras. In addition, we compare our tensor product to other tensor products already known in the literature, namely Deligne's tensor product and the tensor product of locally presentable categories.

In a second part of the talk we focus on algebraic well-generated triangulated categories, which are known since work of Porta to be the analogues of Grothendieck categories in the triangulated world. Considering dg enhancements of such categories and their localization theory together with the homotopy category of dg categories developed by Tabuada and Toën, we construct a tensor product of such enhancements similarly as we have done in the abelian case.

## Bertrand Toën: Matrix factorizations, motives and conductor formula

Matrix factorizations, motives and conductor formula. Short abstract: In this series of lecture we propose an approach to the so-called Bloch's conductor formula by means of non-commutative methods. We will start by some reminders on cohomology of non-commutative spaces and eventually will explain how these can be constructed by methods from the stable homotopy theory of schemes à la Voevodsky-Morel. We then focus on the statement and the proof of the trace formula for non-commutative spaces. In the last lecture we apply this trace formula for non-commutative spaces coming from matrix factorizations and use this to prove the Bloch's conductor formula.