• Hausdorff School 2016/2017
• Derived Noncommutative Geometry

• Schedule
• Poster
• Participants

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.

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.

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 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.

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 , a sufficiently nice scheme, , 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 , we can identify with a form of twisted topological K-theory of .

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.

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.