Algebraic Analysis and its Environs June 9, 2008 - Padova - Italy

Venue

Torre Archimede (map) - room 1BC/45
via Trieste, 63 - Padova - Italy

Schedule

Abstracts

Damien Calaque, Tangent (co)homology, Duflo's isomorphism and Caldararu's conjecture

In late '70 Michel Duflo proved that one can twist the PBW isomorphism in Lie theory so that it restricts to an algebra isomorphism on invariants. There exists an analogous result for complex manifolds, stating that the HKR isomorphism can be twisted by an appropriate element so that it induces an algebra isomorphism in cohomology. The main goal of this talk is to present this result together with its homological version, known as Caldararu's conjecture. We will also try to catch a few ingredients of the proof. The talk will be based on a joint work with Michel Van den Bergh, and on works in progress with Carlo Rossi and Michel Van den Bergh.

Luisa Fiorot, On the Gauss-Manin connection

We propose a brief history of the Gauss-Manin connection. We would translate the Katz-Oda definition (which makes use of the Leray spectral sequence) in terms of a distinguished triangle in the derived category. We propose a comparison between the notion of Gauss-Manin connection and the derived direct image for D-modules in the case of a smooth morphism between smooth varieties over the complex numbers. This is a joint work with Maurizio Cailotto.

Stéphane Guillermou, Microlocalization for chains complexes

The sheaf of microdifferential operators, E, on a complex manifold can be defined via Sato's microlocalization. This procedure also produces microlocal complexes of sheaves, naturally endowed with an action of E. It was proved some years ago by Kashiwara, Schapira, Ivorra, Waschkies that these complexes can be naturally defined as objects of the derived category of E-modules. We will see that this is also true in the "tempered case". The main point in our proof is to lift, in some cases, the six operations on derived categories of sheaves to operations on dg-modules over the de Rham algebra.

Marco Hien, Periods for flat algebraic connections

Given a flat algebraic connection on a smooth variety over the complex numbers, we prove a local duality involving the associated meromorphic de Rham complex and another de Rham complex with asymptotically flat coefficients. Additionally, we define a complex of rapid decay chains and obtain a perfect pairing between the resulting rapid decay homology groups and the algebraic de Rham cohomology in terms of period integrals.

Behrang Noohi, Uniformization of Deligne-Mumford curves

I will discuss the trichotomy hyperbolic/euclidean/spherical for (non-singular) Deligne-Mumford analytic curves. I will explain the classification of such curves by their uniformization types and use this to give an explicit presentation of a Deligne-Mumford curve as a quotients stack. The homotopy of crossed-modules plays an important role in this classification. This is joint work with Kai Behrend.