L4

In this talk, two concepts are brought together: Algebras with triangular decomposition (as studied by Bazlov & Berenstein) and pointed Hopf algebra. The latter are Hopf algebras for which all simple comodules are one-dimensional (there has been recent progress on classifying all finite-dimensional examples of these by Andruskiewitsch & Schneider and others). Quantum groups share both of these features, and we can obtain possibly new classes of deformations as well as a characterization of them.

We study the nonhomogeneous boundary value problem for Navier--Stokes equations of steady motion of a viscous incompressible fluid in  a plane or spatial exterior domain with multiply connected boundary. We prove that this problem has a solution for axially symmetric case (without any restrictions on fluxes, etc.)  No restriction on the size of fluxes are required. This is a joint result with K.Pileckas and R.Russo.

In this talk I will introduce cluster categories and report on some new results on cluster categories of type E_6.

I will explain some ongoing work on understanding algebraic D-moldules via their reduction to positive characteristic. I will define the p-cycle of an algebraic D-module, explain the general results of Bitoun and Van Den Bergh; and then discuss a new construction of a class of algebraic D-modules with prescribed p-cycle.

In this talk I will give a definition of cluster algebra and state some main results.

Moreover, I will explain how the combinatorics of certain cluster algebras can be modeled in geometric terms.

The Contou-Carrère symbol has been introduced in the 90's in the study of local analogues of autoduality of Jacobians of smooth projective curves. It is closely related to the tame symbol, the residue pairing, and the canonical central extension of loop groups. In this talk we will a discuss a K-theoretic interpretation of the Contou-Carrère symbol, which allows us to generalize this one-dimensional picture to higher dimensions. This will be achieved by studying the K-theory of Tate objects, giving rise to natural central extensions of higher loop groups by spectra. Using the K-theoretic viewpoint, we then go on to prove a reciprocity law for higher-dimensional Contou-Carrère symbols. This is joint work with O. Braunling and J. Wolfson.

In 2000 Eliashberg-Polterovich introduced the natural notion of orderability of contact manifolds; that is, the existence of a natural partial order on the group of contactomorphisms. I will explain how one can study orderability questions using the machinery of Rabinowitz Floer homology. We establish a link between orderable and hypertight contact manifolds, and show that the Weinstein Conjecture holds (i.e. there exists a closed Reeb orbit) whenever there exists a positive (not necessarily contractible) loop of contactomorphisms.

Joint work with Peter Albers and Urs Fuchs.