Gerbes and Brauer groups over stacks

The aim of this paper is to develop the theory of Brauer groups for stacks, which are not necessarily algebraic, using gerbes as foundamental tools. As an application, we focus our attention on Brauer theory for mixed motives: in particular, over a normal base scheme, we prove the generalized Theorem of the Cube for 1-motives and that a torsion class of the H^2_et(M,G_m) of a 1-motive M, whose pull-back via the unit section is zero, comes from an Azumaya algebra. Over an algebraically closed field, all classes of H^2_et(M,G_m) come from Azumaya algebras.Comment: We add a section about 2-descent theory for stack

Algebraic Anosov actions of Nilpotent Lie groups

In this paper we classify algebraic Anosov actions of nilpotent Lie groups on closed manifolds, extending the previous results by P. Tomter. We show that they are all nil-suspensions over either suspensions of Anosov actions of Z^k on nilmanifolds, or (modified) Weyl chamber actions. We check the validity of the generalized Verjovsky conjecture in this algebraic context. We also point out an intimate relation between algebraic Anosov actions and Cartan subalgebras in general real Lie groups.Comment: 40 page

From Topology to Noncommutative Geometry: KK-theory

We associate to each unital $C^*$-algebra $A$ a geometric object---a diagram of topological spaces representing quotient spaces of the noncommutative space underlying $A$---meant to serve the role of a generalized Gel'fand spectrum. After showing that any functor $F$ from compact Hausdorff spaces to a suitable target category can be applied directly to these geometric objects to automatically yield an extension $\tilde{F}$ which acts on all unital $C^*$-algebras, we compare a novel formulation of the operator $K_0$ functor to the extension $\tilde K$ of the topological $K$-functor.Comment: 14 page

On finite-dimensional Hopf algebras

This is a survey on the state-of-the-art of the classification of finite-dimensional complex Hopf algebras. This general question is addressed through the consideration of different classes of such Hopf algebras. Pointed Hopf algebras constitute the class best understood; the classification of those with abelian group is expected to be completed soon and there is substantial progress in the non-abelian case.Comment: 25 pages. To be presented at the algebra session of ICM 2014. Submitted versio