Abstracts for MPI-Oberseminar

My talk is dedicated to weight structures. Weight structures are natural counterparts of  $t$-structures (for triangulated categories); the simplest examples of weight structures come from stupid truncations of complexes (whereas $t$-structures are related with canonical truncations). Weight structures yield (functorial) weight complexes,  weight filtrations, and  weight spectral sequences.  An example: a conservative exact weight complex functor from the Voevodsky's category of geometric motives to $K^b(Chow)$.

Coisotropic submanifolds generalize energy level sets of time-independent Hamiltonian systems. A leafwise fixed point of the time-one map of a time-dependent perturbation of the system corresponds to a trajectory for which the perturbation results in a phase shift. In this talk, I will illustrate this in the example of the harmonic oscillator, using a short computer animation. The main result of the talk is that under suitable hypotheses the number of leafwise fixed points is bounded below by the sum of the Betti numbers of the coisotropic submanifold.

A Lie superalgebra is a generalization of Lie algebra to include a $Z_2$ grading. Definitions of, such as, a Lie superalgebra, homomorphism, and module, etc., resemble the Lie algebra case but include a suitable sign twist everywhere. The complexity of representation theory of Lie superalgebras far exceeds that of Lie algebras. For example, the finite-dimensional representations /C is not complete reducible for most complex simple Lie superalgbras.

We survey recent results concerning local rigidity of lattices in semisimple Lie groups and characterize the deformability of surface groups in semisimple Lie groups in terms of balanced conditions on root spaces. This is a joint work with Pierre Pansu.