Abstracts for MPI-Oberseminar

Deligne's weights are very important for various cohomology theories. It is conjectured that weights for cohomology could be lifted to weights for the abelian categories of mixed motives (and mixed motivic sheaves, i.e. mixed motives over a base scheme S). Since the existence of mixed motives is very much conjectural, the attempts to define weights for them previously didn't yield any interesting results.

I will give an introduction to the worldsheet geometry underlying two-dimensional, holomorphic superconformal field theory, and indicate how the worldsheet geometry of propagating superstrings gives rise to the notion of vertex operator superalgebra. I will discuss some applications of this understanding of the relationship between the geometric and algebraic aspects of superconformal field theory, such as the construction of twisted modules for a vertex operator algebra (also known in physics as twisted sectors).

We carry out the global qualitative analysis of planar polynomial dynamical systems and suggest a new geometric approach to solving Hilbert’s Sixteenth Problem on the maximum number and relative position of their limit cycles in two special cases of such systems. First, using geometric properties of four field rotation parameters of a new constructed canonical system, we present the proof of our earlier conjecture stating that the maximum number of limit cycles in a quadratic system is equal to four and their only possible distribution is (3:1).

In this talk, I will give an introductory overview of some
categories of modules for a simple Lie algebra over C: the category of
finite dimensional modules, BGG category O and its parabolic versions,  the
category of weight modules and some parabolic versions.