Abstracts for MPI-Oberseminar

The SL(2,C) character variety of a finite volume hyperbolic 3-manifold M can be thought of as the moduli space of hyperbolic structures on M. This affine algebraic set encodes much of the topology of M, but many basic properties of this set are unknown.  I will introduce the character variety and discuss several ways in which the geometry of the character variety is related to the topology of M and the structure of the fundamental group of M.

 

Hyperbolic orbifolds of small volume tend to be arithmetic, and given by
Coxeter groups. In this talk I will present new results that confirm
that arithmetic hyperbolic orbifolds of small volume tend to be given by
Coxeter groups (for reasonable dimensions).

In many areas of number theory, and especially in the theory of
modular forms, the computations which are necessary to produce new and
interesting examples today are usually very complex (in both a precise
and a naive sense).
Due both to this increased complexity as well as the increase in
available computational power, the importance of computer assisted
calculations in number theory has been growing steadily. As a result,
computer programs are an essential tool for research in number theory
and modular forms today.

We prove that for $n\geq2$ there exists a compact subset $X$  of the
closed ball in $R^{2n}$ of radius $\sqrt{2}$, such that $X$ has
Hausdorff dimension $n$ and does not symplectically embed into the
standard open symplectic cylinder. The proof involves a certain
Lagrangian submanifold of linear space, which was considered by M.
Audin and L. Polterovich. (Joint work with Jan Swoboda)