Abstracts for MPI-Oberseminar

We give a brief survey of the theory of elliptic hypergeometric functions - a new class of special functions discovered around ten years ago. Main topics to be discussed: the elliptic gamma function and elliptic beta integral, generalizing Euler's gamma function and beta integral; an elliptic analogue of the Gauss hypergeometric function; an elliptic extension of the Selberg integral; elliptic biorthogonal functions, generalizing classical orthogonal polynomials.

The trajectories of an integrable Hamiltonian system are, in general of a simple nature : in some special coordinates they become just straight lines on an abelian manifold, or on a torus if the situation is real. Such motions are called quasi-periodic. The KAM theorem states that under reasonable assumptions on the initial Hamiltonian, most quasi-periodic motions persist under perturbation, where most as to be understood in the sense of Lebesgue measure.

The conjugate $\sigma(f)$ of a Hilbert modular form $f$ by an automorphism
$\sigma$ of the field of complex numbers is obtained by applying $\sigma$
to the Fourier coefficients of $f$. A classical Theorem of Shimura states
that the conjugate of a Hilbert modular form $f$ of weight $k$ again is a
Hilbert modular form of conjugate weight $\sigma(k)$. We want to describe
an approach to Shimura's Theorem, which deduces the Theorem from a
comparison of trace formulas.
 

The construction of compactifications of Teichmüller spaces in the approach of Morgan and Shalen has close relationships with tropical geometry. By studying the general properties of the logarithmic limit set of real semi-algebraic sets, it is possible to generalize their construction and to understand some of its properties. When applied to Teichmüller spaces, this gives the compactification of Thurston, and the natural piecewise linear structure of the boundary appears automatically, showing clearly how this structure is related with the semi-algebraic structure of the interior part.