Talk

A split Kac-Moody group $G$ over a topological ring $R$ carries a natural topology defined by Kac and Peterson. In case the underlying topological ring $R$ is $k_\omega$, this turns the Kac-Moody group $G$ into a $k_\omega$ group, which allows for a certain amount of control over this topology. Each $\sigma$-compact locally compact ring is a $k_\omega$ ring, hence the above topology allows to study $S$-arithmetic subgroups of topological Kac-Moody groups over local fields.

We construct some representations of braid groups, arising from the representations of quantum groups.

In series of our papers with Carlo Madonna (2002--2008) we described self-correspondences via moduli of sheaves with primitive isotropic Mukai vectors for K3 surfaces with Picard number one or two. Here we give a natural and functorial answer to the same problem for arbitrary Picard number of K3 surfaces. As an application, we characterize in terms of self-correspondences via moduli of sheaves K3 surfaces with reflective Picard lattices, that is when the automorphism group of the lattice is generated by reflections up to finite index.

In this talk we address the following strengthening of the Inverse Galois problem over $\mathbb{Q}$, introduced by B. Birch around 1994: Let $G$ be a finite group. Is there a tamely ramified Galois extension of $\mathbb{Q}$ with Galois group $G$? When $G$ is a linear group, this problem can be approached through the study of Galois representations attached to arithmetic-geometric objects. Let $\ell$ be a prime number.

In this talk, I will describe a family of embeddings of the Teichmüller space of a cusped Riemann surface Sigma in the $PU(2,1)$-representation variety of the fundamental group of Sigma. These embeddings are obtained by a bending process which allows an explicit description of an equivariant mapping from the universal cover of Sigma into the complex hyperbolic plane.

We discuss addition-theorems-type functional equations which naturally arise in various problems of mathematical physics and topology  (Toda and Calogero-Moser dynamical systems, rational and pole solutions of the KdV equation; elliptic genera associated with the string-inspired Witten index).