Events

Given a hyperbolic curve $Y$ defined over the integers and a finite set of primes $S$, the set of $S$-integral points $Y(\mathbb{Z}_S)$ is finite by theorems of Siegel, Mahler, and Faltings. Determining this set in practice is a difficult problem for which no general method is known. In this talk I report on joint work in progress with Martin Lüdtke in which we develop a Chabauty--Coleman method for finding $S$-integral points on affine curves.

In the past decade, a major triumph for the Tate conjecture (over finite fields) is its resolution for K3 surfaces. However, all known proofs rely crucially on the Kuga-Satake construction—effectively outsourcing certain difficulties to the theory of abelian varieties. There is an alternative approach which seeks to prove the conjecture by linking it to finiteness statements in arithmetic geometry, and using only the geometry of K3 surfaces.

We present two important actions of free groups on hyperbolic 3-space and Hermitian symmetric space via Kleinian group theory

and surface group representations into Hermitian Lie groups. The first one is related to the Thurston's conjecture of generalization of the double limit 

theorem to 3-manifolds with compressible boundary, and the second one is related to the Atiyah-Patodi-Singer index theory on flat bundles over surfaces with boundary.

Seminar webpage:  https://guests.mpim-bonn.mpg.de/bianchi/wiqi.html