Modul: MAT760 Ergodic Theory and Dynamical Systems Seminar

## Effective solution count in intrinsic Diophantine approximation

Prof. Dr. Amos Nevo talk

Speaker invited by: Prof. Dr. Alexander Gorodnik

**Date:** 08.10.18 **Time:** 15.30 - 16.30 **Room:** Y27H25

In his 1965 "Report on Diophantine approximation" Serge Lang raised the problem of establishing the approximation properties of rational points on homogeneous algebraic varieties, singling out in particular the questions of establishing Diophantine approximation exponents, an analog of Khinchin's dichotomy theorem and an analog of W. Schmidt's solution counting theorem. In recent years a systematic approach to Lang's problems has been developed for varieties homogeneous under an action of semisimple groups, and some progress towards answering the questions mentioned above has been obtained, with the answers in certain special cases being optimal. The methods involve lattice actions, ergodic theorems and spectral estimate in the automorphic representation. Based on joint work with Anish Ghosh and Alex Gorodnik.