Past lectures

Abstract

In the late 50's Y. V. Linnik obtained remarkable results on the distribution of the representations of integers as sums of three squares and, more generally, on the distribution of integers by other ternary quadratic forms. The main tool was what he called the "ergodic method". It is only in the late 80's that Linnik's results have been substantially improved, by Duke, using a completely different approach based on harmonic analysis and especially on the theory of automorphic forms.

Nowadays, Linnik's results and problems may be reinterpreted in terms of distribution properties of torus orbits on arithmetic homogeneous spaces; as such, these problems are not accessible, at least directly, by other, well developped methods, eg. the works of Margulis, Ratner and others on unipotent flows on homogeneous spaces. In these lectures, we will present Linnik's ergodic method as well as Duke's "automorphic" approach from this perspective. This will lead us to somewhat remote fields such as Waldspurger's formulae, the analytic theory of L-functions, and the subconvexity problem. We will also discuss generalizations of Linnik's type problems along with the variety of techniques invented to approach them. These include, amongst others, works of Einsiedler, Katok, Lindenstrauss, Venkatesh, myself, as well as Holowinsky and Soundararajan.