(Joint with R. Boutonnet, A. Ioana) The notion of local spectral gap for general measure preserving actions will be defined. We prove that the left translation action of a dense subgroup of a simple Lie group has local spectral gap if the subgroup has algebraic entries. This extends to the non-compact setting recent works of Bourgain-Gamburd and Benoist-de Saxce. We also extend Bourgain-Yehudayoff’s result. The application of this result to Banach-Ruziewicz problem, delayed random-walk, and monotone expanders will be explained. 

The talk will be concerned with a property of lattices in the geometry of numbers which I refer to as "almost sure grid-DV with respect to a function F". The goal of the talk will be three-fold:

(1) Discuss a theorem relating this property to the dynamics of the lattice under the invariance group of the function.

(2) Apply the theorem to various classical discussions and see what it gives (it relates nicely to Minkowski's conjecture and to diophantine exponents).

(3) Discuss a mistake I have in the published paper and some open problems