Values of Quadratic Forms and Effective Equidistribution

Ratner's landmark equidstribution results for unipotent flows have had dramatic applications in many mathematical areas. Recently there has been considerable progress in the long sought for goal of getting effective equidistribution results for these flows.

One of the famous applications of unipotent flow has been the proof by Margulis of the longstanding Oppenheim conjecture (independently of Ratner's work) by using unipotent dynamics. We show how using the new effective equidistribution result one can get a quantitative and effective results regarding values of indefinite nonintegral quadratic forms. We also discuss a problem pioneered by Sarnak regarding pair correlation of eigenvalues on flat tori.

The works I will present and joint with Amir Mohammadi, Zhiren Wang and Lei Yang.