Final Report Summary - GMODGAMMADYNAMICS (Dynamics on homogeneous spaces, spectra and arithmetic)
The project's focus has been dynamics of actions on homogeneous spaces of algebraic groups, and tackled central problems in the area. Significant progress was made in the study of diagonal groups on homogeneous spaces, including a completely new approach to classifying measures invariant under such actions using quantitative recurrence that allows handling non-maximal tori as well as the fisrt results in positive characteristic. Quantitative understanding of equidistribution properties of unipotent flows and groups generated by unipotents was obtained, with number theoretic implications regarding values of quadratic forms.
Connections with arithmetic combinatorics were established. In particular, in the framework of the project a quantitative central-local limit theorem on random walks by isometries of d-dimensional Euclidean space and a classification of positive dimensional measurable subgroups of simple Lie groups were obtained.
Significant results were also obtained related to self similar sets and mathematical physics (in particular, quantum ergodicity and unique ergodicity).
An emphasis was given in combining tools from different mathematical disciplines and building new bridges between these areas.
Connections with arithmetic combinatorics were established. In particular, in the framework of the project a quantitative central-local limit theorem on random walks by isometries of d-dimensional Euclidean space and a classification of positive dimensional measurable subgroups of simple Lie groups were obtained.
Significant results were also obtained related to self similar sets and mathematical physics (in particular, quantum ergodicity and unique ergodicity).
An emphasis was given in combining tools from different mathematical disciplines and building new bridges between these areas.