The research to be conducted lies at the interface of numerous mathematical disciplines in which there has been exciting progress recently. It involves in particular the theory of dynamical systems (specifically the study of concrete and natural group actions on locally homogeneous spaces), algebraic group theory, analytic number theory, automorphic forms and spectral theory, arithmetic combinatorics and mathematical physics (the quantum mechanical behavior of classically chaotic systems). By using the rich toolbox given by the mathematical disciplines above with particular emphasis on the tools from the theory of dynamical systems, we plan to tackle fundamental questions in the theory of group actions on homogeneous spaces, Diophantine approximations (e.g. the Littlewood Conjecture and effective versions of Oppenheim Conjecture), arithmetic, and the theory of automorphic forms.
We hope that the research conducted in this project will build new bridges and strengthen existing bridges between seemingly disperse mathematical disciplines. Unlike other sciences, mathematical knowledge does not become obsolete, and by discovering new connections between different subjects one can bring to bear deep results, both new and old, to the fundamental questions mathematicians investigate. A key part of the proposal is the investigation of the space of lattices in d-dimensional Euclidean space --- a space whose investigation began more than 100 years ago by the mathematician Hermann Minkowski to better understand the basic properties of number fields and other purely theoretical questions, and which is now used in an essential way in many algorithms to tackle real life problems. While it is hard to predict what practical applications the improved understanding of the space of lattices we hope to attain via this proposal may have many (or maybe not so many) years down the road, the algorithmic success of lattice space techniques suggests such applications are quite likely.
The application of methods from homogeneous dynamics, especially ergodic theory and the classification of invariant measures associated with such dynamics, coupled with techniques from automorphic forms and number theory have led to progress and even the solution of a number of central problems. A number of these were goals of this grant or emerged from developments connected with this grant. These include a generalization to higher degree number fields of Duke's celebrated theorem concerning the distribution in the space of lattices of tori corresponding to ideal classes of real quadratic fields, the distribution of eigenfunctions of the Laplacian in arithmetic manifolds in the semiclassical limit (the strongest form of Quantum Unique ergodicity known), the computation of the quantum variance for such surfaces, a conditional proof (under the Riemann Hypothesis) that the number of nodal domains of eigenfunctions on such a surface goes to infinity with the eigenvalue, the classification of invariant and stationary measures under a nonabelian linear action on the torus (that is solution of the stiffness conjecture in this setting), the proof the linear disjointness of the Mobius system from skew product (of zero entropy) systems on the torus whose orbits are not stochastic. The results obtained continue to demonstrate the rich and powerful interplay between arithmetic and homogeneous dynamics and form a basis for the expectation that many more striking results will be found in this direction.
