The problems to be investigated are in the area of the theory of Lie groups and their discrete subgroups. One of the main objectives is to continue the program of establishing a homogeneous space approach as a powerful tool in number theory. Special attention will be given to the problem of "effectivization" in Oppenheim conjecture and its quantitative generalizations. It is also proposed to continue to study recurrence properties of random walks on Lie groups and their discrete subgroups on homogeneous spaces, manifolds and general metric spaces.
The theory of Lie groups and their discrete subgroups is one of the central fields in mathematics. During the last few decades, it was realized that some aspects of the theory can be applied to solve certain problems in number theory and related topics, which could not be tackled by other methods. This proposal is related to rigidity theory that studies phenomena when rather weak data about geometric and mathematical objects determines completely or almost completely the structure of those objects.
is one of the central areas in mathematics. The project established new connections between the theory of Lie groups and their discrete subgroups, number theory, geometry, dynamical systems, and ergodic theory, and in general between discrete and continuous in mathematics. More specifically G.Margulis, in collaboration with his former graduate student A.Mohammadi, proved the quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms. G.Margulis, in collaboration with J.Athreya, established a completely new phenomenon: random Minkowsky theorem. This theorem says that if A is a measurable set in the Euclidean space of dimension greater thah or equal to two and the measure of A is "large", then the probability that a random unimodular lattice does not intersect A is "small". Special attention was given to the problem of "effectivization" in the Oppenheim conjecture and its quantitative generalizations and to the problem of effective density and effective equidistribution. In particular G.Margulis, in collaboration with E.Lindenstrauss obtained effective estimates for the size of the smallest solution in the Oppenheim conjecture (the Oppenheim conjecture is about solvability in integers of inequalities related to irrational quadratic forms). Jointly with D.Kleinbock and W.Jumbo, G.Margulis obtained new results about metric Diophantine approximation for systems of linear forms via dynamics. G.Margulis also continued his work on discrete groups of affine transformations in two different directions: (1) Auslander conjecture and classification problems related to it (joint work with H.Abels and G.Soifer: (2) the study of the moduli spaces of the proper affine deformation of the Fuchsian groups and the study of the boundaries of these moduli spaces (in collaboration with W.Goldman, F.Labourie, and Y.Minsky). A.Mohammadi (currently holding a tenure track position at the University of Texas, Austin) and I.Rapinchuk (currently a postdoc at Harvard) completed their dissertations under the direction of G.Margulis. Han Li, another graduate student of G.Margulis, has just submitted his dissertation. A part of this disssertation is based on a joint work with G.Margulis. G.Margulis taught courses and ran seminars for graduate students on topics related to the project. G.Margulis continued to participate in international conferences and workshops and presented several colloquium talks and invited lecture series. He also continued his collaboration with researchers from France, Germany, Israel, and UK.
