Research Interests

As a mathematician I work in several fields: dynamical systems, Lie groups and their discrete subgroups, algebraic and arithmetic groups, number theory, and combinatorics. I am mostly interested in problems that allow me to apply methods and results from different fields of mathematics. In my Ph.D. thesis I used methods from the theory of dynamical systems to the study of asymptotics of the number of closed geodesics on compact manifolds of negative curvature and to other geometric problems. Later I was able to use ergodic theory to prove structural results about discrete subgroups of Lie groups, in particular arithmeticity, superrigidity, and the finiteness of nontrivial quotients. More recently I have been involved in applications of homogeneous dynamics to number theory and diophantine approximation. In particular, using homogeneous dynamics and partially in collaboration with other mathematicians, I was able to prove some conjectures about values of irrational quadratic forms and about diophantine approximation on manifolds.

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Section 11: Mathematics