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 and, in particular in Diophantine approximation. Special attention will be given to the problem of the effective density and the effective equidistribution in homogeneous dynamics. There will also be emphasis on logarithm laws for unipotent flows and applications of homogeneous dynamics to metric theory of Diophantine approximation. It is also proposed to continue the work on discrete groups of affine transformations in two directions: (a) Auslander conjecture and classification problems related to it; (2) the boundary of the set of proper affine transformations of Fuchsian groups.
The theory of Lie groups and their discrete subgroups is one if the central areas in mathematics. During the last few decades, it was realized that some aspects of the theory can be applied to solve longstanding problems in number theory, quantum chaos and related topics, which could not be tackled by other methods. The 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. The proposal should establish new connections between theory of Lie groups and their discrete subgroups, number theory, geometry, dynamical systems and ergodic theory, probability theory, theoretical computer science, mathematical physics, and in general between discrete and continuous in mathematics.
