Effective and sparse equidistribution problems on homogeneous spaces and linear dynamics of semisimple groups

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The PI (Nimish A. Shah) proposes to investigate, together with his graduate students, a wide range of limit distribution problems on homogenous spaces of Lie groups with an aim of applications to questions in number theory and arithmetic geometry. The following three areas will be focused on: (1) effective density and effective equidistribution of orbits of unipotent flows, (2) description of limit distributions of stretching translates of sub-manifolds on homogeneous spaces, and (3) equidistribution of relatively sparse set of points on unstable leaves under the action of a partially hyperbolic flow. This study will combine techniques and results from diverse areas of mathematics: theory of semi-simple Lie groups and their finite and infinite dimensional representations, ergodic theory and dynamical systems, probability theory, number theory and automorphic forms, algebraic geometry, etc. The work will further the development of bridges between these areas, and yield interesting new results and techniques, of dynamical, number theoretical, geometrical and combinatorial nature. The proposed research links several important areas of mathematics: we apply concepts of ergodic theory and dynamical systems (the area with origin in physics) to solve a class of problems in number theory, using techniques of Lie groups, algebraic and differential geometry, and probability theory. All of these areas have deep connections with other sciences such as physics, astronomy, statistics, and computer science. The broad purpose of the proposed project is to investigate a range of ideas and create techniques that have the potential to impact many problems of interest in all of these fields. An important aspect of this project will be training of graduate students and post doctoral fellows to conduct research where ideas of one filed are applied to solve problem in another field.