Aspects of Unipotent Dynamics on Homogenous Spaces

Ohio State University, The, Columbus OH

Investigators

Abstract

Many questions in number theory and geometry involve multiple infinite groups of symmetries in their structures. Exploring the interactions between these groups of symmetries using ideas from dynamical systems, applied to suitably created algebraic models, is at the core of mathematical work involved in this project. The primary goal of the project is to make new observations that relate, and to develop new techniques applicable for solving, a variety of dynamical, number theoretical, and geometrical problems. The work combines concepts from several different areas of mathematics, and it is anticipated that the project will build conceptual bridges between these different disciplines. Graduate students and post-doctoral researchers will participate in the research project, preparing them to work on some of the forefront areas of mathematics. The investigator and his graduate students will study a wide range of dynamical questions on unipotent flows on homogeneous spaces that arise from number theoretic or geometrical questions. The project investigates four types of problems: (1) Describe closures of unipotent orbits and totally geodesic immersions in infinite-volume geometrically-finite hyperbolic manifolds; (2) Control the non-divergence rate of unipotent trajectories or expanding translates of short unipotent curves under suitable Diophantine conditions on the base point; (3) Find geometric conditions under which expanding translates of a curve in a homogeneous space are uniformly distributed in the limit; (4) Compute Hausdorff dimensions of the sets of points with specified excursion rates for flows on non-compact homogeneous spaces. This study will combine techniques from various areas of mathematics, including dynamical systems, Lie groups, representation theory, algebraic geometry, probability theory, number theory, and combinatorics. The research aims to obtain new results and develop new techniques of dynamical, number theoretical, and combinatorial nature.