Homogeneous and Teichmuller Dynamics: A Quantitative Viewpoint

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Dynamical systems, which originated from classical mechanics, have become an indispensable part of modern mathematics. Roughly speaking, a dynamical system is a rule which predicts the motions of a point in a geometric space; examples of mathematical models of dynamical systems include the study of fluid dynamics, airflow dynamics, billiard trajectories, etc. As a concrete example, consider a polygonal billiard table, e.g., an L-shaped billiard table; and to simplify the situation, let us assume that the table has no friction and the angle of incidence equals the angle of reflection. Behavior of billiard trajectories have long been fascinating to mathematicians and physicists--note that different trajectories can have different behaviors, e.g., some trajectories are periodic and some trajectories are dense (they make a visit near every point in the table). For instance, one can ask whether a billiard table has any periodic trajectories, or more quantitatively, one can ask how many periodic trajectories of a certain length exist? Surprisingly, questions of this kind have proven to be notoriously difficult to answer and are connected to several areas of modern mathematics. It turns out that even to understand billiard trajectories on one single table one is led to the study of families of billiard tables which have same combinatorics, e.g., same number of edges for the table. These families turn out to have new symmetries and enjoy certain rigidity properties which help to unravel the original problem. This project studies questions with similar philosophical point of view.The project seeks extensions and extensions of certain rigidity results where a rather weak initial data about an object yields an almost complete classification of the object; the goal is to provide a more quantitative account of these rigidity results with applications in mind. The following will be the main objectives: (i) Dynamical systems have become a major player in several unexpected areas in modern mathematics. Proofs relying on dynamical arguments, however, are usually existential and not constructive. For various applications, quantitative and effective accounts of dynamical arguments are much sought after and challenging. (ii) Discrete subgroups of Lie groups with finite covolume are central objects in mathematics and have been extensively studied. In some natural settings -- e.g., Kleinian groups, several monodromy groups, Apollonian circle packings, and Zaremba's conjecture -- however, one needs to consider Zariski dense, discrete subgroups (and semigroups) of semisimple Lie groups with infinite covolume. These investigations often lead to the study of group actions on infinite volume spaces. (iii) The moduli space of a compact Riemann surface is one of the hubs of modern mathematics. Despite the very inhomogeneous nature of the moduli space, there are several similarities between this space and homogeneous spaces. Dynamics on moduli spaces presents a very exciting avenue for research with many outstanding and challenging problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.