Totally Geodesic Subvarieties in the Moduli Space of Riemann Surfaces

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In this NSF funded research, the principal investigator seeks to address fundamental challenges in the study of dynamical systems in general, and trajectories of a ball moving in polygons in particular. Dynamical systems are mathematical objects which evolve in time, and they are ubiquitous in the applications of mathematics. Whenever mathematics is used to predict the future, e.g. to predict the weather, the stock market, or the behavior of particles in a solution, a dynamical system is involved. The dynamical systems in the real world, like those just mentioned, are often extraordinarily complex. The principal investigator will study a simple class of dynamical systems modeled on ideal billiards in polygons with a view towards understanding the complicated dynamical systems that occur in applications. The principal investigator will work to uncover and categorize the range of dynamical behaviors possible in billiard systems, and will continue his research exploring connections between billiards and the theory of numbers. In addition, the principal investigator will continue his educational, mentoring, and outreach activities to promote the broader impacts of his work. This project seeks to address fundamental challenges in the study of dynamics on Riemann surfaces and their moduli spaces. These subjects have connections with many areas of mathematics and important applications to the classification of mapping classes, the dynamics of rational maps and trajectories of a ball moving in polygons in polygons. Of particular importance are the special subvarieties of moduli space which are invariant under the geodesic flow. Such subvarieties are rare and their origins are mysterious. The principal investigator will pursue two lines of research. First, the principal investigator and his coauthors will give new constructions of special subvarieties. Second, the principal investigator will investigate connections between special subvarieties and number theory, with the particular goal of understanding the arithmetic geometry of these spaces.