CAREER: Totally Geodesic Varieties in Moduli Space: Arithmetic and Classification

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In the research funded by this CAREER award, the PI will seek to address fundamental challenges in the study of dynamical systems in general, and billiards in polygons in particular. Dynamical systems are mathematical objects which evolve in time, and they are ubiquitous in the applications of mathematics. Whenever math 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 PI 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 PI 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. Using the resources allotted by this grant, the PI will also create a graduate course on topics relevant to his research. The course will culminate in a workshop hosted at Rice University during which leaders in the field have a chance to interact with students in the class. Finally, the PI will use funding from this grant to create a curriculum in mathematical biology for the Say STEM Camp, a camp aimed at engaging high school students in groups typically underrepresented in STEM fields.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 billiards 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 PI will pursue two lines of research in this proposal. First, the PI will investigate connections between special subvarieties and number theory, with the particular goal of understanding the arithmetic geometry of these spaces. Second, the PI will seek to develop a theory for higher dimensional special subvarieties with a view towards eventual classification.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.