CAREER: Moduli Space of Curves and Teichmueller Dynamics

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An Abelian differential defines a flat metric on the underlying Riemann surface. Varying the flat structure induces an action on moduli spaces of Abelian differentials; this is called Teichmueller dynamics. A number of questions about the geometry of Riemann surfaces boils down to the study of the orbits under such dynamics. The proposed project aims to explore Teichmueller dynamics using tools in algebraic geometry and to develop applications to the geometry of the moduli space of Riemann surfaces. The ultimate goal is to establish a correspondence between dynamical properties of these orbits and the intersection theory of their closures in the moduli space. Moreover, the intersection calculation can determine the cycle class of an orbit closure in the moduli space, which in turn provides crucial information towards understanding the cone of effective divisors, cone of curves, Chow ring structure, and birational models for the moduli space. Algebraic geometry and dynamical systems are two important branches of modern mathematics. The former uses algebraic (polynomial) equations to study geometrical structures, while the latter applies analytical tools to describe the time dependence of a moving point. Despite the fact they initially seem unrelated, the principal investigator plans to explore their inner connections by constructing algebraic equations to measure the behavior of Teichmueller dynamics. In a sense this is analogous to introducing coordinates in Descartes geometry. The proposed project also opens many avenues for student and postdoctoral research. The principal investigator will continue to integrate his research with undergraduate, graduate and post-graduate training as well as workshop organization. More precisely, he plans to develop a student mathematics symposium, advise student research projects, design new courses in algebraic geometry, create a junior scholar visiting program, and organize a series of conferences and workshops with a focus on students and young researchers.