Moduli of Differentials

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The concept of differentials dates back to the origin of calculus. For instance, integration of differentials can tell us about distance, area, volume, etc, for the physical world. Nowadays the study of differentials has broad connections to a number of fields in and outside of mathematics. It has been proven extremely useful that one should collect differentials with similar structures into a parameter space (called moduli space) and study the global properties of this space, which can in turn determine crucial information about each individual differential. The principal investigator plans to explore new, fascinating properties of differentials as well as their moduli spaces. Moreover, he plans to discover hidden connections between initially unrelated subjects in which differentials appear from different aspects. The proposed project also opens many gates for student and postdoctoral research. The principal investigator will continue to integrate his research with undergraduate, graduate and postdoctoral training as well as conference organizations. In particular, he plans to advise student research projects, design new courses, and organize a series of workshops, with a focus on increasing diversity and supporting underrepresented groups. The moduli space of differentials on Riemann surfaces can be stratified according to the types of zeros and poles. Each stratum is equipped with a group action by varying the polygonal structure induced by a differential. Various questions about surface geometry reduce to understanding the strata of differentials and orbits under the action. The principal investigator plans to employ his expertise in algebraic geometry, combined with mixing techniques from analysis, dynamics and topology, to study these strata and orbits, such as their volumes, geodesics, and boundary behavior. An ultimate goal is to establish a correspondence between dynamical invariants of differentials and intersection theory on moduli spaces. Moreover, the PI plans to use the obtained results to analyze algebro-geometric properties of moduli spaces, such as birational types, compactifications, and tautological rings.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.