Algebraic geometry and Arithmetic geometry

The main theme of my research is to investigate moduli spaces classifying differential equations and connections on algebraic varieties from an arithmetic point of view. For example, linear differential equations and their solutions in positive characteristic behave in a special way, unlike in the complex domain, and must be understood by developing a theory specific to positive characteristic (including p-adic methods). On the other hand, due to these peculiarities, "good" linear differential equations in positive characteristic (and their generalizations) is closely related to combinatorics, representation theory, and enumerative geometry with physics backgrounds; and one can find some very interesting connections among them. In recent years, I have been working on deepening the moduli theory of such objects, making full use of these connections and building on (part of) a theory of arithmetic geometry called p-adic Teichmüller theory.