9870126 Hain The set of all conformal structures on a given surface of genus g is called the moduli space of Riemann surfaces of genus g. Hain is studying various questions regarding the topology, arithmetic and geometry of such moduli spaces. In genus 1 (i.e., the boundary of a doughnut), there is a classical theory of modular forms that is of great importance in number theory. (In fact, it was one of the main tools used by Andrew Wiles in his proof of Fermat's Last Theorem.) At present, there is no good theory of modular forms in higher genus. One of Hain's projects is to find natural examples of modular forms in higher genus, especially ones that generalize the classical modular form of weight 12 given by the discriminant. His basic tool is Hodge theory, especially that of fundamental groups of algebraic varieties. He is further developing the Hodge theory of fundamental groups of varieties, especially those aspects related to modular forms and Galois theory. A closed surface of genus g is the surface bounding a doughnut with g holes. Each such surface has additional structure, often called a conformal structure or a complex structure. The study of surfaces and these enriched structures on them dates back to the middle of the 19th century, when they were first studied by Bernhard Riemann. Their study is still a very active and central area of mathematics. It is important in number theory, topology and geometry. It is also a central tool in String Theory, the physical theory that attempts to unite general relativity and quantum mechanics. ***
