- Professor of Mathematics
- Managing Editor of the Duke Mathematical Journal
Research Areas and Keywords
Algebra & Combinatorics
Geometry: Differential & Algebraic
algebraic geometry, Hodge theory, arithmetic geometry, topology of varieties
topology of varieties, mapping class groups
I am a topologist whose main interests include the study of the topology of complex algebraic varieties (i.e. spaces that are the set of common zeros of a finite number of complex polynomials). What fascinates me is the interaction between the topology, geometry and arithmetic of varieties defined over subfields of the complex numbers, particularly those defined over number fields. My main tools include differential forms, Hodge theory and Galois theory, in addition to the more traditional tools used by topologists. Topics of current interest to me include:
- the topology and related geometry of various moduli spaces, such as the moduli spaces of smooth curves and moduli spaces of principally polarized abelian varieties;
- the study of fundamental groups of algebraic varieties, particularly of moduli spaces whose fundamental groups are mapping class groups;
- the study of various enriched structures (Hodge structures, Galois actions, and periods) of fundamental groups of algebraic varieties;
- polylogarithms, mixed zeta values, and their elliptic generalizations, which occur as periods of fundamental groups of moduli spaces of curves.
Hain, R. “Notes on the Universal Elliptic KZB Equation.” Pure and Applied Mathematics Quarterly, vol. 12, no. 2, International Press.
Hain, Richard. Hodge theory of the Goldman bracket.
Hain, Richard. Johnson Homomorphisms.