Research Interests for Richard M Hain

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 and mixed zeta values which occur as periods of fundamental groups of moduli spaces of curves.

My primary collaborator is Makoto Matsumoto of the University of Tokyo.

Areas of Interest:

topology
algebraic geometry
arithmetic geometry

Recent Publications

1. Richard Hain, Monodromy of codimension-one sub-families of universal curves, Duke Math. Journal (Accepted, November, 2011) [arXiv:1006.3785]
2. Richard Hain, Normal Functions and the Geometry of Moduli Spaces of Curves, in Handbook of Moduli, edited by Gavril Farkas, Ian Morrison (Accepted, August, 2011) [arXiv:1102.4031]
3. Richard Hain, Rational points of universal curves, J. Amer. Math. Soc. 24 (2011), 709-769 [arXiv:1001.5008]
4. Alexandru Dimca, Richard Hain, Stefan Papadima, The abelianization of the Johnson kernel (Submitted, January, 2011) [arXiv:1101.1392]
5. Richard Hain, Remarks on non-abelian cohomology of proalgebraic groups, J. Algebraic Geom. (Accepted, 2011) [arXiv:1009.3662]