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 Hiroshima University.

Recent Publications

1. Richard Hain, Relative weight filtrations on completions of mapping class groups, in Groups of Diffeomorphisms, Advanced Studies in Pure Mathematics, vol. XX (Accepted, May, 2008), Mathematical Society of Japan [arXiv:0802.0814]
2. Richard Hain, Makoto Matsumoto, Relative pro-l completions of mapping class groups (Submitted, February, 2008) [arXiv:0802.0806]
3. Richard Hain, Finiteness and Torelli Spaces, in Problems on Mapping Class Groups and Related Topics, Proc. Symp. Pure Math. 74, edited by Benson Farb (September, 2006), pp. 57-70, American Mathematical Society [math.GT/0508541]
4. Richard M. Hain and Makoto Matsumoto, Galois actions on fundamental groups of curves and the cycle C-C^-, J. Inst. Math. Jussieu, vol. 4 (2005), pp. 363-403 [math.NT/0306037]
5. Richard M. Hain, David Reed, On the Arakelov Geometry of Moduli Spaces of Curves, J. Differential Geom., vol. 67 (2004), pp. 195-228 [math.AG/0211097]