## Research Interests for Richard Hain

**Research Interests:** *Topology of Algebraic Varieties, Hodge Theory, and Moduli of Curves*
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.

**Areas of Interest:**
topology

algebraic geometry

arithmetic geometry

**Recent Publications**
(search)
- Hain, R,
*Deligne-Beilinson Cohomology of Affine Groups*,
in Hodge Theory and $L^2$-analysis, edited by Ji, L
(Submitted, July, 2015), International Press, ISBN 1571463518 [arXiv:1507.03144] [abs]
- Arapura, D; Dimca, A; Hain, R,
*On the fundamental groups of normal varieties*,
Communications in Contemporary Mathematics, vol. 18 no. 04
(August, 2016),
pp. 1550065-1550065, ISSN 0219-1997 [doi]
- Hain, R,
*Notes on the Universal Elliptic KZB Equation*,
Pure and Applied Mathematics Quarterly, vol. 12 no. 2
(July, 2016), International Press [arXiv:1309.0580], [1309.0580v3] [abs]
- Hain, R,
*The Hodge-de Rham theory of modular groups*,
in Recent Advances in Hodge Theory Period Domains, Algebraic Cycles, and Arithmetic, edited by Kerr, M; Pearlstein, G, vol. 427
(January, 2016),
pp. 422-514, Cambridge University Press, ISBN 110754629X
- Hain, R; Matsumoto, M,
*Universal Mixed Elliptic Motives*,
Journal of the Institute of Mathematics of Jussieu
(Submitted, December, 2015) [arxiv:1512.03975] [abs]