# Richard Hain

- Professor of Mathematics
- Managing Editor of the Duke Mathematical Journal

**External address:**107 Physics Bldg, Durham, NC 27708

**Internal office address:**Box 90320, Durham, NC 27708-0320

**Phone:**(919) 660-2819

### Research Areas and Keywords

##### Algebra & Combinatorics

algebraic geometry

##### Geometry: Differential & Algebraic

algebraic geometry, Hodge theory, arithmetic geometry, topology of varieties

##### Number Theory

arithmetic

##### Topology

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.

My primary collaborators are Francis Brown of Oxford University and Makoto Matsumoto of Hiroshima University.

Hain, R. M., and S. Zucker. “Unipotent variations of mixed Hodge structure.” *Inventiones Mathematicae*, vol. 88, no. 1, Feb. 1987, pp. 83–124. *Scopus*, doi:10.1007/BF01405093.
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Hain, Richard M. “On the Indecomposable Elements of the Bar Construction.” *Proceedings of the American Mathematical Society*, vol. 98, no. 2, JSTOR, Oct. 1986, pp. 312–312. *Crossref*, doi:10.2307/2045704.
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Hain, R. M. “On the indecomposable elements of the bar construction.” *Proceedings of the American Mathematical Society*, vol. 98, no. 2, Jan. 1986, pp. 312–16. *Scopus*, doi:10.1090/S0002-9939-1986-0854039-5.
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Hain, Richard M. “On a generalization of Hilbert's 21st problem.” *Annales Scientifiques De L’École Normale Supérieure*, vol. 19, no. 4, Societe Mathematique de France, 1986, pp. 609–27. *Crossref*, doi:10.24033/asens.1520.
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Hain, R. M. “Mixed hodge structures on homotopy groups.” *Bulletin of the American Mathematical Society*, vol. 14, no. 1, Jan. 1986, pp. 111–14. *Scopus*, doi:10.1090/S0273-0979-1986-15410-8.
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Hain, R. M. “Iterated integrals, intersection theory and link groups.” *Topology*, vol. 24, no. 1, Jan. 1985, pp. 45–66. *Scopus*, doi:10.1016/0040-9383(85)90044-8.
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HAIN, R. M. “ITERATED INTEGRALS AND HOMOTOPY PERIODS.” *Memoirs of the American Mathematical Society*, vol. 47, no. 291, AMER MATHEMATICAL SOC, Jan. 1984, pp. 1–98.

“Primitive elements in rings of holomorphic functions.” *Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)*, vol. 1984, no. 346, Walter de Gruyter GmbH, Jan. 1984, pp. 199–220. *Crossref*, doi:10.1515/crll.1984.346.199.
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Hain, R. M. “Twisting Cochains and Duality Between Minimal Algebras and Minimal Lie Algebras.” *Transactions of the American Mathematical Society*, vol. 277, 1983, pp. 397–411. *Manual*, doi:10.2307/1999363.
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Hain, R. M. “A Characterization of Smooth Functions Defined on a Banach Space.” *Proceedings of the American Mathematical Society*, vol. 77, 1979, pp. 63–67. *Manual*, doi:10.2307/2042717.
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