# 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.

### Selected Grants

(98-0294) Modular Forms and Topology awarded by National Science Foundation (Principal Investigator). 1998 to 1999

(97-0515) Mathematical Sciences: Representations of Braid and Mapping Class Groups awarded by National Science Foundation (Principal Investigator). 1995 to 1998

(95-0224) Representation of Braid and Mapping Class Groups awarded by National Science Foundation (Principal Investigator). 1995 to 1997

(96-0949) Representations of Braid and Mapping Class Groups awarded by National Science Foundation (Principal Investigator). 1995 to 1997

(94-0057) Mathematical Sciences: Topology and Geometry of Algebraic Varieties awarded by National Science Foundation (Co-Principal Investigator). 1992 to 1995

## Pages

Dupont, J., et al. “Regulators and Characteristic Classes of Flat Bundles.” *The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998)*, vol. 24, American Mathematical Society, 2000, pp. 47–92.

Hain, R. “Moduli of Riemann Surfaces, Transcendental Aspects, Moduli Spaces.” *ALgebraic Geometry*, edited by L. Gottsche, vol. 1, Abdus Salam Int. Cent. Theoret. Phys., 2000, pp. 293–353.

Hain, R. “Locally Symmetric Families of Curves and Jacobians.” *Moduli of Curves and Abelian Varieties*, edited by C. Faber and E. Looijenga, Friedr. Vieweg, 1999, pp. 91–108.

FREEDMAN, M. I. C. H. A. E. L., et al. “BETTI NUMBER ESTIMATES FOR NILPOTENT GROUPS.” *World Scientific Series in 20th Century Mathematics*, CO-PUBLISHED WITH SINGAPORE UNIVERSITY PRESS, 1997, pp. 413–34. *Crossref*, doi:10.1142/9789812385215_0045.
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Hain, R., and E. Looijenga. “Mapping Class Groups and Moduli Spaces of Curves.” *Algebraic Geometry—Santa Cruz 1995*, vol. 62, American Mathematical Society, 1997, pp. 97–142.

Hain, R. M. “Torelli Groups and Geometry of Moduli Spaces of Curves.” *Current Topics in Complex Algebraic Geometry*, edited by C. H. Clements and J. Kollar, vol. 28, Cambridge Univ. Press, 1995, pp. 97–143.

Hain, R. M. “Classical Polylogarithms, Motives.” *Motives (Seattle, WA, 1991)*, vol. 55, American Mathematical Society, 1994, pp. 3–42.

Hain, Richard, and Robert MacPherson. “Introduction to higher logarithms.” *Structural Properties of Polylogarithms*, American Mathematical Society, 1991, pp. 337–53. *Crossref*, doi:10.1090/surv/037/15.
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Hain, Richard M. *Algebraic cycles and extensions of variations of mixed Hodge structure*. American Mathematical Society, 1991, pp. 175–221. *Crossref*, doi:10.1090/pspum/053/1141202.
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Carlson, J. A., and R. M. Hain. *Extensions of Variations of Mixed Hodge Structure*. Theorie de Hodge, 1987, pp. 39–65.

## Pages

Hain, R., and M. Matsumoto. “Galois Actions on Fundamental Groups of Curves and the Cycle $C-C^-$.” *Journal of the Institute of Mathematics of Jussieu*, vol. 4, Cambridge University Press (CUP): STM Journals, 2005, pp. 363–403. *Manual*, doi:10.1017/S1474748005000095.
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Kim, Minhyong, and Richard M. Hain. “A De Rham–Witt approach to crystalline rational homotopy theory.” *Compositio Mathematica*, vol. 140, no. 05, Wiley, Sept. 2004, pp. 1245–76. *Crossref*, doi:10.1112/s0010437x04000442.
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Hain, R., and D. Reed. “On the arakelov geometry of moduli spaces of curves.” *Journal of Differential Geometry*, vol. 67, no. 2, Jan. 2004, pp. 195–228. *Scopus*, doi:10.4310/jdg/1102536200.
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Hain, R., and M. Matsumoto. “Weighted completion of galois groups and galois actions on the fundamental group of ℙ1 -{0, 1, ∞}.” *Compositio Mathematica*, vol. 139, no. 2, Nov. 2003, pp. 119–67. *Scopus*, doi:10.1023/B:COMP.0000005077.42732.93.
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Hain, R. “The rational cohomology ring of the moduli space of abelian 3-folds.” *Mathematical Research Letters*, vol. 9, no. 4, Jan. 2002, pp. 473–91. *Scopus*, doi:10.4310/MRL.2002.v9.n4.a7.
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Hain, R., and D. Reed. “Geometric proofs of some results of Morita.” *Journal of Algebraic Geometry*, vol. 10, no. 2, Apr. 2001, pp. 199–217.

Hain, R. M. “The Hodge De Rham theory of relative Malcev completion.” *Annales Scientifiques De L’Ecole Normale Superieure*, vol. 31, no. 1, Jan. 1998, pp. 47–92. *Scopus*, doi:10.1016/S0012-9593(98)80018-9.
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Hain, R. "Infinitesimal presentations of the Torelli groups." *Journal of the American Mathematical Society* 10.3 (July 1, 1997): 597-651.

Elizondo, E. J., and R. M. Hain. “Chow varieties of Abelian varieties.” *Boletin De La Sociedad Matematica Mexicana*, vol. 2, no. 2, Dec. 1996, pp. 95–99.

Hain, R. M. “The existence of higher logarithms.” *Compositio Mathematica*, vol. 100, no. 3, Dec. 1996, pp. 247–76.