# 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

Universal Teichmuller Motives awarded by National Science Foundation (Principal Investigator). 2014 to 2020

Park City Mathematics Institute awarded by Princeton University (Principal Investigator). 2011 to 2015

Applications of Topology to Arithmetic and Algebraic Geometry awarded by National Science Foundation (Principal Investigator). 2010 to 2013

Topology and motives associated to moduli spaces of curves awarded by National Science Foundation (Principal Investigator). 2007 to 2011

Hodge Theory, Galois Theory and the Topology of Moduli Spaces awarded by National Science Foundation (Principal Investigator). 2004 to 2007

Integrable Systems and Calibrated Geometry awarded by National Science Foundation (Principal Investigator). 2006

The Third DMJ/IMRN Conference awarded by National Science Foundation (Principal Investigator). 2004 to 2005

The Topology, Geometry awarded by National Science Foundation (Principal Investigator). 2001 to 2004

Modular Forms and Topology awarded by National Science Foundation (Principal Investigator). 1998 to 2002

DMJ/IMRN Conference awarded by National Science Foundation (Principal Investigator). 2001 to 2002

## Pages

Farb, B., et al., editors. *Moduli Spaces of Riemann Surfaces*. Vol. 20, American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 2013.

*Contemporary Trends in Algebraic Geometry and Algebraic Topology*. World Scientific Publishing Co. Pte. Ltd., 2002. *Crossref*, doi:10.1142/9789812777416.
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*Mapping Class Groups and Moduli Spaces of Riemann Surfaces*. American Mathematical Society, 1993. *Crossref*, doi:10.1090/conm/150.
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Hain, R. M. *Iterated Integrals and Homotopy Periods*. American Mathematical Society, 1984, pp. iv–98. *Manual*, doi:10.1090/memo/0291.
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Hain, R. “Deligne-Beilinson Cohomology of Affine Groups.” *Hodge Theory and $L^2$-Analysis*, edited by L. Ji, International Press, 2017.

Hain, R. “The Hodge-de Rham theory of modular groups.” *Recent Advances in Hodge Theory Period Domains, Algebraic Cycles, and Arithmetic*, edited by M. Kerr and G. Pearlstein, vol. 427, Cambridge University Press, 2016, pp. 422–514.

Hain, R. “Normal Functions and the Geometry of Moduli Spaces of Curves.” *Handbook of Moduli*, edited by G. Farkas and I. Morrison, vol. 1, International Press, 2013, pp. 527–78.

Hain, R. “Lectures on Moduli Spaces of Elliptic Curves.” *Transformation Groups and Moduli Spaces of Curves: Advanced Lectures in Mathematics*, edited by L. Ji and S. T. Yau, vol. 16, Higher Education Press, 2010, pp. 95–166.

Hain, R. “Relative Weight Filtrations on Completions of Mapping Class Groups.” *Groups of Diffeomorphisms: Advanced Studies in Pure Mathematics*, vol. 52, Mathematical Society of Japan, 2008, pp. 309–68.

Hain, R. “Finiteness and Torelli Spaces.” *Problems on Mapping Class Groups and Related Topics*, edited by B. Farb, vol. 74, Amererican Mathematics Societty, 2006, pp. 57–70. *Manual*, doi:10.1090/pspum/074/2264131.
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Hain, R. “Periods of Limit Mixed Hodge Structures.” *CDM 2002: Current Developments in Mathematics in Honor of Wilfried Schmid & George Lusztig*, edited by D. Jerison et al., International Press, 2003, pp. 113–33.

Hain, R., and M. Matsumoto. “Tannakian Fundamental Groups Associated to Galois Groups.” *Galois Groups and Fundamental Groups*, edited by L. Schneps, vol. 41, Cambridge Univ. Press, 2003, pp. 183–216.

Hain, Richard, and Philippe Tondeur. “THE LIFE AND WORK OF KUO-TSAI CHEN.” *Contemporary Trends in Algebraic Geometry and Algebraic Topology*, WORLD SCIENTIFIC, 2002, pp. 251–66. *Crossref*, doi:10.1142/9789812777416_0012.
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Hain, R. “Iterated Integrals and Algebraic Cycles: Examples and Prospects.” *Contemporary Tends in Algebraic Geometry and Algebraic Topology*, vol. 5, World Scientific Publishing, 2002, pp. 55–118. *Manual*, doi:10.1142/9789812777416_0004.
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## Pages

Brown, F., and R. Hain. “Algebraic de Rham theory for weakly holomorphic modular forms of level one.” *Algebra and Number Theory*, vol. 12, no. 3, Jan. 2018, pp. 723–50. *Scopus*, doi:10.2140/ant.2018.12.723.
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Arapura, D., et al. “On the fundamental groups of normal varieties.” *Communications in Contemporary Mathematics*, vol. 18, no. 4, Aug. 2016. *Scopus*, doi:10.1142/S0219199715500650.
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Hain, R., and M. Matsumoto. “Universal Mixed Elliptic Motives.” *Journal of the Institute of Mathematics of Jussieu*, 2016.

Hain, Richard. “Genus 3 mapping class groups are not Kähler.” *Journal of Topology*, vol. 8, no. 1, Wiley, Mar. 2015, pp. 213–46. *Crossref*, doi:10.1112/jtopol/jtu020.
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Dimca, A., et al. “The abelianization of the Johnson kernel.” *Journal of the European Mathematical Society*, vol. 16, no. 4, Jan. 2014, pp. 805–22. *Scopus*, doi:10.4171/JEMS/447.
Full Text Open Access Copy

Hain, Richard. “Remarks on non-abelian cohomology of proalgebraic groups.” *Journal of Algebraic Geometry*, vol. 22, no. 3, American Mathematical Society (AMS), Mar. 2013, pp. 581–98. *Crossref*, doi:10.1090/s1056-3911-2013-00598-6.
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Hain, R. “Rational Points of Universal Curves.” *Journal of the American Mathematical Society*, vol. 24, 2011, pp. 709–69. *Manual*, doi:10.1090/S0894-0347-2011-00693-0.
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Hain, R., and M. Matsumoto. “Relative Pro-$l$ Completions of Mapping Class Groups.” *Journal of Algebra*, vol. 321, 2009, pp. 3335–74. *Manual*, doi:10.1016/j.jalgebra.2009.02.014.
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Kim, M., and R. M. Hain. “The Hyodo-Kato theorem for rational homotopy types.” *Mathematical Research Letters*, vol. 12, no. 2–3, Mar. 2005, pp. 155–69.
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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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## Pages

Hain, R., and P. Tondeur, editors. “Chen Memorial Volume.” *Illinois Journal of Mathematics*, vol. 34, 1990.

Hain, R. M., and S. Zucker. “Truncations of Mixed Hodge Complexes.” *Proceedings of the U.S. Spain Workshop*, vol. 1246, Spring-Verlag, 1987. *Manual*, doi:10.1007/BFb0077533.
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Hain, R. M. “Iterated Integrals and Mixed Hodge Structures on Homotopy Groups.” *Proceedings of the U.S. Spain Workshop*, vol. 1246, Springer-Verlag, 1987, pp. 75–83. *Manual*, doi:10.1007/BFb0077530.
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Hain, R. M. “Higher Albanese Manifolds.” *Proceedings of the U.S. Spain Workshop*, vol. 1246, Springer-Verlag, 1987. *Manual*, doi:10.1007/BFb0077531.
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Hain, R. M., and S. Zucker. “A Guide to Unipotent Variations of Mixed Hodge Structure.” *Proceedings of the U.S. Spain Workshop*, vol. 1246, Springer-Verlag, 1987. *Manual*, doi:10.1007/BFb0077532.
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Hain, R. M. *Iterated Integrals, Minimal Models and Rational Homotopy Theory*. 1980.

Hain, R. *The de Rham homotopy theory of complex algebraic varieties*. 1984.

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The Department of Mathematics, Duke University, and Duke University Press are pleased to announce that Richard Hain has been appointed managing editor for the *...* read more »

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Six graduate students participated in the May 15, 2016 graduation ceremonies to celebrate earning their PhDs in Mathematics. Their thesis topics were impressive and varied, and reflected the breadth of study in the department. Their advisors and... read more »