# Richard Hain

- Professor of Mathematics

**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

##### Geometry: Differential & Algebraic

##### Number Theory

##### Topology

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 2018

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

* Moduli Spaces of Riemann Surfaces.* Ed. B Farb, R Hain, and E Looijenga. American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 2013.

* Contemporary Trends in Algebraic Geometry and Algebraic Topology.* Ed. S-S Chern, L Fu, and R Hain. World Scientific Publishing Co., Inc., River Edge, NJ, 2002.
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* Mapping Class Groups and Moduli Spaces of Riemann Surfaces.* Ed. C-F Bödigheimer and R Hain. American Mathematical Society, Providence, RI, 1993.
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Hain, RM.* Iterated Integrals and Homotopy Periods.* Providence, RI: American Mathematical Society, 1984.
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Hain, R. "Deligne-Beilinson Cohomology of Affine Groups." *Hodge Theory and $L^2$-analysis.* Ed. L Ji. International Press, 2017. (Chapter)

Hain, R. "The Hodge-de Rham theory of modular groups." *Recent Advances in Hodge Theory Period Domains, Algebraic Cycles, and Arithmetic.* Ed. M Kerr and G Pearlstein. Cambridge University Press, January 31, 2016. 422-514. (Chapter)

Hain, R. "Normal Functions and the Geometry of Moduli Spaces of Curves." *Handbook of Moduli.* Ed. G Farkas and I Morrison. Somerville, MA: International Press, 2013. 527-578.

Hain, R. "Lectures on Moduli Spaces of Elliptic Curves." *Transformation Groups and Moduli Spaces of Curves: Advanced Lectures in Mathematics.* Ed. L Ji and ST Yau. Beijing: Higher Education Press, 2010. 95-166.

Hain, R. "Relative Weight Filtrations on Completions of Mapping Class Groups." *Groups of Diffeomorphisms: Advanced Studies in Pure Mathematics.* Mathematical Society of Japan, 2008. 309-368.

Hain, R. "Finiteness and Torelli Spaces." *Problems on Mapping Class Groups and Related Topics.* Ed. B Farb. Providence, RI: Amererican Mathematics Societty, 2006. 57-70.
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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.* Ed. D Jerison, G Lustig, B Mazur, T Mrowka, W Schmid, R Stanley, and ST Yau. Somerville, MA: International Press, 2003. 113-133.

Hain, R, and Matsumoto, M. "Tannakian Fundamental Groups Associated to Galois Groups." *Galois Groups and Fundamental Groups.* Ed. L Schneps. Cambridge: Cambridge Univ. Press, 2003. 183-216.

Hain, R. "Iterated Integrals and Algebraic Cycles: Examples and Prospects." *Contemporary Tends in Algebraic Geometry and Algebraic Topology.* River Edge, NJ: World Scientific Publishing, 2002. 55-118.
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Hain, R, and Tondeur, P. "The Life and Work of Kuo-Tsai Chen [ MR1046561 (91b:01072)]." *Contemporary trends in algebraic geometry and algebraic topology (Tianjin, 2000).* World Sci. Publ., River Edge, NJ, 2002. 251-266.
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## Pages

Arapura, D, Dimca, A, and Hain, R. "On the fundamental groups of normal varieties." *Communications in Contemporary Mathematics* 18.04 (August 2016): 1550065-1550065.
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Hain, R, and Matsumoto, M. "Universal Mixed Elliptic Motives (Submitted)." *Journal of the Institute of Mathematics of Jussieu* (2016).

Hain, R. "Genus 3 mapping class groups are not Kähler." *Journal of Topology* 8.1 (March 2015): 213-246.
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Dimca, A, Hain, R, and Papadima, S. "The Abelianization of the Johnson Kernel." *Journal of the European Mathematical Society (JEMS)* 16 (2014): 805-822.
Full Text Open Access Copy

Hain, R. "Remarks on non-abelian cohomology of proalgebraic groups." *Journal of Algebraic Geometry* 22.3 (March 21, 2013): 581-598.
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Hain, R. "Rational Points of Universal Curves." *Journal of the American Mathematical Society* 24 (2011): 709-769.
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Hain, R, and Matsumoto, M. "Relative Pro-$l$ Completions of Mapping Class Groups." *Journal of Algebra* 321 (2009): 3335-3374.
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Kim, M, and Hain, RM. "The Hyodo-Kato theorem for rational homotopy types." *Mathematical Research Letters* 12.2-3 (2005): 155-169.
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Hain, R, and Matsumoto, M. "Galois Actions on Fundamental Groups of Curves and the Cycle $C-C^-$." *Journal of the Institute of Mathematics of Jussieu* 4 (2005): 363-403.
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Kim, M, and Hain, RM. "A De Rham–Witt approach to crystalline rational homotopy theory." *Compositio Mathematica* 140.05 (September 2004): 1245-1276.
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## Pages

"Chen Memorial Volume." Ed. R Hain and P Tondeur. *Illinois Journal of Mathematics* 34 (1990). (Journal issue)

Hain, RM, and Zucker, S. "Truncations of Mixed Hodge Complexes." Hodge Theory. 1985 - 1985. Sant Cugat, Spain. Spring-Verlag, 1987. Full Text

Hain, RM. "Iterated Integrals and Mixed Hodge Structures on Homotopy Groups." Hodge Theory. 1985 - 1985. Sant Cugat, Spain. Berlin: Springer-Verlag, 1987. Full Text

Hain, RM. "Higher Albanese Manifolds." Hodge Theory. 1985 - 1985. Sant Cugat, Spain. Berlin: Springer-Verlag, 1987. Full Text

Hain, RM, and Zucker, S. "A Guide to Unipotent Variations of Mixed Hodge Structure." Hodge Theory. 1985 - 1985. Sant Cugat, Spain. Berlin: Springer-Verlag, 1987. Full Text

Hain, RM.* Iterated Integrals, Minimal Models and Rational Homotopy Theory.* 1980. (PhD Thesis)

## Pages

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