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.

Office Location: | 107 Physics Bldg, Durham, NC 27708 |

Office Phone: | (919) 660-2819 |

Email Address: | |

Web Pages: | http://fds.duke.edu/db/aas/math/faculty/hain/http://www.thecostofknowledge.com/ |

**Teaching (Fall 2021):**

- MATH 221.01,
*LINEAR ALGEBRA & APPLICA*Synopsis- Physics 119, TuTh 01:45 PM-03:00 PM

- MATH 721.01,
*LINEAR ALGEBRA & APPLICA*Synopsis- Physics 119, TuTh 01:45 PM-03:00 PM

- MATH 721.11,
*LINEAR ALGEBRA & APPLICA*Synopsis- Online ON, TuTh 01:45 PM-03:00 PM

**Office Hours:**- Monday and Wednesday 2 to 3, or by appointment

**Education:**Ph.D. University of Illinois, Urbana-Champaign 1980 M.Sc. Australian National University (Australia) 1977 B.Sc. (hons) University of Sydney (Australia) 1976

**Specialties:**-
Algebra

Topology

Geometry

**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

**Current Ph.D. Students**- Eric Hopper
- Ma Luo

**Postdocs Mentored**- Ding Ma (2017 - present)
- Gregory Pearlstein (2005 - 2006)
- Amir Jafari (2005 - 2007)
- Eric Katz (2004 - 2007)
- Owen Patashnick (2000-2003)
- Andreas Rosenschon (2000-2003)
- David Reed (1995-1998)

**Undergraduate Research Supervised**- Alex Pieloch (2016 - 2017)

Thesis:*The topology of moduli spaces of real algebraic curves* - Aaron Pollack (2007 - 2009)

Thesis:*Relations between derivations arising from modular forms* - Melanie Wood (2001 - 2003)

Thesis:*Invariants and relations on the action of Gal(Qbar/Q) on dessins d' enfant and pi_1^(P^1-{0,1,infty})* - Andrew Dittmer (1997 - 1999)

Thesis:*The circumradius and area of cyclic polygons* - Garrett Mitchener (1997 - 1999)

Thesis:*Lattices and sphere packing* - Sang Chin (1992 - 1993)

Thesis:*The action of the Torelli group on a 3-fold cover of a g-holed torus*

- Alex Pieloch (2016 - 2017)

**Recent Publications**(search)- Hain, R,
*Hodge theory of the Goldman bracket*, Geometry & Topology, vol. 24 no. 4 (November, 2020), pp. 1841-1906, Mathematical Sciences Publishers [doi] - Hain, R; Matsumoto, M,
*Universal Mixed Elliptic Motives*, Journal of the Institute of Mathematics of Jussieu, vol. 19 no. 3 (May, 2020), pp. 663-766 [arxiv:1512.03975], [doi] [abs] - Hain, R,
*Notes on the universal elliptic KZB connection*, Pure and Applied Mathematics Quarterly, vol. 16 no. 2 (January, 2020), pp. 229-312 [doi] [abs] - Brown, F; Hain, R,
*Algebraic de Rham theory for weakly holomorphic modular forms of level one*, Algebra & Number Theory, vol. 12 no. 3 (January, 2018), pp. 723-750 [doi] [abs] - Hain, R,
*Deligne-Beilinson Cohomology of Affine Groups*, in Hodge Theory and $L^2$-analysis, edited by Ji, L (2017), International Press, ISBN 1571463518 [arXiv:1507.03144] [abs]

- Hain, R,

**Recent Grant Support***Universal Teichmuller Motives*, National Science Foundation, DMS-1406420, 2014/08-2020/07.

**Conferences Organized**- The Fourth Duke Mathematical Journal Conference, Organizer, April 26-29, 2018
- Hot Topics Workshop:
*Galois Theory of Periods and Applications*, Co-organizer (with Francis Brown et al), March 27-31, 2017 *Local zeta functions and the arithmetic of moduli spaces: A conference in memory of Jun-Ichi Igusa*, Program committee, March 22-26, 2017- Workshop on Multiple Zeta Values, Modular Forms and Elliptic Motives II, (co-organizer with J. Burgos (Madrid), K. Ebrahimi-Fard (Madrid), H. Gangl (Durham, UK), J. Kramer (Berlin), O. Patashnick (Bristol, UK), L. Scneps (P, December 1-5, 2014
- Moduli Spaces of Riemann Surfaces, Organizer (co-organizers Benson Farb and Eduard Looijenga), July 3-23, 2011
- Multiple zeta values, modular forms and elliptic motives, Co-organizer (with Herbert Gangl), May 2-6, 2011
- (with Jonathan Wahl) The Third Duke Mathemtical Journal Conference, April 23-25, 2004
- (with Jonathan Wahl) The Second Duke Mathematical Journal Conference, April 27-29, 2001
- (with Jonathan Wahl) The Duke Math. Journal Conference, May 1-2, 1998
- Torellifest, a conference at Duke, March, 1996
- Mapping Class Groups and Moduli Spaces of Curves, Seattle, August 1991

- Patterns in Cohomology of Moduli Spaces, which is part of Clay Mathematics Conference and Workshops, Oxford University, September 29 to October 4, 2019.
- Master Class on Elliptic Motives, Stockholm University, May 20 to 24, 2019.
*Modular forms, periods and scattering amplitudes*, Institute for Theoretical Studies, ETH Zurich, February to April, 2019.-
*The Johnson homomorphism and its generalizations*, Seminaire Groupes de Lie et espaces des modules, University of Geneva, April 5, 2019. *Periods in Number Theory, Algebraic Geometry and Physics*, Hausdorff Institute of Mathematics, Bonn, January to April, 2018.*Johnson homomorphisms and related topics*, University of Tokyo, May 22-26, 2017.*GRT, MZVs and associators*, Les Diablerets, Switzerland, August 29-September 3, 2016.- Topology of Complex Algebraic Varieties, CIRM, Luminy, France, May 30-June 3, 2016.
- Moduli Spaces in Geometry, CIRM, Luminy, France, October 26-30 2015.
- Workshop on Multiple Zeta Values, Modular Forms and Elliptic Motives, II, ICMAT, Madrid, December 1-5, 2014.
- Course on
*Universal Mixed Elliptic Motives*at IHES, Paris, May, 2014.(web page) - Fundamental groups in arithmetic and algebraic geometry, De Giorgi Center, Pisa, Italy, December 16-20, 2013.
- Texas Geometry and Topology Conference, Texas A&M University, October 11-13, 2013.
- Recent advances in Hodge theory, Pacific Institute for the Mathematical Sciences, Vancouver BC, June 10-20, 2013.
- Heights and moduli spaces, Lorentz Center, Leiden, Netherlands, June 10-14, 2013.
- Grothendieck-Teichmüller theory and Multiple Zeta Values, Newton Institute, Cambridge, UK, April 8-12, 2013.
- Workshop on the Topology of Varieties, Luminy, France, June 4-8, 2012.
- Moduli Spaces of Riemann Surfaces, IAS/Park City Mathematics Institute, Park City, UT, July-23, 2011.
- Motivic Fundamental Groups, Lorentz Center, University of Leiden, May 23-27, 2011.
- Multiple zeta values, modular forms and elliptic motives, Heilbronn Institute, University of Bristol, UK, May 3-6, 2011.