Richard Hain, Professor

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:	+1 919 660 2819
Email Address:
Web Pages:	https://fds.duke.edu/db/aas/math/faculty/hain/ https://sites.math.duke.edu/~hain/talks/

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

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

Postdocs Mentored
Ding Ma (2017 - 2018)
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

Recent Publications (search)

Cox, D; Esnault, H; Hain, R; Harris, M; Ji, L; Saito, M-H; Saper, L, Remembering Steve Zucker, edited by Cox, D; Harris, M; Ji, L, Notices of the American Mathematical Society, vol. 68 no. 7 (August, 2021), pp. 1156-1172, American Mathematical Society
Hain, R, Hodge theory of the Turaev cobracket and the Kashiwara-Vergne problem, Journal of the European Mathematical Society, vol. 23 no. 12 (January, 2021), pp. 3889-3933 [doi] [abs]
Hain, R, Johnson homomorphisms, EMS Surveys in Mathematical Sciences, vol. 7 no. 1 (January, 2021), pp. 33-116 [doi] [abs]
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]

Recent Grant Support

RTG: Linked via L-functions: training versatile researchers across number theory, National Science Foundation, 2023/10-2028/09.
Topics in the topology, geometry and arithmetic of moduli spaces associated to curves, Simons Foundation, 2023/09-2028/08.

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
(co-organizer with J. Burgos (Madrid), K. Ebrahimi-Fard (Madrid), H. Gangl (Durham, UK), J. Kramer (Berlin), O. Patashnick (Bristol, UK), L. Scneps (P, (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
Organizer (co-organizers Benson Farb and Eduard Looijenga). Moduli Spaces of Riemann Surfaces. IAS/Park City Mathematics Institute. July 03, 2011 - Ju, Organizer (co-organizers Benson Farb and Eduard Looijenga), July 3-23, 2011
Co-organizer (with Herbert Gangl). Multiple zeta values, modular forms and elliptic motives. Heilbronn Institute, University of Bristol, UK. May 02, 2, 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

Recent and Future Conferences and Talks

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.