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.  Contact Info:
Teaching (Fall 2016):
Teaching (Spring 2017):
 MATH 612.01, ALGEBRAIC TOPOLOGY II
Synopsis
 Physics 227, TuTh 01:25 PM02:40 PM
 Office Hours:
 3:00 to 4:00 MWF, or by appointment
 Education:
Ph.D.  University of Illinois  UrbanaChampaign  1980 
M.Sc.  Australian National University (Australia)  1977 
B.Sc. (hons)  University 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
 Curriculum Vitae
 Current Ph.D. Students
(Former Students)
 Postdocs Mentored
 Recent Publications
(More Publications)
(search)
 Arapura, D; Dimca, A; Hain, R, On the fundamental groups of normal varieties,
Communications in Contemporary Mathematics, vol. 18 no. 04
(August, 2016),
pp. 15500651550065, ISSN 02191997 [doi]
 R. Hain and Makoto Matsumoto, Universal mixed elliptic motives
(Submitted, December, 2015) [arxiv:1512.03975]
 Richard Hain, DeligneBeilinson cohomology of affine groups
(Submitted, July, 2015) [arXiv:1507.03144]
 Hain, R, Genus 3 Mapping Class Groups are not Kähler,
Journal of Topology, vol. 8 no. 1
(2015),
pp. 213246, Oxford University Press, ISSN 17538416 [arXiv:1305.2052], [2052]
 D. Arapura, A. Dimca, R. Hain, On the fundamental groups of normal varieties
(Submitted, December, 2014) [arXiv:1412.1483]
 Recent Grant Support
 Universal Teichmuller Motives, National Science Foundation, DMS1406420, 2014/082017/07.
 Park City Mathematics Institute, Princeton University, 745223104&5, 2011/122015/04.
 Conferences Organized
 Workshop on Multiple Zeta Values, Modular Forms and Elliptic Motives II, (with J. Burgos (Madrid), K. EbrahimiFard (Madrid), H. Gangl (Durham, UK), J. Kramer (Berlin), O. Patashnick (Bristol, UK), L. Scneps (Paris)), December 15, 2014
 organizer (coorganizers Benson Farb and Eduard Looijenga) : Moduli Spaces of Riemann Surfaces. July 3, 2011  July 23, 2011, organizer (coorganizers Benson Farb and Eduard Looijenga) : Moduli Spaces of Riemann Surfaces, July 323, 2011
 coorganizer (with Herbert Gangl) : Multiple zeta values, modular forms and elliptic motives. May 2, 2011  May 6, 2011, coorganizer (with Herbert Gangl) : Multiple zeta values, modular forms and elliptic motives, May 26, 2011
 (with Jonathan Wahl) The Third Duke Mathemtical Journal Conference, April 2325, 2004
 (with Jonathan Wahl) The Second Duke Mathematical Journal Conference, April 2729, 2001
 (with Jonathan Wahl) The Duke Math. Journal Conference, May 12, 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
 Topology of Complex Algebraic Varieties, CIRM, Luminy, France, May 30June 3, 2016.
 Moduli Spaces in Geometry, CIRM, Luminy, France, October 2630 2015.
 Workshop on Multiple Zeta Values, Modular Forms
and Elliptic Motives, II, ICMAT, Madrid, December 15, 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 1620, 2013.
 Texas Geometry and Topology Conference, Texas A&M University, October 1113, 2013.
 Recent advances in Hodge theory, Pacific Institute for the Mathematical Sciences, Vancouver BC, June 1020, 2013.
 Heights and moduli spaces, Lorentz Center, Leiden, Netherlands, June 1014, 2013.
 GrothendieckTeichmüller theory
and Multiple Zeta Values, Newton Institute, Cambridge, UK, April 812, 2013.
 Workshop on the Topology of Varieties, Luminy, France, June 48, 2012.
 Moduli Spaces of Riemann Surfaces, IAS/Park City Mathematics Institute, Park City, UT, July23, 2011.
 Motivic Fundamental Groups, Lorentz Center, University of Leiden, May 2327, 2011.
 Multiple zeta values, modular forms and elliptic motives, Heilbronn Institute, University of Bristol, UK, May 36, 2011.
