Math @ Duke

Richard Hain, Professor Emeritus
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.
 Contact Info:
 Office Hours:
 Monday and Wednesday 2 to 3, or by appointment
 Education:
Ph.D.  University of Illinois, UrbanaChampaign  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
 Curriculum Vitae
 Current Ph.D. Students
(Former Students)
 Postdocs Mentored
 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 3fold cover of a gholed torus
 Recent Publications
(More Publications)
(search)
 Cox, D; Esnault, H; Hain, R; Harris, M; Ji, L; Saito, MH; 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. 11561172, American Mathematical Society
 Hain, R, Hodge theory of the Turaev cobracket and the KashiwaraVergne problem,
Journal of the European Mathematical Society, vol. 23 no. 12
(January, 2021),
pp. 38893933 [doi] [abs]
 Hain, R, Johnson homomorphisms,
EMS Surveys in Mathematical Sciences, vol. 7 no. 1
(January, 2021),
pp. 33116 [doi] [abs]
 Hain, R, Hodge theory of the Goldman bracket,
Geometry & Topology, vol. 24 no. 4
(November, 2020),
pp. 18411906, 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. 663766 [arxiv:1512.03975], [doi] [abs]
 Recent Grant Support
 RTG: Linked via Lfunctions: training versatile researchers across number theory, National Science Foundation, 2023/102028/09.
 Topics in the topology, geometry and arithmetic of moduli spaces associated to curves, Simons Foundation, 2023/092028/08.
 Conferences Organized
 The Fourth Duke Mathematical Journal Conference, Organizer, April 2629, 2018
 Hot Topics Workshop: Galois Theory of Periods and Applications, Coorganizer (with Francis Brown et al), March 2731, 2017
 Local zeta functions and the arithmetic of moduli spaces: A conference in memory of JunIchi Igusa, Program committee, March 2226, 2017
 (coorganizer with J. Burgos (Madrid), K. EbrahimiFard (Madrid), H. Gangl (Durham, UK), J. Kramer (Berlin), O. Patashnick (Bristol, UK), L. Scneps (P, (coorganizer with J. Burgos (Madrid), K. EbrahimiFard (Madrid), H. Gangl (Durham, UK), J. Kramer (Berlin), O. Patashnick (Bristol, UK), L. Scneps (P, December 15, 2014
 Organizer (coorganizers Benson Farb and Eduard Looijenga). Moduli Spaces of Riemann Surfaces. IAS/Park City Mathematics Institute. July 03, 2011  Ju, Organizer (coorganizers Benson Farb and Eduard Looijenga), July 323, 2011
 Coorganizer (with Herbert Gangl). Multiple zeta values, modular forms and elliptic motives. Heilbronn Institute, University of Bristol, UK. May 02, 2, Coorganizer (with Herbert Gangl), 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
 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 2226, 2017.
 GRT, MZVs and associators, Les Diablerets, Switzerland, August 29September 3, 2016.
 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.


dept@math.duke.edu
ph: 919.660.2800
fax: 919.660.2821
 
Mathematics Department
Duke University, Box 90320
Durham, NC 277080320

