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Richard Hain, Professor Emeritus

Richard Hain

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 Location:  107 Physics Bldg, Durham, NC 27708
Email Address: send me a message
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-Champaign1980
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 3-fold cover of a g-holed torus 
    Recent Publications   (More Publications)   (search)

    1. 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
    2. 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]
    3. Hain, R, Johnson homomorphisms, EMS Surveys in Mathematical Sciences, vol. 7 no. 1 (January, 2021), pp. 33-116 [doi]  [abs]
    4. Hain, R, Hodge theory of the Goldman bracket, Geometry & Topology, vol. 24 no. 4 (November, 2020), pp. 1841-1906, Mathematical Sciences Publishers [doi]
    5. 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

    Recent and Future Conferences and Talks

     

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    ph: 919.660.2800
    fax: 919.660.2821

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