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Richard Hain, Professor

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.

My primary collaborators are Francis Brown of Oxford University and Makoto Matsumoto of Hiroshima University.

Contact Info:
Office Location:  107
Office Phone:  (919) 660-2819
Email Address: send me a message
Web Page:  http://www.math.duke.edu/faculty/hain/

Teaching (Fall 2016):

  • MATH 333.01, COMPLEX ANALYSIS Synopsis
    Physics 235, TuTh 11:45 AM-01:00 PM
  • MATH 690-10.01, TOPOLOGY Synopsis
    Physics 205, TuTh 03:05 PM-04:20 PM
Office Hours:

2:30 to 3:30 MWF, or by appointment
Education:

Ph.D.University of Illinois -- Urbana-Champaign1980
M.Sc.Australian National University (Australia)1977
B.Sc. (hons)University Sydney Australia1976
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)

  1. D Arapura, A Dimca and R Hain, On the fundamental groups of normal varieties, Communications in Contemporary Mathematics, vol. 18 no. 04 (August, 2016), pp. 1550065-1550065, ISSN 0219-1997 [doi]
  2. R. Hain and Makoto Matsumoto, Universal mixed elliptic motives (Submitted, December, 2015) [arxiv:1512.03975]
  3. Richard Hain, Deligne-Beilinson cohomology of affine groups (Submitted, July, 2015) [arXiv:1507.03144]
  4. R Hain, Genus 3 Mapping Class Groups are not Kähler, Journal of Topology, vol. 8 no. 1 (2015), pp. 213-246, Oxford University Press, ISSN 1753-8416 [arXiv:1305.2052], [2052]
  5. 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, DMS-1406420, 2014/08-2017/07.      
  • Park City Mathematics Institute, Princeton University, 7452-2310-4&5, 2011/12-2015/04.      
Conferences Organized

Recent and Future Conferences and Talks

 

dept@math.duke.edu
ph: 919.660.2800
fax: 919.660.2821

Mathematics Department
Duke University, Box 90320
Durham, NC 27708-0320