Richard Hain

Richard Hain
  • Professor of Mathematics
  • Managing Editor of the Duke Mathematical Journal
External address: 107 Physics Bldg, Durham, NC 27708
Internal office address: Box 90320, Durham, NC 27708-0320
Phone: (919) 660-2819

Research Areas and Keywords

Algebra & Combinatorics
algebraic geometry
Geometry: Differential & Algebraic
algebraic geometry, Hodge theory, arithmetic geometry, topology of varieties
Number Theory
topology of varieties, mapping class groups

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.

Education & Training
  • Ph.D., University of Illinois -- Urbana-Champaign 1980

  • M.Sc., Australian National University (Australia) 1977

  • B.Sc. (hons), University Sydney Australia 1976

Hain, RM. "The Geometry of the Mixed Hodge Structure on the Fundamental Group." Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985). Providence, RI.: American Mathematical Society. 1987. 247-282.


Hain, RM. "Completions of Mapping Class Groups and the Cycle C-C." Contemporary Mathematics 150 (1993): 75-105. Full Text


Hain, R, and Tondeur, P. "The life and work of Kuo-Tsai Chen." Illinois Journal of Mathematics 34.2 (June 1990): 175-190. Full Text

Hain, R. "Biextensions and heights associated to curves of odd genus." Duke Mathematical Journal 61.3 (January 1, 1990): 859-898. Full Text

Hain, R, and Tondeur, P. "The life and work of Kuo-Tsai Chen." Illinois Journal of Mathematics 34.2 (January 1, 1990): 175-190.

Hain, RM, and MacPherson, R. "Higher logarithms." Illinois Journal of Mathematics 34.2 (January 1, 1990): 392-475.

Durfee, AH, and Hain, RM. "Mixed Hodge Structures on the Homotopy of Links." Mathematische Annalen 280 (1988): 69-83. Full Text

Hain, RM. "The de Rham homotopy theory of complex algebraic varieties II." K Theory 1.5 (September 1, 1987): 481-497. Full Text

Hain, RM. "The de rham homotopy theory of complex algebraic varieties I." K Theory 1.3 (May 1, 1987): 271-324. Full Text

Hain, RM, and Zucker, S. "Unipotent variations of mixed Hodge structure." Inventiones Mathematicae 88.1 (February 1, 1987): 83-124. Full Text