Richard Hain

  • Professor of Mathematics
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. "On the indecomposable elements of the bar construction." Proceedings of the American Mathematical Society 98.2 (February 1, 1986): 312-312. Full Text

Hain, RM. "Mixed Hodge structures on homotopy groups." Bulletin of the American Mathematical Society 14.1 (January 1, 1986): 111-115. Full Text

Hain, RM. "On a generalization of Hilbert's 21st problem." Annales scientifiques de l'École normale supérieure 19.4 (1986): 609-627. Full Text


Duchamp, T, and Hain, RM. "Primitive Elements in Rings of Holomorphic Functions." Journal für die Reine und Angewandte Mathematik. [Crelle’s Journal] 346 (1984): 199-220. Full Text

Hain, RM. "Twisting Cochains and Duality Between Minimal Algebras and Minimal Lie Algebras." Transactions of the American Mathematical Society 277 (1983): 397-411. Full Text

Hain, RM. "A Characterization of Smooth Functions Defined on a Banach Space." Proceedings of the American Mathematical Society 77 (1979): 63-67. (Academic Article) Full Text

Eades, P, and Hain, RM. "On Circulant Weighing Matrices." Ars Combinatoria 2 (1976): 265-284.

Hain, R. "Notes on the Universal Elliptic KZB Equation (Accepted)." Pure and Applied Mathematics Quarterly 12.2.