- Professor of Mathematics
Research Areas and Keywords
Algebra & Combinatorics
Geometry: Differential & Algebraic
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.
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. "NIL-MANIFOLDS AS LINKS OF ISOLATED SINGULARITIES." COMPOSITIO MATHEMATICA 84.1 (October 1992): 91-99.
Hain, R. "Biextensions and heights associated to curves of odd genus." Duke Mathematical Journal 61.3 (December 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.
HAIN, R, and TONDEUR, P. "THE LIFE AND WORK OF CHEN,KUO,TSAI." ILLINOIS JOURNAL OF MATHEMATICS 34.2 (1990): 175-190.
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 (1987): 481-497. Full Text
Hain, RM. "The de rham homotopy theory of complex algebraic varieties I." K-Theory 1.3 (1987): 271-324. Full Text
HAIN, RM. "ON THE INDECOMPOSABLE ELEMENTS OF THE BAR CONSTRUCTION." PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 98.2 (October 1986): 312-316. Full Text