Research Interests for Richard Hain
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
 Recent Publications
(search)
 Cox, D; Esnault, H; Hain, R; Harris, M; Ji, L; Saito, MH; 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. 11561172, American Mathematical Society
 Hain, R, Hodge theory of the Turaev cobracket and the KashiwaraVergne problem,
Journal of the European Mathematical Society, vol. 23 no. 12
(January, 2021),
pp. 38893933 [doi] [abs]
 Hain, R, Johnson homomorphisms,
Ems Surveys in Mathematical Sciences, vol. 7 no. 1
(January, 2021),
pp. 33116 [doi] [abs]
 Hain, R, Hodge theory of the Goldman bracket,
Geometry & Topology, vol. 24 no. 4
(November, 2020),
pp. 18411906, Mathematical Sciences Publishers [doi]
 Hain, R; Matsumoto, M, Universal Mixed Elliptic Motives,
Journal of the Institute of Mathematics of Jussieu, vol. 19 no. 3
(May, 2020),
pp. 663766 [arxiv:1512.03975], [doi] [abs]
