- Professor of Mathematics
- Managing Editor of the Duke Mathematical Journal
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.
Universal Teichmuller Motives awarded by National Science Foundation (Principal Investigator). 2014 to 2019
Park City Mathematics Institute awarded by Princeton University (Principal Investigator). 2011 to 2015
Applications of Topology to Arithmetic and Algebraic Geometry awarded by National Science Foundation (Principal Investigator). 2010 to 2013
Topology and motives associated to moduli spaces of curves awarded by National Science Foundation (Principal Investigator). 2007 to 2011
Hodge Theory, Galois Theory and the Topology of Moduli Spaces awarded by National Science Foundation (Principal Investigator). 2004 to 2007
Integrable Systems and Calibrated Geometry awarded by National Science Foundation (Principal Investigator). 2006
The Third DMJ/IMRN Conference awarded by National Science Foundation (Principal Investigator). 2004 to 2005
The Topology, Geometry awarded by National Science Foundation (Principal Investigator). 2001 to 2004
Modular Forms and Topology awarded by National Science Foundation (Principal Investigator). 1998 to 2002
DMJ/IMRN Conference awarded by National Science Foundation (Principal Investigator). 2001 to 2002
Moduli Spaces of Riemann Surfaces. Ed. B Farb, R Hain, and E Looijenga. American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 2013.
Contemporary Trends in Algebraic Geometry and Algebraic Topology.: World Scientific Publishing Co. Pte. Ltd., 2002. Full Text
Mapping Class Groups and Moduli Spaces of Riemann Surfaces.: American Mathematical Society, 1993. Full Text
Hain, R. "Deligne-Beilinson Cohomology of Affine Groups." Hodge Theory and $L^2$-analysis. Ed. L Ji. International Press, 2017. (Chapter)
Hain, R. "The Hodge-de Rham theory of modular groups." Recent Advances in Hodge Theory Period Domains, Algebraic Cycles, and Arithmetic. Ed. M Kerr and G Pearlstein. Cambridge University Press, January 31, 2016. 422-514. (Chapter)
Hain, R. "Normal Functions and the Geometry of Moduli Spaces of Curves." Handbook of Moduli. Ed. G Farkas and I Morrison. Somerville, MA: International Press, 2013. 527-578.
Hain, R. "Lectures on Moduli Spaces of Elliptic Curves." Transformation Groups and Moduli Spaces of Curves: Advanced Lectures in Mathematics. Ed. L Ji and ST Yau. Beijing: Higher Education Press, 2010. 95-166.
Hain, R. "Relative Weight Filtrations on Completions of Mapping Class Groups." Groups of Diffeomorphisms: Advanced Studies in Pure Mathematics. Mathematical Society of Japan, 2008. 309-368.
Hain, R, and Matsumoto, M. "Tannakian Fundamental Groups Associated to Galois Groups." Galois Groups and Fundamental Groups. Ed. L Schneps. Cambridge: Cambridge Univ. Press, 2003. 183-216.
Hain, R. "Periods of Limit Mixed Hodge Structures." CDM 2002: Current Developments in Mathematics in Honor of Wilfried Schmid & George Lusztig. Ed. D Jerison, G Lustig, B Mazur, T Mrowka, W Schmid, R Stanley, and ST Yau. Somerville, MA: International Press, 2003. 113-133.
Hain, R. "Iterated Integrals and Algebraic Cycles: Examples and Prospects." Contemporary Tends in Algebraic Geometry and Algebraic Topology. River Edge, NJ: World Scientific Publishing, 2002. 55-118. Full Text
Hain, R, and Tondeur, P. "The Life and Work of Kuo-Tsai Chen [ MR1046561 (91b:01072)]." Contemporary trends in algebraic geometry and algebraic topology (Tianjin, 2000). World Sci. Publ., River Edge, NJ, 2002. 251-266. Full Text
Brown, F, and Hain, R. "Algebraic de Rham theory for weakly holomorphic modular forms of level one." Algebra & Number Theory 12.3 (2018): 723-750. Full Text
Hain, R, and Matsumoto, M. "Universal Mixed Elliptic Motives (Submitted)." Journal of the Institute of Mathematics of Jussieu (2016).
Hain, R. "Remarks on non-abelian cohomology of proalgebraic groups." Journal of Algebraic Geometry 22.3 (March 21, 2013): 581-598. Full Text
Hain, R, and Matsumoto, M. "Relative pro-ℓ completions of mapping class groups." Journal of Algebra 321.11 (June 2009): 3335-3374. Full Text
Hain, R, and Matsumoto, M. "Galois Actions on Fundamental Groups of Curves and the Cycle $C-C^-$." Journal of the Institute of Mathematics of Jussieu 4 (2005): 363-403. Full Text
Kim, M, and Hain, RM. "The Hyodo-Kato theorem for rational homotopy types." Mathematical Research Letters 12.2-3 (2005): 155-169. Open Access Copy
Hain, RM. "Iterated Integrals and Mixed Hodge Structures on Homotopy Groups." Hodge Theory. 1985 - 1985. Sant Cugat, Spain. Berlin: Springer-Verlag, 1987. Full Text
Hain, RM, and Zucker, S. "A Guide to Unipotent Variations of Mixed Hodge Structure." Hodge Theory. 1985 - 1985. Sant Cugat, Spain. Berlin: Springer-Verlag, 1987. Full Text
Program committee (with Valery Alexeev, and François Loeser) for conference:. Local zeta functions and the arithmetic of moduli spaces: A conference in memory of Jun-Ichi Igusa. Johns Hopkins University. March 22, 2017 - March 26, 2017
(co-organizer with J. Burgos (Madrid), K. Ebrahimi-Fard (Madrid), H. Gangl (Durham, UK), J. Kramer (Berlin), O. Patashnick (Bristol, UK), L. Scneps (Paris)). Workshop on Multiple Zeta Values, Modular Forms and Elliptic Motives II. ICMAT, Madrid, Spa...
April 26 - 29, 2018 The goal of this conference is to bring young mathematicians together, both as speakers and as participants. The talks will cover an array of subject areas that are well-represented in the Duke Journal. There... read more »
The main result of this thesis is the construction of Massey triple products of Eisenstein series. Massey triple products are a generalization of the ordinary notion of multiplication; instead of multiplying two objects together, the Massey triple... read more »