Richard M Hain, Professor

Office Location:	107 Physics Bldg
Office Phone:	(919) 660-2819, (919) 660-2800
Email Address:

Office Hours:

Mon & Wed, 3:00 to 4:00, or by
appointment

Education:

Ph.D.	University of Illinois, Urbana-Champaign	1980
M.Sc.	Australian National University	1977
B.Sc.(Hons)	University of Sydney	1976

Specialties:

Geometry
Topology

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.

Current Ph.D. Students  

Postdocs Mentored

• Gregory Pearlstein (2005 - 2006)  
• Amir Jafari (2005 - 2007)  
• Eric Katz (2004 - 2007)  
• Owen Patashnick (2000-2003)  
• Andreas Rosenschon (2000-2003)  
• David Reed (1995-1998)  

Recent Publications

1. Richard Hain, Relative weight filtrations on completions of mapping class groups, in Groups of Diffeomorphisms, Advanced Studies in Pure Mathematics, vol. XX (Accepted, May, 2008), Mathematical Society of Japan [arXiv:0802.0814]
2. Richard Hain, Makoto Matsumoto, Relative pro-l completions of mapping class groups (Submitted, February, 2008) [arXiv:0802.0806]
3. Richard Hain, Finiteness and Torelli Spaces, in Problems on Mapping Class Groups and Related Topics, Proc. Symp. Pure Math. 74, edited by Benson Farb (September, 2006), pp. 57-70, American Mathematical Society [math.GT/0508541]
4. Richard M. Hain and Makoto Matsumoto, Galois actions on fundamental groups of curves and the cycle C-C^-, J. Inst. Math. Jussieu, vol. 4 (2005), pp. 363-403 [math.NT/0306037]
5. Richard M. Hain, David Reed, On the Arakelov Geometry of Moduli Spaces of Curves, J. Differential Geom., vol. 67 (2004), pp. 195-228 [math.AG/0211097]

Recent Grant Support

• Topology and motives associated to moduli spaces of curves, NSF, DMS-0706955, 2007/08-2010/07.      
• Hodge Theory, Galois Theory and the Topology of Moduli Spaces, National Science Foundation, DMS-0405440, 2004/07-2007/06.      

Conferences Organized

• (with Jonathan Wahl) The Third Duke Mathemtical Journal Conference, April 23-25, 2004  
• (with Jonathan Wahl) The Second Duke Mathematical Journal Conference, April 27-29, 2001  
• (with Jonathan Wahl) The Duke Math. Journal Conference, May 1-2, 1998  
• Torellifest, a conference at Duke, March, 1996  
• Mapping Class Groups and Moduli Spaces of Curves, Seattle, August 1991  

• Non-Abelian Fundamental Groups in Arithmetic Geometry, Newton Institute, Cambridge University, July-December, 2009.
• Geometry of Teichmuller spaces and moduli spaces of curves, Zhejiang University, Hangzhou, China, July 14 to 20, 2008.
• Motifs et fonctions multizêtas, Séminaire de géométrie algébrique Rennes-Nantes-Paris, IHES, Paris, April 3, 2007
• Hodge Theory, BIRS, Banff, Canada, April 6-11, 2008.
• Moduli Spaces, University and Max Planck Institute for Mathematics, Bonn, Germany, January 7-11, 2008.
• Geometry and Lie Theory, Sydney University, Australia, July, 2007