Richard M Hain, Professor

Contact Info:

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

Teaching (Fall 2013):

• MATH 221.02, LINEAR ALGEBRA & APPLICA Synopsis

  Physics 119, TuTh 01:25 PM-02:40 PM

• MATH 501.01, ALGEBRAIC STRUCTURES I Synopsis

  Physics 205, TuTh 11:45 AM-01:00 PM

Office Hours:

3:00 to 4:30 Mondays and Wednesdays, 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:

Algebra
Topology
Geometry

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 the University of Tokyo.

Areas of Interest:

topology
algebraic geometry
arithmetic geometry

Curriculum Vitae

Current Ph.D. Students   (Former Students)

• Kevin Kordek  
• Tatsunari Watanabe  
• Anil Venkatesh  

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   (More Publications)

1. Richard Hain, Genus 3 Mapping Class Groups are not Kähler (Submitted, May, 2013) [arXiv:1305.2052]
2. Richard Hain, Normal Functions and the Geometry of Moduli Spaces of Curves, in Handbook of Moduli, edited by Gavril Farkas, Ian Morrison, vol. I (March, 2013), pp. 527-578, International Press [arXiv:1102.4031]
3. Richard Hain, Remarks on non-abelian cohomology of proalgebraic groups, J. Algebraic Geom., vol. 22 (2013), pp. 581-598 [arXiv:1009.3662]
4. Richard Hain, Monodromy of codimension-one sub-families of universal curves, Duke Math. Journal, vol. 161 no. 7 (June, 2012), pp. 1351-1378 [arXiv:1006.3785]
5. Alexandru Dimca, Richard Hain, Stefan Papadima, The abelianization of the Johnson kernel, Journal of the European Mathematical Society (Accepted, June, 2012) [arXiv:1101.1392]

Recent Grant Support

• Applications of Topology to Arithmetic and Algebraic Geometry, National Science Foundation, DMS-1005675, 2010/07-2013/06.      
• Park City Mathematics Institute, Princeton University, 7451-2302 & 7451-2310, 2010/04-2011/03.      

Conferences Organized

• Moduli Spaces of Riemann Surfaces, organizer (co-organizers Benson Farb and Eduard Looijenga), July 3-23, 2011  
• Multiple zeta values, modular forms and elliptic motives, co-organizer (with Herbert Gangl), May 2-6, 2011  
• (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  

• The main PCMI web site
• The upcoming summer program: Geometric Analysis
• Institute for Advanced Study (IAS)
• PCMI's mathematics education web pages on the Math Forum
• Math for America (MfA)

• Fundamental groups in arithmetic and algebraic geometry, De Giorgi Center, December 16-20, 2013.
• Recent advances in Hodge theory, Pacific Institute for the Mathematical Sciences, Vancouver BC, June 10-20, 2013.
• Heights and moduli spaces, Lorentz Center, Leiden, Netherlands, June 10-14, 2013.
• Grothendieck-Teichmüller theory and Multiple Zeta Values, Newton Institute, Cambridge, UK, April 8-12, 2013.
• Workshop on the Topology of Varieties, Luminy, France, June 4-8, 2012.
• Moduli Spaces of Riemann Surfaces, IAS/Park City Mathematics Institute, Park City, UT, July-23, 2011.
• Motivic Fundamental Groups, Lorentz Center, University of Leiden, May 23-27, 2011.
• Multiple zeta values, modular forms and elliptic motives, Heilbronn Institute, University of Bristol, UK, May 3-6, 2011.