Mini Course Math 378.02: DeRham Theory of the Fundamental Group

Spring 2010

This mini course will be an introduction to K.-T. Chen's de Rham theory of the fundamental group. In plain language, I will explain how one can use differential forms to detect information about non-abelian quotients of the fundamental group of a manifold. Applications will be given to Multiple Zeta Values and, if time permits, to Manin's non-abelian modular symbols of classical modular forms.

Topics to be covered include:

• review of the classical de Rham Theorem
• Chen's iterated line integrals
• Chen's de Rham Theorem for the fundamental group
• multiple zeta values --- shuffle relations
• Kontsevich's integral formula --- shuffle relations

• relative unipotent completion of SL₂(Z)
• a brief introduction to classical modular forms
• Manin's iterated Shimura integrals

1. R. Hain: The geometry of the mixed Hodge structure on the fundamental group, Algebraic Geometry, 1985, Proc. Symp. Pure Math. 46 (1987), 247-282. (pdf)
2. R. Hain: Lectures of the Hodge-de Rham Theory of the Fundamental Group of P¹ - {0,1,infty}, Arizona Winter School, 2005. (pdf)
3. Y. Manin: Iterated integrals of modular forms and noncommutative modular symbols, Algebraic geometry and number theory, 565-597, Progr. Math., 253, Birkhauser, 2006. (math.NT/0502576/)
4. Y. Manin: Iterated Shimura integrals, Mosc. Math. J. 5 (2005), 869-881, 973. (math.NT/0507438)