The topic of my lectures will be:

An introduction to the mixed Hodge structure on the fundamental group
of a smooth variety.

In these, I will:

* give an introduction to mixed Hodge structures (MHSs) and their
extensions (unless you tell me that this is unnecessary).

* give an introduction to Chen's iterated integrals and prove his pi_1
de Rham theorem. This will include a discussion of polylogs.

* sketch the construction of the MHS on the fundamental group of a smooth
quasi-projective variety, and give the construction in detail in the case 
of an open subset of P^n.

* give a very brief introduction to limits MHS on fundamental groups --- 
mainly focused on P^1 - {0,1,infty}. This will include a discussion of
asymptotic base points.

* give a computation of the extensions that appear in the limit MHS on the
polylog variation of MHS over P^1 -{0,1,infty}. This will include a 
discussion of the Drinfeld associator and mixed zeta numbers.


A good introductory reference is:

Richard M. Hain, The geometry of the mixed Hodge structure on the 
fundamental group, Algebraic Geometry, 1985, Proc. Symp. Pure Math. 46 
(1987), 247--282 .


Other references include:

Richard M. Hain, Periods of Limit Mixed Hodge Structures, in CDM 2002: 
Current Developments in Mathematics in Honor of Wilfried Schmid & George 
Lusztig, edited by David Jerison, George Lustig, Barry Mazur, Tom Mrowka, 
Wilfried Schmid, Richard Stanley & S.-T. Yau (2003), International Press 
[math.AG/0305090].

Richard M. Hain, Iterated Integrals and Algebraic Cycles: Examples and 
Prospects, Contemporary Trends in Algebraic Geometry and Algebraic 
Topology , Nankai Tracts in Mathematics, vol. 5, World Scientific, 2002 
[math.AG/0109204].

Richard M. Hain, Classical Polylogarithms, Motives, Proc. Symp. Pure Math. 
55 part 2 (1994), 3--42 .


Idea for project:

Compute in detail the limit MHS on J/J^4, where J denotes the augmentation
ideal of Q pi_1(P^1-{0,1,infty},10) and 01 denotes the tangent vector
-del/del z at 1 in the direction of 0 given by the natural parameter z on
P^1 satisfying z(1) = 1 and z(0) = 0.