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.