Professor Wilberd van der Kallen is visting IMSc during the summer of 2013 (tentative dates of visit: 15 June to 15 July; 01 August to 31 August). He is giving a series of ten lectures on cohomological finite generation during his visit.
Tentative schedule of the lectures: June 18 (Tue), 19 (Wed), 21 (Fri),
24 (Mon), 27 (Thu), July 3 (Wed), 5 (Fri), 8 (Mon), 09 (Tue), 12 (Fri).
Time: 1100am on all days; with a break for tea in between, each lecture is expected to last 75--90 minutes (not counting the break time)
Exception: On the first day (18th June) the lecture will start at 1130am.
The lectures will be centered around the following theorem: if a reductive algebraic group, defined over a field of positive characteristic, acts algebraically on a commutative finitely generated algebra, then the cohomology algebra of the group with coefficients in the algebra is finitely generated as an algebra over the field. We will encounter many themes such as good filtrations, strict polynomial bifunctors, Frobenius twist, Schur functors, equivariant vector bundles, spectral sequences associated with the Grosshans deformation, ...
The audience is expected to be familiar with algebraic groups and the classification of irreducible representations by highest weight. The idea is not so much to provide a proof of the cohomological finite generation theorem. Rather, the theorem and its proof will serve to motivate a selection of topics. We will often take facts for granted, those from algebraic geometry for instance. We will spend time on later developments that shed a new light on the existence of the universal classes of Touzé.
The introduction to the following paper is a good place to start reading: A. Touzé and Wilberd van der Kallen, Bifunctor cohomology and cohomological finite generation, Duke Mathematical Journal Vol. 151 (2010), 251-278
The first half of the following article (pdf or tif) provides a quick introduction to relevant representation theory of reductive groups in positive characteristic: Olivier Mathieu, Tilting modules and their applications. Analysis on homogeneous spaces and representation theory of Lie groups, Okayama-Kyoto (1997), 145-212, Adv. Stud. Pure Math., 26, Math. Soc. Japan, Tokyo, 2000. MR1770721
For an introduction to strict polynomial bifunctors and the formality problem, see this paper: Wilberd van der Kallen, Nantes lectures on bifunctors and CFG.
We will need the spectral sequence of a filtered complex. We use a little bit about derived categories. For that it would help to know about the homotopy category of (co)chain complexes. See for instance the book by Weibel on homological algebra.
For prerequisite homological algebra, one may want to see these notes by Touzé.
Concurrent to the lecture course, there will be other talks addressed mainly at graduate students on background and supplementary material. Those interested may look out for the following list that will be updated frequently:
Organizers: Upendra Kulkarni upendra-at-cmi-dot-ac-dot-in and K N Raghavan knr-at-imsc-dot-res-dot-in