- Course of P. Gille and A. Pianzola
- Title: Torsors and infinite
dimensional Lie theory
- Abstract: Recently some
interesting connections have been discovered between non-abelian
Galois cohomology of Laurent polynomial rings on the one hand,
while on the other, a class of infinite dimensional Lie algebras
which, as rough approximations, can be thought off as higher
nullity analogues of the affine Kac-Moody Lie algebras.
Though the algebras in question are in general infinite
dimensional over the given base field (say the complex numbers),
they can be thought as being finite provided that the base
field is now replaced by a ring (in this case the centroid of
the algebras, which turns out to be a Laurent polynomial ring).
This leads us to the theory of reductive group schemes as
developed by M. Demazure and A. Grothendieck. Once this point of
view is taken, the language of torsors arise naturally. This
geometrical approach has lead to unexpected interplay between
infinite dimensional Lie theory and the theory of algebraic
groups, such as the work of Raghunathan and Ramanathan on torsors
over the affine space, isotriviality questions for Laurent
polynomial rings, Azumaya algebras, and Serre's Conjecture I and
II.
- Course of B. Totaro
- Title: The birational geometry of
quadrics
- Abstract: I will begin by describing the general theory of quadratic
forms over fields, as created by Witt in the 1930s and
enriched by Pfister in the 1960s. In particular, Pfister
defined "Pfister forms", the simplest of all quadratic
forms. An important role in Pfister's theory is played
by the field of rational functions on a quadric hypersurface.
These "function fields" were used even more fundamentally in the
1970s developments of quadratic form theory
by Arason-Pfister and Knebusch, as I will describe.
As a result of that work, it has become a central problem
in quadratic form theory to try to classify quadrics over a field
up to stable birational
equivalence. (Two varieties over a field are "birational" if their
function fields are isomorphic, and "stably birational"
if they become birational after multiplying by some projective
space.) A lot is known about stable birational equivalence
of quadrics, in a fairly large
range of dimensions. I will discuss many of the results and
methods, including the results of Izhboldin
and Karpenko.
Much less is known about the problem of classifying quadrics
up to birational (rather than stable birational) equivalence.
There is no general machinery available for this problem:
to show that two different quadrics are birational, we have
to write down a birational map by some clever formula.
I will describe the known results in this direction,
by Ahmad-Ohm, Roussey, and me. I will conclude with
Macdonald's geometric analysis of the most important
birational maps between quadrics.
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