- 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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