Mini Course 2009


10h-11h: Gille: Torsors and infinite dimensional Lie theory I
11h-1h15: coffee
11h15-12h15: Gille: Torsors and infinite dimensional Lie theory II
12h15-14h15: Lunch
14h15-15h15: Pianzola: Torsors and infinite dimensional Lie theory III
15h15-15h30: coffee
15h30-16h30: Pianzola: Torsors and infinite dimensional Lie theory IV

10h-11h: Totaro: The birational geometry of quadrics I
11h-1h15: coffee
11h15-12h15: Totaro: The birational geometry of quadrics II
12h15-14h15: Lunch
14h15-15h45: Pianzola: Torsors and infinite dimensional Lie theory V
15h45-16h: coffee
16h-17h: Zainoulline

9h30-11h: Gille: Torsors and infinite dimensional Lie theory VI
11h-11h15: coffee
11h15-12h45: Totaro: The birational geometry of quadrics III

The Wednesday afternoon is free!

9h30-11h: Totaro: The birational geometry of quadrics IV
11h-11h15: coffee
11h15-12h15: Wouters
12h15-14h15: Lunch
14h15-15h45: Pianzola: Torsors and infinite dimensional Lie theory VII
15h45-16h: coffee
16h-17h: Haution

9h30-10h30: Steinmetz
10h30-10h45: coffee
10h45-12h15: Gille: Torsors and infinite dimensional Lie theory VIII

Abstracts of courses:

  1. 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.

  2. 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.

Abstracts of talks:

  1. Talk of O. Haution

    • Title: Adams operations and the first Steenrod square
    • Abstract: I will discuss the interaction between the Adams operations and the topological filtration on Grothendieck groups of schemes. This gives rise to Steenrod operations on the reduced Chow groups of geometrically cellular varieties. It requires resolution of singularities, however one particular operation, the first Steenrod square, can be constructed without this hypothesis. I will provide an application to quadratic forms over fields of characteristic two.

  2. Talk of A. Steinmetz

    • Title: Torsors over the Laurent polynomial ring in two variables over C
    • Abstract: As will be seen in the course of Gille/Pianzola, certain infinite dimensional Lie algebras define torsors over this Laurent polynomial ring, called "loop torsors". I will show that in the case of groups of classical type of large enough rank, all torsors are in fact "loop torsors". We also obtain a variant of Serre's Conjecture II for such groups over this ring.

  3. Talk of T. Wouters

    • Title: Suslin's invariant in positive characteristic
    • Abstract: For a central simple k-algebra A, Suslin defined a cohomological invariant for SK_1(A) if the index of A is prime to the characteristic of A. Using Kato's logarithmic differentials and a lift from characteristic p to characteristic 0, I will explain how this invariant can be generalised to arbitrary characteristic.

  4. Talk of K. Zainoulline

    • Title: Towards motivic classification of algebras with orthogonal involutions
    • Abstract: We apply the generalized J-invariant of Vishik to classify algebras with orthogonal involutions. In particular, using the characteristic map for K_0 we relate the indices of the respective Tits algebras with the first values of the J-invariant.