I will discuss joint work with Ramesh Sreekantan on the essential dimension of abelian varieties over number fields. Using results of Bogomolov, Florence and Serre, we showed that it is infinite for non-trivial abelian varieties.

We find minimal elements in three different (but essentially equivalent) partially ordered categories:

(A) finite-volume, noncompact, complete, locally symmetric spaces of higher rank,
(B) nonuniform, irreducible lattices in semisimple Lie groups of higher real rank,
(C) isotropic, simple algebraic Q-groups of higher real rank.

The main interest is in categories (A) and (B), but the proof is carried out using the machinery of category (C).

I will discuss some joint work with Nicole Lemire on calculating the essential dimension of the moduli stack of SL_n-bundles over a smooth projective curve in characteristic zero. Our calculations rely heavily on the recent work of Brosnan-Reichstein-Vistoli. The other ingredient we need is the calculation of the Brauer group of the moduli space which is due to Balaji-Biswas-Gabber-Nagaraj. Both of these ingredients will be described in the talk.

A finite-dimensional representation of an algebraic group G gives a trace symmetric bilinear form on the Lie algebra of G. We give a criterion in terms of root systems for this form to vanish. As a corollary, we show that a Lie algebra of type E₈ over a field of characteristic 5 does not have a so-called "quotient trace form".

This is joint work with J-P. Serre.

I will describe a rather explicit parameter space for a class of Fourier transforms of orbital integrals in a p-adic Lie algebra. This allows to show that the values of Harish-Chandra characters of a class of supercuspidal representations of p-adic classical groups are controlled by a finite number of geometric objects defined over Q.

This is joint work with Clifton Cunningham and Loren Spice.

Dimension essentielle d'un objet algébrique est le nombre minimal de paramètres algébriquement indépendants qui sont nécessaires pour définir l'objet. Cette notion a été introduite par Buhler, Reichstein et Serre. Nous verrons des relations avec la géométrie algébrique cohomologie galoisienne, K-théorie algébrique et cycles algébriques en suivant des pas de la démonstration du résultat (en commun avec Merkurjev) qui dit que la dimension essentielle d'un p-groupe sur un corps contenant une racine primitive p-ième de l'unité coïncide avec le minimum des dimensions des représentations fidèles du groupe sur le corps.

In this talk I will discuss applications of the method of field patching, recently developed by Harbater and Hartmann, towards problems in quadratic forms and division algebras. In particular, I will discuss approaches to the computation of the u-invariant and approaches to the period-index problem for certain fields of the form K(C) where K is a complete discretely valued field and C is a curve over K.

This is joint work with Harbater and Hartmann.

Let G be a reductive group over an algebraically closed field of characteristic zero. We study the question when automorphisms of the categorical quotient V//G may be lifted to V where V is the representation given by several copies (generally at least two) of the adjoint representation of G. As an application we determine the way an automorphism of V//G may act on the Luna stratification of V//G, a question studied earlier jointly with Z. Reichstein in the case of matrices.

For a field F of characteristic 2, let n¢(F) be the smallest integer n such that (IF)ⁿ=0 (if this minimum exists), where IF is the ideal of even dimensional bilinear forms over F. In our talk, we give a complete answer to the behavior of the invariant n¢(F) under finite extensions of F. This completes a previous work by Aravire and Baeza where the same problem was considered for the n-invariant defined by the filtration (IF)ⁿ ÄW_q(F) = 0 of the Witt group W_q(F) of nonsingular quadratic forms over F.

I plan to discuss recent joint work with D. Benson, S. Chebolu, F. Chemotti, I. Efrat, N. Lemire, A. Schultz, and J. Swallow related to the Galois module structure of Galois cohomology and its influence on the structure of certain p-extensions of fields.

Many interesting objects in infinite dimensional Lie theory can be thought as being finite dimensional when viewed not as objects (algebras, groups...) over the base field (usually the complex numbers), but rather as objects over their centroids (usually a Laurent polynomial ring). From this point of view, the language of torsors and descent constructions arise naturally. I will present several examples of how these methods can be used to study a class of algebras intimately related to the affine Kac-Moody Lie algebras.

In the 60's, Pfister proved strong theorems describing quadratic forms of even dimension £ 12 that have trivial discriminant and Clifford invariant, i.e., that are in I³. His results have been extended to quadratic forms of dimension 14 in I³ by Rost in 1999. One knows also extensions of these theorems where quadratic forms are replaced by central simple algebras with orthogonal involution, except in degree 12. Using Rost's argument-that is the fact that a projective representation of Spin₁₂ has an open orbit-we produce a generalization of Pfister's result; the statement is in terms of `quadratic extensions' of algebras with involution.

This talk is based on a joint work with S. Garibaldi.

Let G be a reductive subgroup of a reductive group G¢. We are interested to the irreducible representation V(l) of G which occurs as subrepresentation of a given irreducible representation V(l¢) of G¢. More precisely, we consider the convex cone C generated by the pairs (l,l¢) as above. In fact, these cones have numerous interpretations and a rich history. Here, we will explain how Geometric Invariant Theory allows us to give an almost minimal list of linear inequalities which characterizes C.