Abstracts

Mini Course 2012

Courses:

  1. Course of B. Kahn:

    On expliquera ce que la théorie des motifs de Voevodsky a à dire sur la partie non ramifiée des foncteurs usuels de la géométrie algébrique. En particulier, on étudiera des "foncteurs dérivés de la cohomologie non ramifiée" et leur lien avec des invariants classiques: groupe de Griffiths et (2,1)-cycles indécomposables. Il s'agit en grande partie d'un travail en collaboration avec R. Sujatha. On expliquera aussi le lien entre les motifs de Voevodsky et la R-équivalence sur les tores ou les variétés semi-abéliennes.

  2. Course of R. Parimala:

    In this series of lectures, we shall describe progress towards Conjecture II of Serre which asserts that every principal homogeneous space under a semisimple simply connected linear algebraic group over a perfect field of cohomological dimension at most 2 has a rational point.

  3. Course of H. Petersson:

    Originally designed in the early nineteen-thirties as a tool to understand the foundations of quantum mechanics, Jordan algebras in the intervening decades have grown into a full- edged mathematical theory, with profound applications to various branches of algebra, analysis, and geometry. Cubic Jordan algebras form an important subclass whose significance comes to the fore through the connection with exceptional algebraic groups. In order to exploit this connection to the fullest, a thorough understanding of cubic Jordan algebras is indispensable. The primary purpose of my lecture will be to lay the foundations for such an understanding. More specifically, it will be shown that the main concepts of the theory can be investigated over arbitrary commutative rings. Moreover, a novel approach to the two Tits constructions of cubic Jordan algebras will be presented that works in this generality and yields new insights even when the base ring is a field.

Talks:

  1. Talk of J. Ducoat:

    Let W be a finite Coxeter group. A cohomological invariant of W is a morphism of functors between the first cohomology set functor $H^1(./k_0,W)$ and the Galois cohomology functor $H^i(./k_0,C)$, where $C$ is a finite $\Gamma_{k_0}$-module. We will state a general vanishing principle for the cohomological invariants of W over a base field of characteristic zero and then use it to describe all the cohomological invariants in the particular case of the Weyl groups of classical type.

  2. Talk of Y. Hu:

    Let K be a field of one of the following two types: 1. (the arithmetic case) the function field of an integral algebraic curve a p-adic number field; 2. (the local henselian case) the fraction field of a 2-dimensional henselian excellent local domain with finite residue field. In the arithmetic case, Colliot-Thélène, Parimala and Suresh made two conjectures on the Hasse principle with respect to discrete valuations of K for the existence of rational points on homogeneous spaces under connected linear algebraic groups over K: one for projective homogeneous spaces and another for torsors under semisimple simply connected groups. They proved the Hasse principle for quadratic forms of rank at least 3 and for torsors under quasi-split semisimple simply connected groups without E_8 factors. In this talk, we will discuss the Hasse principle for quadratic forms in the local henselian case as well as the Hasse principle for torsors under some non quasi-split semisimple simply connected groups in both cases.

  3. Talk of R. Lötscher:

    The well-known fiber dimension theorem in algebraic geometry says that for every morphism f : X -> Y of integral schemes of finite type, the dimension of each nonempty fiber of f is at least dim X - dim Y. We will discuss two analogues of this theorem, where schemes are replaced by categories fibered in groupoids and dimension is replaced by essential or canonical dimension. These analogues will be applied in particular to derive new relations between the essential dimension and canonical dimension of group schemes.

  4. Talk of S. Scully:

    An (integral) algebraic variety is called compressible if it admits a degree 0 rational endomorphism. Any variety which has a rational point is compressible, but there are several examples of varieties over non-algebraically closed fields for which this property fails to hold. For instance, it does not hold for any smooth anisotropic quadric with first Witt index equal to 1 (M. Rost, A. Vishik, N. Karpenko). N. Karpenko and A. Merkurjev showed moreover that these quadrics have a stronger property: so-called strong 2-incompressibility. An extension of this result to non-smooth quadrics was later proved by B. Totaro, including the special case of quadrics which have no smooth points at all. Quadrics of the latter type belong to a larger class of nowhere smooth varieties which we call quasilinear hypersurfaces. I will describe an analogue of the Karpenko-Merkurjev theorem for quasilinear hypersurfaces, and discuss some of its applications.