Algebraic groups and homology theories – GATHO

Development of new motivic categories (birational motives, Chow-Witt motives).
Equivariant cohomology theories.
Decomposition of Chow motives, and application to the isotropy of various algebraic structures.
Development of a theory of descent for quadratic forms.

The project having only started a few months ago, it is too early to expect results in the form of publications.
A conference is being organized (mini-course in Lens, June 2013).
The project funds the participation of several of its members in the thematic program «Torsors, Nonassociative Algebras and Cohomological Invariants«, at the Fields Institute, Canada

A doctoral student has been recruited in Lens, under the direction of B. Calmès, starting in Sept. 2013.

See:
bcalmes.perso.math.cnrs.fr/gatho/presentation/
No patent

Looking for symmetries of a geometric objet or of a problem is now a very classical approach in mathematics. In algebraic geometry, it appears as the study of algebraic groups and of their actions on varieties. Interesting such varieties often parametrize structures associated to known algebraic objects, such as central simple algebras or quadratic forms, and any progress towards the understanding of the geometry of these varieties translates as new results on these structures.

A possible technique to obtain some results is to use cohomological invariants, in a wide sense, i.e. abelian groups or more generally categories, naturally associated to the objects under study, suitable to distinguish them or at least to extract certain properties. Chow groups, algebraic K-theory, algebraic cobordism or motivic cohomology are examples of such cohomological theories, and their categorical aspects lie in the various motivic categories or the stable homotopy category of schemes.

The scientific program is divided in several parts. The first one is to contribute to the foundation and development of cohomological or categorical tools, a rapidly growing area. The second one is to understand and compute these tools in the area of algebraic groups. The third one is to use these tools to solve classical problems in algebra, algebraic geometry or number theory. These problems retroactively illuminate and point towards aspects of the techniques that need to be developed. One of the goals of the project is to group specialists of each of the designated parts to make substantial progress in general.

Here is a detailed list of research themes:

Homotopy of schemes: oriented theories and equivariant ones in the homotopy and motivic areas. Classifying spaces and their cohomologies. Link with Witt groups and Chow-Witt groups of schemes.

Chow motives: Decomposition of the motives of projective homogeneous varieties, study of their upper motives. Links with essential dimension, with the splitting of central simple algebras and with isotropy of involutions.

Rationality: Fields of functions of group varieties, Noether's problem for connected linear groups, birational motives.

Number theory: obstruction to strong approximation for points over the integers of families of homogeneous spaces, local-global principle on various fields, weak approximation on homogeneous spaces over global fields, R-equivalence, Hasse principle.

Thus, summarizing, the general theme of the project is the study of algebraic groups and their actions on varieties, mostly by cohomological and categorical methods, partly to be developed, followed by applications in algebra, geometry or number theory. Within this project, to disseminate our results, we will organize a conference, a series of mini-courses in Lens, and seminars. We will also fund a PhD student and a post-doc, to work on some of the themes.