Linear algebraic groups over arbitrary fields
The theory of semisimple linear algebraic groups is well known, up to the so-called anisotropic groups. Examples of anisotropic groups are given by the compact real Lie groups, which are relatively well known. But there are many such groups in more general situations whose properties are totally unknown. In fact it is true that all semisimple linear groups are derived by `specialization` from their `anisotropic forms`.
This project proposes to develop methods in order to classify these anisotropic groups and to get informations about their internal structure. Tools are obtained from Galois cohomology, from generic splitting techniques and from the techniques which are used by the so called underlying `related structures' of linear groups, like quadratic and Hermitean forms, Azumaya, Lie, and Jordan algebras.
Conversely, knowledge about those structures can be obtained from knowledge of these groups. Linear algebraic groups and their underlying structures always have been and still are of importance in many areas of mathematics and other sciences.