Classifying spaces of algebraic groups and A1-homotopy theory.

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We propose to study in a systematic way the classifyings spaces of algebraic groups G from the point of view of motivic homotopy theory, as well as some naturally related problems of G-equivariant motivic homotopy theory. For instance following our proof of the Friedlander-Milnor conjecture, we may attack new results on the homology of groups of points G(F) for not necessarily algebraically closed fields F. Another (related) direction is concerned with Serre's conjectures in Galois cohomology. We do have a precise conjecture that implies most of these type of conjectures, most importantly the one which is up to now untouched, and a new approach to settle it. More generally we are interested in transporting classical problems of (equivariant) homotopy theory to motivic homotopy theory. Like the Adams conjecture, the Segal conjecture, the behaviour of various structured cobordism like, to start with, the orthogonal cobordism.