Special Talk on A^1-Homotopy Theory and Its Recent Developments

In this talk I will recall the statement of the Friedlander conjecture on the cohomology of a reductive group G and give a short recollection on known results. I will present a new approach which relies on the study of the A^1 homotopy type of the classifying space of the Suslin-Voevodsky construction on the group G. This approach allows us to prove the first complete cases of semi-simple groups: mainly SL_2 and SL_3. We will explain the basic strategy of the proof of this result and how to potentially extend it to a complete proof of the conjecture.