A^1-homotopy theory and algebraic groups

Abstract  Within the framework of A^1-homotopy theory, one can study algebraic varieties up to a reasonable notion of homotopy equivalence. In this framework, it is then possible to translate topological statements for compact Lie groups into statements for algebraic groups. As an example, stabilization results for homotopy groups of classical Lie groups translate into stabilization results for algebraic K-theory. I will give an introduction to the ingredients of the proof of this result.