Bott periodicity from Algebraic geometry

A famous theorem in algebraic topology is Bott periodicity: the homotopy groups of the space of orthogonal matrices repeat with period 8:  pi_k(O) = pi_{k+8}(O) . I will give a short overview of Bott periodicity and then I will explain how to formulate and prove a theorem in algebraic geometry which, when specialized to the field of complex numbers, recovers the usual topological Bott periodicity, but makes sense over any field. This is work in progress with Ravi Vakil.