This was a byproduct of my Master’s thesis.
Abstract
Atiyah’s $KR$-theory from his “$K$-Theory and Reality” paper (1966) establishes a unified view of complex and real $K$-theory and in particular gives a quite elegant proof of the periodicity theorem for $KO$. In this expository talk I will try to give a quick overview of the basic properties of this theory and afterwards focus on the proof of Bott periodicity in the real case via $KR$-theory. Contrary to the approach of the original paper we will also encounter a proof that the Atiyah–Bott–Shapiro homomorphism is an isomorphism during this discussion.
The slides are available here