In Math 6190, we followed Adams' work and used Adams' operation on complex K-theory to fully characterize the dimensions n for which R^n can be a division algebra. There is an alternative proof by Milnor and Bott in 1958, building substantially on the works of Chinese mathematician 吴文俊  (Wenjun Wu), that relates this problem to the vanishing of Stiefel-Whitney (SW) classes of real vector bundles over S^n for n != 1, 2, 4, 8. We will survey this approach in the first part of the talk.
In the second part, we will cover Atiyah and Hirzebruch's generalization of Milnor and Bott's result. In particular, Atiyah and Hirzebruch uses real K-theory to prove that the SW classes of any real vector bundle on a ninth suspended finite CW complex must be trivial. This talk is a companion lecture to the ongoing Kan seminar on K-theory in MATH 6190.