Abstract:
(This is "pre-seminar" talk intended to give the background needed in order to understand the afternoon talk)
What's a real or complex vector bundle over a compact space X? What are real and complex K-theory for compact spaces? What is a cohomology theory? What's Bott periodicity? What's a Kunneth formula?
We're sketching all of this as preparation for the afternoon talk, and so we will emphasize comparing the real and the complex settings.
(Side comment for skeptics: why should an analyst care about K-theory of any flavor? Here's an example. Suppose that T is a bounded operator on Hilbert space and T*T - TT* is a compact operator. How can I tell if T is of the form (normal) + (compact) ? It's true if and only if the index of T - lambda is zero for every lambda not in the essential spectrum of T. Try proving that without K-theory! This example won't be in the talk - if you are interested see
for a nice summary.)
