Directed reading on K-theory
Directed reading on K-theory
In the spring semester of 2024, I read topological K-theory following [1][2] under the guidance of Professor Bo Liu. At the same time, Professor Hang Wang taught a course on operator K-theory. Both of them aimed to prove the Bott-periodicity theorem in different approach. After this, I had a quick look at equivariant K-theory accounted in [3].
My interest in K-theory was gained from 2 sources. One is the Serre-Swan theorem which bridges projective modules in algebra and vector bundles in geometry. I found this theorem when I was seeking an understanding of projective modules. And the other is the philosophical insight from Galfand-Naimark theorem, using "space of functions"(commutative C*-Algebras) to reconstruct the geometric informations. These combine to give a useful aspect in study of C* algebras.
I studied vector bundles, and calculated K-rings of basic examples.
I went through the proof of Bott periodicity, and read about one of its application: the classification of Division algebras.
It also coincidentally got me interested in springer theory, and I am knocking around this now.
[1] Hatcher, A. (2003). Vector bundles and K-theory. Unpublished book, available at https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf
[2] Karoubi, M. (1978). K-theory: An introduction. Springer.
[3] Chriss, N., & Ginzburg, V. (1997). Representation theory and complex geometry. Birkhäuser.