Math 691 (Phillips)

This is the home page for N. C. Phillips' Math 691 at the University of Oregon, winter quarter 2023.

Course information

Course description

Modified course description

The description below was originally written for a course starting K-theory for the beginning. However, Boris Botvinnik's Math 690 in Fall 2022 covered a substantial amount of K-theory. (I have not found a website for this course.) Therefore some topics on the list below will be reviewed only very briefly, and other topics, not on the list, will be covered in the course.

Since some students are expected who did not have Math 690, the course will begin with a fast paced review from the beginning, with the opportunity to ask questions. In particular, items (1) and (2) on the list below were done, while items (3) (Bott periodicity) was not, and will be treated carefully in this course.

Original course description

Outline of topics:

  1. Vector bundles.
  2. Vector bundles over compact spaces.
  3. Bott periodicity.
  4. K-theory as a cohomology theory.
  5. Products in K-theory.
  6. The Thom isomorphism and computations of K-theory in other examples.
  7. Pseudodifferential operators and the calculus of symbols.
  8. Application to the Hopf invariant problem..

This is a course on the K-theory of (locally) compact Hausdorff spaces, not on the K-theory of C*-algebras. C*-algebras will be mentioned only occasionally, and no knowledge of them (or interest in them) is assumed. Nevertheless, this course is important background for anyone interested in the K-theory of C*-algebras.

This course contains much of the algebraic topological background relevant to this year's Math 684--685 sequence on elliptic pseudodifferential operators, the analysis background for the Atiyah-Singer Index Theorem. Taking this course will not be necessary for Math 684--685: the material needed there will be summarized there.

Course files

See the comments on the different formats for more information on the formats of files posted below. One warning is important enough to give here: In the winter quarter 1998, somebody printed some of my pdf files somewhere on campus and found that certain mathematical symbols (such as minus signs in exponents) did not print, damaging the meanings.

This page maintained by N. Christopher Phillips, email.

When emailing me, please use 7 bit ASCII plain text only. In particular:

Last significant change 7 January 2023.