Math 691 (Phillips)

Course information

• Course number: Math 691.
• Course title: Topics in K-theory.
• CRN: 23184.
• UO class schedule page for this course.
• Instructor: N. Christopher Phillips.
• Office: 105 Deady [now renamed to "University Hall"]. Please knock. I can't leave my door open, because if I do I get too many people asking to borrow my stapler or pencil sharpener, or where to find the math department office or nonexistent rooms (such as 350 Deady).
• Office hours: Mondays 1:00 am--2:00 pm [except during the first three weeks of classes], Tuesdays 11:00 am--12:00 noon and 3:00--4:00 pm, or by appointment.
• Email. All messages should have a subject starting "Math 691:". Binary files or attachments are accepted only by prior arrangement. I do not ever accept Microsoft Word documents, html (web) files, or encoded text messages. Please send 7 bit ASCII plain text only.
• Time and place: MWF 11:00--11:50 am, room 205 Deady [now renamed to "University Hall"].
• Textbook: M. F. Atiyah, K-Theory. Electronic copy supposedly available. (This has not been checked.)
• Prerequisites: Math 634--636; in principle also Math 690. Eventually some basic knowledge of smooth manifolds will be needed.

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:

• No binary files or attachments (except by prior arrangement).
• No Microsoft Word files. I do not accept these under any circumstances, since I don't have software that reads them. If you really want to send something in a word processor format, use TeX.
• No html encoded messages.
• No mime encoding or other encoding of ordinary text messages.

Last significant change 7 January 2023.