Math 684 (Phillips)

This is the home page for N. C. Phillips' Math 684 at the University of Oregon, fall quarter 2022.

Course information

Course description

This description is for Math 684 and Math 685. It was originally written for a three quater sequence, but the third quarter was not approved. This means that material will need to be covered faster, and some parts will be omitted.

The goal of this course is to develop the theory of elliptic pseudodifferential operators on compact smooth manifolds, to a sufficient extent to cover all the analysis needed for the proof of the Atiyah-Singer Index Theorem for families. This includes:

Time permitting, I will then give a survey of K-theory and a proof (taking some algebraic topology on faith) of the Atiyah-Singer Index Theorem for families.

Material on compact operators and Fredholm theory is put first, since it is important for many functional analysts whose interests are unrelated to the Atiyah-Singer Index Theorem, or even to C*-algebras. As an inducement to C*-algebraists: the only proof I know of the general Bott periodicity theorem in equivariant K-theory, when the group is not abelian, depends on the material above, although backwards: instead of computing the index of a family of elliptic operators using algebraic topological data, it constructs a class in equivariant K-theory by constructing an equivariant family of elliptic operators whose index is the desired class.

Comments about C*-algebras will be made in passing, as appropriate, but little time will be spent on them. (For example, the index of a family of Fredholm operators is a special case of the index of a Fredholm operator between Hilbert modules over a C*-algebra, but almost nothing will be said about this theory beyond several definitions and a pointer to further reading.)

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 fall 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.

Exercises for this course, as of 17 October 2022 (pdf); AMSLaTeX. This file will be updated throughout the course. Little proofreading has been done.


This page maintained by N. Christopher Phillips, email.

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Last significant change 27 September 2022.