Math 684 (Phillips)

Course information

• Course number: Math 684.
• Course title: Real Analysis.
• CRN: 12541.
• 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 11 am--12 noon, Tuesdays 3--4 pm, Wednesdays 11 am--12 noon, or by appointment.
• Email. All messages should have a subject starting "Math 684:". 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 10:00--10:50 am, room 210 Deady [now renamed to "University Hall"].
• Textbook: No official text. References for various topics are listed here; this list will be updated throughout the course.
  • Compact operators and Fredholm theory
    • Conway, A Course in Functional Analysis, Section 6.3 and Chapter 9.
    • Murphy, C*-Algebras and Operator Theory, Section 1.4.
    • Douglas, Banach Algebra Techniques in Operator Theory, Chapter 5.
  • Differential operators
    • Palais, Seminar on the Atiyah-Singer Index Theorem, Chapter 4.
    • Erik van den Ban and Marius Crainic, Analysis on Manifolds, Utrecht University "Master Class", Fall 2009, Lecture 1.
  • Distributions and Sobolev spaces
    • Taylor, Pseudodifferential Operators, Chapter 1.
    • Hörmander, The Analysis of Linear Partial Differential Operators I, Chapters 1--7. (This contains far more material than needed.)
    • Conway, A Course in Functional Analysis, Chapter 4. (This has all the material on functiuonal analysis needed to properly define the space of distributions, and the definition of this space, but goes no farther.)
    • Palais, Seminar on the Atiyah-Singer Index Theorem, Chapters 8--10. (Sobolev spaces done without using distributions.)
    • Erik van den Ban and Marius Crainic, Analysis on Manifolds, Utrecht University "Master Class", Fall 2009, Lecture 2, Lecture 3, Lecture 4, and Notes on Rellich's lemma.
  • Pseudodifferential Operators
    • Taylor, Pseudodifferential Operators, Chapter 2.
    • Hörmander, The Analysis of Linear Partial Differential Operators III, Chapter 18.
    • Palais, Seminar on the Atiyah-Singer Index Theorem, Chapter 9.
    • Erik van den Ban and Marius Crainic, Analysis on Manifolds, Utrecht University "Master Class", Fall 2009, Lecture 5, Lecture 6, Appendix to Lecture 6, Lecture 7, and Lecture 8.
• Prerequisites: Math 616--618. Eventually some basic knowledge of smooth manifolds will be needed.

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:

• Basics of compact operators on Hilbert space.
• Basics of vector bundles. (Not analysis, but an essential ingredient for the analysis.)
• Basics of Fredholm operators and families of Fredholm operators on Hilbert space, and the Fredholm index.
• Brief reminder of the basics of the Fourier transform on ${\mathbb{R}}^n$.
• Sobolev spaces associated to ${\mathbb{R}}^n$ and to smooth vector bundles on manifolds.
• Basics of differential operators and their symbols, on ${\mathbb{R}}^n$ and on smooth vector bundles on manifolds.
• Pseudodifferential operators and the calculus of symbols.
• Maps on Sobolev spaces induced by pseudodifferential operators, including boundedness theorems and compactness theorems (Rellich's Lemma).
• Elliptic pseudodifferential operators on compact smooth manifolds are Fredholm.

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.

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 27 September 2022.