MATH748I
Selected Topics in Geometry and Topology:
Index Theory

Course web site:

Meeting times:

MWF, 10:00am-10:50am (MTH 0104). Important note: The course will not meet on Wednesday and Friday, September 9th or 11th, since I will be at an international conference on Noncommutative Geometry, nor on Monday, September 28th, because of Yom Kippur. If paticipants are interested, I will arrange makeup times.

Instructor:

Professor Jonathan Rosenberg. His office is room 2114 of the Math Building, phone extension 55166, or you can contact him by email. Office hours are Wednesdays and Fridays after class, or by appointment.

Capsule Description:

The Atiyah-Singer Index Theorem is one of the great accomplishments of 20th century mathematics. It relates PDE theory, functional analysis, geometry, and topology in a very deep way. Today, many proofs of this theorem are known, and there are also literally dozens of variants and generalizations, as well as a huge stock of interesting applications. The purpose of this course will be to explore the Atiyah-Singer Theorem and its variants.

Text:

Mikio Furuta, Index Theorem, I, American Mathematical Society, 2007, ISBN-10: 0-8218-2097-4, ISBN-13: 978-0-8218-2097-1.

At some point you might also want to see the original papers of Atiyah and Singer, especially:

• The original announcement in the Bulletin
• The Index of Elliptic Operators: I
• The Index of Elliptic Operators: III

Other useful books:

• Peter Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, 2nd edition, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.
• Richard Melrose, The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics, 4. A K Peters, Ltd., Wellesley, MA, 1993.
• John Roe, Elliptic operators, topology and asymptotic methods, 2nd edition, Pitman Research Notes in Mathematics Series, 395. Longman, Harlow, 1998.
• Patrick Shanahan, The Atiyah-Singer index theorem: an introduction, Lecture Notes in Mathematics, 638. Springer, Berlin, 1978.

Prerequisites:

We will develop a lot of the background material along the way, though any previous exposure to elliptic PDE, functional analysis, Riemannian geometry, and characteristic classes, such as developed in MATH 631, 632, 673, 674, or 740, will be useful. The minimal prerequisites are MATH 630 (real analysis) and MATH 734 (algebraic topology).

Course Requirements:

There will probably be a few homework assignments, but there will be no exams.

Homework #1, due 10/23/09

At some point participants may be asked to present some material to the rest of the class.

Course Outline:

1. Introduction and a few motivating examples
2. Basics of compact and Fredholm operators
3. Background from elliptic PDE theory
4. Spinors and Dirac operators
5. Basics of K-Theory
6. The Atiyah-Singer Theorem following Furuta
7. Applications of the Index Theorem (as time permits)
8. Variants of the Index Theorem (as time permits)