This is a lecture-based class on the celebrated Atiyah-Singer index theorem, proved in the 60's by Sir Michael Atiyah and Isadore Singer. Their work on this theorem lead to a joint Abel prize in 2004.
Requirements: Knowledge of topology and manifolds, although we'll try to review some of this.
Here are my two favorite books on the subject in case you'd like a book on the subject:
Heat Kernels and Dirac Operators by Nicole Berline, Ezra Getzler, Michèle Vergne. (The paperback is a beautiful book, although a little sophisticated for beginners.)
Elliptic Operators, Topology, and Asymptotic Methods, Second Edition (Paperback) by John Roe. (Nicely written and understandable!)
However, we will not be following any of these books because I would like to approach this class assuming less background in analysis and geometry.
Office: LN-2225, Extension: 7-3506, Email: paul (at sign) math binghamton edu
Office Hours: M: 2-3, R: 4-5, and by appointment.
Lecture notes on "An introductory course in differential geometry and the Atiyah-Singer index theorem" (Any comments, typos, suggestions, basically anything you want to say about these notes, PLEASE let me know!)
1) Introduction to the Atiyah-Singer theorem and manifolds
2) Smooth functions and pou's (Version 2/7.)
3) Tangent and cotangent vectors (Version 2/9.) In the notes I use a simpler definition of C_p than in the lectures. (It does not effect any of the theorems.)
4) Vector bundles I (Version 2/15.)
5) Tensors (together with an intro.)
6) The exterior algebra with an appendix on permutations (Version 3/1.)
7) Vector bundles and ext. derivative (Version 3/19.)
8) de Rham cohomology I (Version 3/22.)
9) de Rham cohomology II (Version 3/30.)
1) Turn in one (of course, feel free to do more on your own if you wish :) but only turn in one) of the Problems 1, 2, 3, or 4 in Exercises 2.4. (Do not turn in the "projects")
2) Turn one of Problems 2 or 3 in Exercises 2.5 or Problems 1 or 2 in Exercises 2.6.
1) Turn in one of the Problems 3 or 5 in Exercises 2.8.
2) Turn in Problem 5 or 6 in Exercises 2.9.
We meet MW 4:40-6:15 LN2205.
Here is an approximate schedule of the class. We will go much slower!
Just show-up and look interested! Homework will be to try and make sense of the lectures ... that is, just review your notes and rewrite them in your own words so you can understand them the way you think. I'll also try and give some exercises during lecturers.
CHANGE IN GRADING (Recall the "subject to change below"): For students required to hand in homework (see above), your grade will be based on homework and attendance.
Class attendance is required to get a grade.
If you have difficulties, don't wait until the last minute. Make use of my office hours and my email, and the best of all your own classmates.