MATH 524: Linear Analysis on Manifolds
Spring 2012

Instructor:Pierre Albin
Office:Illini Hall 237
E-mail:palbin(at)uiuc.edu
Lectures: TTh 9:30-10:50 in 445 Altgeld
Assignments: There will be homework each week.You are allowed (and encouraged) to work with other students while trying to understand the homework problems. However, the homework that you hand in should be your work alone. Late homework will not be accepted, but the lowest score will be dropped.
Grading percentages: Problem sets (100%)
Web page: https://faculty.math.illinois.edu/~palbin/Math524.Spring2012/home.html

Lecture Notes

There is no required text for the course and a plethora of recommended texts. There will, however, be lecture notes:
Lecture Notes (last updated May 1, 2012)

Among the recommended texts, let me mention:
Lawson-Michelsohn, Spin Geometry
Rosenberg, The Laplacian on a Riemannian manifold
Berline-Getzler-Vergne, Heat kernels and Dirac operators
Further recommendations can be found in the lecture notes.

Homeworks

HW1 (due Tuesday January 24 in class): Exercises 1-5 from Lecture 1 of the lecture notes
HW2 (due Tuesday January 31 in class): Exercises 6 and 8 from Lecture 1 and Exercises 2, 4, and 5 from Lecture 2 of the lecture notes
HW3 (due Tuesday February 7 in class): Exercises 6, 8 from Lecture 2 of the lecture notes
HW4 (due Tuesday February 14 in class): Exercises 12-15 from Lecture 2 of the lecture notes
HW5 (due Tuesday February 21 in class): Exercises 1-4 from Lecture 3 of the lecture notes
HW6 (due Tuesday February 28 in class): Exercise 5 from Lecture 3 of the lecture notes
HW7 (due Tuesday March 6 in class): Exercises 6-8 from Lecture 3 of the lecture notes
HW8 (due Tuesday March 13 in class): Exercises 1-4 from Lecture 4 of the lecture notes
HW9 (due Tuesday March 27 in class): Exercises 5-8 from Lecture 4 of the lecture notes
HW10 (due Tuesday April 3 in class): Exercises 9 and 11 from Lecture 4 of the lecture notes
HW11 (due Tuesday April 10 in class): Exercises 1-4 from Lecture 5 of the lecture notes
HW12 (due Tuesday April 17 in class): Exercises 5-7 from Lecture 5 of the lecture notes
HW13 (due Tuesday April 24 in class): Exercises 9-11 from Lecture 5 of the lecture notes
There will not be a HW14

Holidays

This semester we will not have classes on
Spring Break March 17 - March 26

Syllabus

The Laplace operator is the simplest differential operator that reflects the geometry of a space. It occurs in all sorts of applications including in celestial mechanics, electromagnetism, wave propagation, quantum mechanics, and relativity. In this course, we will use Laplace-type operators on manifolds to study some interesting topics in analysis, geometry, and topology.

From an analytic point of view, the Laplacian is the quintessential elliptic operator and so we will study some incredible properties of elliptic differential operators: they can always be `almost' inverted, and solutions to elliptic equations have as much regularity as one could possibly hope for.
From a geometric point of view, we will see that the Laplacian has an infinite sequence of eigenvalues and that these eigenvalues encode a lot of geometric information: such as the dimension, the volume, and the total scalar curvature.
From a topological point of view, the Laplacian is at the core of Hodge theory and the results of de Rham cohomology: two analytic ways of approaching the topological cohomology groups of a manifold.

The topics treated in this course will be useful for students looking to do research in mathematical physics, analysis and/or geometry. The prerequisites are a course in geometry including vector bundles and differential forms on manifolds, and a course in analysis including Lp spaces. (A course in topology would be useful but is not necessary.)

A possible syllabus, subject to time constraints: (modified Jan 18)
Review of Differential Geometry
Brief introduction to Riemannian Geometry
Dirac operators
Pseudodifferential operators and the resolvent
Elliptic regularity
De Rham cohomology and Hodge theory
The Heat Equation on a Riemannian Manifold
An Introduction to Spectral Geometry
The Chern-Gauss-Bonnet Theorem
The index theorem for Dirac operators