Spring 2012

Instructor: | Pierre Albin |
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Office: | Illini Hall 237 |

E-mail: | palbin(at)uiuc.edu |

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://math.uiuc.edu/~palbin/Math524.Spring2012/home.html |

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.

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

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

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