Math 425, Bundles and Hodge Theory, Fall 2002

Meetings:
TTh 9-10:20am in Altgeld 241
Web address:
Course information is available online at http://torus.math.uiuc.edu/jms/m425
Professor:
John M. Sullivan, jms@uiuc.edu, 326 Illini Hall,
244-5930 (with answering machine); mailbox in 250 Altgeld.
Office hours:
To be determined, or by appointment.
Prerequisites:
Math 423, Differentiable Manifolds
Required Text:
Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer GTM 94
Outline:
The official title of this course is Linear Analysis on Manifolds. It covers fundamental results about smooth functions on curved spaces. After a brief review of differentiable manifolds, this course will consider sheaves and vector bundles. We will prove the de Rham theorem which expresses the cohomology of a manifold in terms of differential forms, and the celebrated Hodge theorem, which finds a harmonic representative for each cohomology class. These results are central to the further study of manifolds. We will mainly follow chapters 5 and 6 of Warner's book.