Math 490 -- Student Seminar (Chern-Weil Theory)



Time: Spring 2004 -- MWF 11:00-11:50
Location: HB 423 (Herman Brown)
Faculty Supervisor: Robin Forman
Organizer: Jer-Chin Chuang
47 Herman Brown Hall, (713) 348-2867
Email: lukec [at] math [dot] rice [dot] edu

Course Description

Basic manifold theory, differential forms, DeRham cohomology, vector bundles, connections, characteristic classes via Chern-Weil theory.

The pre-requisites are algebraic topology (Math 445), and to a lesser extant, differential geometry (Math 401). Specifically, participants should be familiar with basic manifold theory and basic cohomology theory. Exposure to homological algebra and multi-linear algebra are helpful but not necessary.

Tentative Daily Schedule

Texts

The primary text is Bott/Tu. After some background material (see any introductory text on manifolds), we will cover the first fifty or so pages in Bott/Tu on DeRham cohomology. There will be an interlude on vector bundles and connections, and then we will cover Chern-Weil theory using the appendix in Milnor/Stastaff.

R. Bott and L. Tu: Differential Forms in Algebraic Topology. Springer-Verlag. ISBN 0387906134

J. Milnor and J. Stasheff: Characteristic Classes. Princeton UP

Format

This is a student seminar. Participants will take turns presenting the material. Please do not take this responsibility lightly! Hopefully, these lecture assignments will be determined with a week's notice. Participants: please keep apace of the material so you will be ready to lecture. The anticipated load for this seminar is no less than that of an upper-level graduate course. Participants (only) can receive credit for this seminar (see the organizer about this). Auditors are welcome, but asked to respect the pace of the seminar.


Homework

There will be regular problem sets. Perhaps there may also be regular problem solving sessions, but regardless, participants are responsible for keeping abreast of the material and seeking help if needed. A list with the contact info of the participants will be available shortly into the term.


Getting help

If you have questions, check with other participants. One of the goals of the seminar is to encourage participants to help each other learn mathematics. Dr. Forman has kindly offered us assistance, if there are still unresolved questions.