Vector Bundles

Homework 1
Homework 2
First-Day Handout  (Note that the syllabus is too ambitious!)

M 392C  519015
Tu Th 2:00-3:15
Instructor:   Karen Uhlenbeck   9.160     

This will be an introductory course in geometry  for a first and second year graduate  and advanced undergraduate student audience. The course supplements but does not assume as a prerequisite the prelim course in differential topology. It will be helpful but not necessary to have taken or be enrolled in the algebraic topology course   at the same time. It never hurts to know something about differential equations.  The subject matter has some overlap with Riemannian geometry, and is a very good preparitory course for more advanced courses in geometry. 

A vector bundle is a space which is a twisted product of a manifold M with a vector space V. Special examples are the tangent and cotangent bundles and the line bundles we start with are even simpler. In geometry we study the "fields" which are sections of these bundles, and study interesting equations, many of which come from physics. The course starts with Maxwell's equations equations in three space and ends with the self-dual Yang-Mills equations on a (complex) manifold.  In the middle we will pass by many fundamental ideas in geometry and topology.Below is a tentative outline for the course. I will try to hand out notes from basic sources which will cover some of the material, but I have not found an appropriate text yet. Most of the sources for material in this subject are more advanced. Note that you will not be expected to know what a manifold will learn at least some basic examples. Dan Freed, who will be teaching the differential topology prelim course, points out that you will be super-prepared for his course if you attend this one!

The syllabus is a bit ambitious, especially since we will take time to review some linear algebra, and to discuss questions like "What does it mean for an equation to be invariant under a transformation?" Students who are interested in a later topic may  volunteer to do a project on it, since we probably will not make it through the entire syllabus.

Exercises and problems will be handed out.  In order to receive a grade of B for this course, a student must see Professor Uhlenbeck  to discuss  his or her work at some time before the last week of the semester.  In order to receive an A, students must in addition either work some of the problems  and hand them in, or complete a report (or a project) on a related area.  Suggestions for topics for the report will be given  in class.

1. Maxwell's equations in their Lorentz invariant form on R(1,3); coupling to an electron; Yang-Mills equations on R4; the gauge changes and the topological invariant; first order equations
2. Vortex and monopole equations.;Winding numbers.
3. Definition of a manifold; Examples (Riemann surfaces, complex projective spaces, matrix groups).
4. Line bundles and vector bundles over a manifold.
5. Pullbacks and topological equivalence.
6. Metrics, compatible connections and curvature.
7. Principal and associated bundles
8. Pull-backs and geometric equivalences; the group of gauge transformations
9. Chern-Weil theory (especially the formulas for first and second chern classes)
10. A brief overview of K-theory (no proofs)
11. Introduction to the calculus of variations; the Yang-Mills functional, the Euler-Lagrange equations and the first order equations
12. Holomorphic bundles; the holomorphic Yang-Mills equations

Preliminary reference list: 
M. Atiyah; Geometry of Yang-Mills Fields, Academia Nazionale dei Lincei,
scuola Normale Superiore  (l979)
D. Freed and K. Uhlenbeck; Instantons and Four-manifolds, MSRI publications, Springer-Verlag (1991)
A. Jaffe and C.H. Taubes; Vortices and Monopoles, Progress in Physics 2, Birkhauser(l980)
H.B. Lawson; The Theory of Gauge Fields in Four Dimensions, CBMS regional conferences series, AMS,  (l985).

A more advanced version of this course was taught a few years ago by a former students of mine, Steve Bradlow, at the University of Illinois.  The homepage and notes from this course may be useful.