First-Day Handout (Note that the syllabus is too ambitious!)
M 392C 519015
Tu Th 2:00-3:15
Instructor: Karen Uhlenbeck
This will be an introductory course in geometry for
and second year graduate and advanced undergraduate student
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
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
you will not be expected to know what a manifold is...you will learn at
least some basic examples.
Dan Freed, who will be teaching the differential topology prelim
points out that you will be super-prepared for his course if you attend
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
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
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
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
9. Chern-Weil theory (especially the formulas for first and second
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.