Geometry and Physics (259C) graduate course, Spring 2004

Mondays 4.00-5.20 in APM 6438 and Fridays 9.30-10.50 in APM 7421.

Lecturer: Justin Roberts

Geometry and physics have always been fundamentally interrelated. A few obvious historical highpoints (off the top of my head) are:

- Newton's explanation of the elliptical orbits of the planets using classical geometry
- Hamilton's discovery of symplectic geometry and its role in classical mechanics
- Maxwell's theory of electromagnetism, which amount to the theory of de Rham cohomology and harmonic forms
- Einstein's general relativity, which explains gravity in terms of Riemannian geometry

More recently, physicists working on gauge theories and string theories have incorporated some of the most esoteric objects of mathematics (exceptional Lie groups such as "E_8", the Monster simple group, etc.) as well as its most esoteric methods (homological algebra, derived categories). Moreover, there has been an amazing flow in the other direction: Witten has demonstrated many times that the (unfortunately not yet all rigorous) techniques of quantum field theory, supersymmetry and so on have amazing implications for geometry, topology, and the unity of mathematics.

For some further background and philosophy, look at these links:

THE UNREASONABLE EFFECTIVENESS OF MATHEMATICS IN THE NATURAL SCIENCES, a famous article by Eugene Wigner.

On teaching mathematics, a provocative article by Vladimir Arnold.

Geometry and Physics - A marriage made in heaven, a video lecture by Sir Michael Atiyah.

The course

This is essentially a course on differential geometry with a view to applications in physics. I don't intend to explain the physics itself in great detail - for one thing, my own knowledge of physics is very rusty, and I might be better off inviting physics students to do this - but rather to try to show how the mathematical language we develop is useful for describing it (and thereby justify the mathematical formalism).

In 259A we covered smooth manifolds, differential forms and de Rham cohomology, a little bit on Lie groups, vector calculus done properly, and Maxwell's equations.

In 259B we covered symplectic geometry and Hamiltonian mechanics, vector bundles, connections and curvature, Riemannian metrics, geodesics and the Gauss-Bonnet theorem.

259C will, I hope, cover
1. Characteristic classes (Chern-Weil theory)
2. Relativity (More on curvature, Minkowski space, Einstein's equations)
3. Gauge theory (Lie groups, connections on principal bundles, Maxwell's equations revisited, Yang-Mills equations)
4. Clifford algebras, spinors, the Dirac operator, index theory

Pre-requisites

This course is traditionally a "second-year" graduate course. The great thing about that is that we don't have to have exams, so can concentrate on trying to learn and understand instead!

However,I want it to be accessible also to first years and to physicists, so I am not going to insist on any previous knowledge of geometry or topology. I can't promise that the course will be entirely self-contained, but I will try to make sure that I give at least a rough explanation of anything "external" I need, and encourage people to ask about or read up on things they aren't familiar with. Needless to say, this is going to be a broad course, and I want to encourage the participants to approach it with an open mind and to be as active as possible in discussing it with me and with each other.

Books

I don't intend to follow any single book too closely, but here are the main sources I will be using:

The geometry of physics, by Ted Frankel (Cambridge)

Notes on differential geometry, by Ko Honda, available here

Differential forms in algebraic topology, by Bott and Tu (Springer). (Especially the first bit of the course)

Additional Information

My office hours are Fridays 11-12 after the lecture. -->

justin@math.ucsd.edu

"Event horizon" by Rudie Berkhout