Math 545 - Geometry of Physics

Fall 2008

Course Plan: Our goal is to understand the geometric structure of classical (and semiclassical) gauge fields on general spacetimes. A good part of the course will be about general relativity and Einstein-Maxwell theory, paying special attention to electromagnetism as a model of gauge theory. Other (non-abelian) semiclassical gauge fields will be regarded more broadly and in less detail. The following is a somewhat ambitious wish list of topics we hope to cover:

• 1. Spacetimes and Lorentzian geometry;
• 2. Electromagnetism and matter models;
• 3. Einstein's field equation;
• 4. Simple cosmological models and black holes;
• 5. Spinors and Dirac operators;
• 6. Electromagnetism as a gauge theory;
• 7. Non-abelian gauge theories;
• 8. Unification of gauge fields and gravitation;

Texts: the following texts should be useful at various points. I plan to rely strongly on the first, at least for half the course.

• 1. General Relativity for Mathematicians, by R.K. Sachs and H. Wu. Springer, 1977.
• 2. General Relativity, by R.M. Wald. The University of Chicago Press, 1984.
• 3. Gauge Theory and Variational Principles, by D. Bleecker. Dover, 2005.
• 4. Topology, Geometry, and Gauge Fields (Foundations), by G.L. Naber. Springer, 1997.
• 5. Relativistic Electrodynamics and Differential Geometry, by S. Parrott. Springer, 1987.

Grades: will be based on a moderate amount of homework assignments and/or lecture presentation at the end of the semester on a topic of your choice.