This semester I’m teaching the first semester of Modern Geometry, our year-long course on differential geometry aimed at our first-year Ph.D. students. A syllabus and some other information about the course is available here.
In the spring semester Simon Brendle will be covering Riemannian geometry, so this gives me an excuse to spend a lot of time on aspects of differential geometry that don’t use a metric. In particular, I’ll cover in detail the general theory of connections and curvature, rather than starting with the Levi-Civita connection that shows up in Riemannian geometry. I’ll be starting with connections on principal bundles, only later getting to connections on vector bundles. Most books do this in the other order, although Kobayashi and Nomizu does principal bundles first. In some sense a lot of what I’ll be doing is just explicating Kobayashi and Nomizu, which is a great book, but not especially user-friendly.
A major goal of the course is to get to the point of writing down the main geometrically-motivated equations of fundamental physics and a few of their solutions as examples. This includes the Einstein eqs. of general relativity, although I’ll mostly be leaving that topic to the second semester course.
Ideally I think every theoretical physicist should know enough about geometry to appreciate the geometrical basis of gauge theories and general relativity. In addition, any geometer should know about how geometry gets used in these two areas of physics. I’ve off and on thought about writing an outline of the subject aimed at these two audiences, and thought about writing something this semester. Thinking more about it though, at this point I’m pretty sick of expository writing (proofs of my QM book are supposed to arrive any moment…). In addition, I just took a look again at the 1980 review article by Eguchi, Gilkey and Hanson (see here or here) from which I first learned a lot of this material. It really is very good, and anything I’d write would spend a lot of time just reproducing that material.
Update: Steve Bryson sent me another excellent suggestion for a book covering these topics, aimed at the physical applications: David Bleecker’s Gauge Theory and Variational Principles.