In the Fall I'll be starting teaching again, after a semester away on sabbatical and then enjoying teaching relief during my first semester at Penn. I'll be teaching a course that I truly love, and that I've taught a number of times before - Mathematical Methods of Physics I, to a class of beginning graduate students, and some interested seniors. The backbone of this course, as I teach it, is rather traditional, since the topics involved are things that form the basis of the toolbox that professional physicists need. From year to year I have added various extra topics (some differential geometry, some topology, some group theory, ...), but I always cover

Analysis of Complex Functions

Exact and Approximate Evaluation of Sums and Integrals

Exact and Approximate Solution of Ordinary Differential Equations

Transform Calculus

Sturm-Liouville Theory

The Calculus of Variations

One challenge in a course like this is to maintain the connection with actual applications of the techniques one is covering. Since I was originally taught this material in a set of courses as a mathematics undergraduate, my own take on the material can be rather formal, and I have worked over the years to balance this out. However, as you might guess, my own examples are predominantly drawn from those areas of physics with which I am most familiar - for example, supersymmetry, and the restrictions that holomorphy places on superpotentials, is a nice illustration of the power of complex analysis. But this course is supposed to provide a basis for all graduate students, including those with interests in other branches of theoretical physics or, indeed, experimental physics or observational astrophysics and cosmology. There are, of course, rather general things that one can do that should be of use to everyone, such as the use of Fourier and Laplace transforms in solving heat, diffusion, and other equations. And the calculus of variations appears everywhere already. There are also, incidentally, lots of cute things one can do in the opposite direction, like cooking up examples of oscillating systems in which the sum over all modes gives the total energy, which is easy to calculate another way, and using this to provide a way to compute infinite sums. Nevertheless, what I really yearn for are even more examples illustrating the use of some of the above topics from other branches of physics. I could, of course, annoy my colleagues with this question, but I thought that opening it up to Cosmic Variance readers might provide some novel suggestions. So, if you have some unusual example, brief enough to be useful in a class, of the use of any of the above in any branch of physics (even particle physics and cosmology - there's plenty I don't know there also), I'd appreciate you filling me in in the comments. And if any of my students-to-be are reading this - beware; it's possible that good suggestions you see here, that don't make it into class, may turn up on exams - who knows?