There’s a neat way to introduce 4D twistors through standard 2D complex analysis & group theory.

- on CP^1 there are homogeneous and inhomogeneous coordinates. If you study the transformations the conformal transformations are only linear on the homogeneous coordinates; same is not true for the inhomogeneous ones. You can derive the transformations.

key equation: z_1 = w z_2

So far so good.

- The key observation is that ‘twistors’ are to the usual x^\mu as what homogeneous coordinates are compared to the inhomogeneous ones: twistors transform linearly under the conformal group.

Now you will need 2 vectors on C^2 (2 spinors) and the w is just x^{\mu} \sigma_\mu. The above key equation turns in to the coincidence relation, you can discuss SO(4) \sim SU(2) x SU(2), spinors, etc, etc

I’m not saying it merits major discussion, but I think it’s something to be pointed out, and the tensor formulation of the theory is really nice – you end up with an object a bit like a metric representing the overlap integral between the non-orthogonal functions.

In my experience, the more mathematical the subject matter, the greater the temptation for the professor to basically spend the class just kind of working through the algebra and writing one equation after the next. This adds very little to what can be gained from just reading the textbook. What I wish professors would do is plan each lecture around a set of concepts they want the students to learn, and structure the lectures with the goal that, if nothing else, the students will understand X at the end of the hour. And, moreover, that it will be clear to the students themselves that X is the point of the lecture. I can’t tell you how many lectures I attended that consisted of just furiously scribbling down equations, at the end of which I understood how one equation followed from the next but had no idea which equations were important or why.

E.g., instead of diving into “here is how you evaluate a contour integral”, one can start with some bullet points like “contour integrals are integrals along paths in the complex plane”, “there are a number of useful theorems for evaluating them, such as the residue theorem or the Cauchy integral formula”, and “we can translate some real integrals into contour integrals, which allows us to evaluate them by means of these tricks”. It might be good to actually start the lecture by writing these bullet points on the board, because (as some professors seem not to have realized) students who are busy taking notes won’t necessarily hear every word that comes out of your mouth. Alternatively, one could hand out a paper (or post one online) with “Here are the key points you should learn from this lecture.”

Maybe that’s obvious advice, but believe me when I say that not everyone follows it.

That being said, ask Ron Donagi for good Sturm-Liouville examples. When I took his Math 240 class he did a neat example of vibrations in a metal beam.

I had one quarter of mathematical methods first quarter of my first year of grad school. I had one quarter of statistics and numerical methods at the end of my second year. After my qualifying exam, I have never, ever used anything I learned in MM. Statistics and programming is my bread and butter (although the class came too late and was taught too poorly to be of much use).

So I’m actually all for some more esoteric, fun examples. I love math (got an undergrad degree in it), and would have enjoyed such examples. No offense, but the experimentalists/observationalists will probably not use anything you teach them. They don’t actually care much about your class. So throw in some entertaining examples at least.

Waveguides as an example of modal expansions?

Far-field antenna patterns using fourier transforms?