If the spacetime manifold admits non-trivial bundles, you get instantons
Because you essentially have to sum over all possible bundle structures in the path integral.
It's actually very difficult to find someone discussing this in detail, I pieced my picture of gauge theories together from various sources when I wrote my Bachelor's thesis
Most either never talk about the bundle structures, or they somehow assume you already know it, and just give the shortest intro possible
These US admissions departments are much better than the UK. I got a reply in an hour, whereas all my UK choices took 5-7 days to reply to e-mails, yet they had less applicants :)
@DavidZ it's unbearable what you do to me. You promised help, and after that you said "I told you how to make the formulas visible". I am trying to help. Can you understand this simple good will?
I understand that you're trying to help someone through a calculation. And in light of that, I've offered to help you view formulas in chat so that you can work through that calculation in the chat room.
@JamalS I'm not sure. It is an amazing thing when you see it at first, but when you think about it, what you found so amazing is really rather defined into it, so... shrug
The Ramanujan equation is kinda ghastly, and not horribly interesting. He's done better stuff, like the nested root thing he submitted that no one could solve.
@Jiminion Number theory just completely escapes me. I don't think it is uninteresting as such, but every time someone shows me "cool" number theory, all I can think is "Why would you even care?".
@JamalS Euler's? Well, because it has three math-derived constants in it, e,i, and pi, and it is kind of an odd result, as you don't think of them relating that way. But in the end, it's kind of a trick....
I suppose some simple physical statement such as the Einstein field equations or fluctuation theorem might be cool. Realistically, though, I'd probably put something that looks more intimidating and has appeal in the calligraphic sense.
