I'm looking to put together a 1 hour talk about monodromy aimed at undergraduate math and physics majors. In terms of prerequisites, I only want to assume the students have seen multivariable calculus, intro ODE's, and intro linear algebra.

I'd like to have an example from physics (preferably mechanics) that I can refer to throughout the talk, but I'm having trouble thinking of one. Any ideas?

• Do you mean monodromy like in stroboscopic or Poincaré map? – Evgeny Oct 12 '17 at 9:24
• @Evgeny I don't know what those are. – Avi Steiner Oct 12 '17 at 17:44
• Okay! I mean, for example, you have a linear system $\dot{x} = - x + f(t)$ where $f(t)$ is a periodic forcing. You can study it with monodromy map when you take $x(0)$, start trajectory from it and get $x(T)$. The map $F\colon x(0) \mapsto x(T)$ is also called sometimes a monodromy map. Or stroboscopic map. Is it the same concept that you had in mind? – Evgeny Oct 12 '17 at 19:43
• I believe this is better suited for matheducator.SE, too. @Avi Steiner, would you mind if I migrate your question there? – Jack D'Aurizio Oct 20 '17 at 13:15
• @JackD'Aurizio I don't mind at all – Avi Steiner Oct 20 '17 at 14:04

https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect

The Aharanov–Bohm effect is then a manifestation of the fact that a connection with zero curvature (i.e. flat), need not be trivial since it can have monodromy along a topologically nontrivial path fully contained in the zero curvature (i.e. field free) region. By definition this means that sections that are parallelly translated along a topologically non trivial path pick up a phase, so that covariant constant sections cannot be defined over the whole field free region.

• I'd need to spend a bunch of time parsing these definitions. I know a few people I might be able to ask, though. – Avi Steiner Oct 21 '17 at 3:14
1. I appreciate your desire for a practical application, especially one from mechanics (more conceptual).

2. I wasn't able to find any from that field, exactly. ;-( [But you said that was nice to have, not strict requirement.]

3. Google does show a lot of examples from solid state physics. Essentially monodromy has to do with periodicity and crystal lattices are...well...periodic.