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?

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  • 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

How about the Aharanov–Bohm effect?

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.

https://www.google.com/search?q=monodromy+solid+state+physics&sourceid=ie7&rls=com.microsoft:en-US:IE-Address&ie=&oe=

  1. Given your target audience and the prereq level and then the inherent difficulty of the topic, I recommend to try to make your talk conceptual rather than rigourous. Diagrams, not theorems. "This is what it is about and what it is good for" rather than "this is a proof". Do that and maybe it motivates someone to learn the topic properly. Or at least to have conceptual awareness of what the heck it is. So even if they never learn the topic, they have an idea of what the topic is about. I don't think you can do more than that, in an hour, given the students, and their lack of group theory prereqs.

  2. Good luck. Fight the good fight.

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