Dear mathematicians,
I will graduate soon, and as part of the defense, my department has a tradition of a kind of second examination. The idea is that after you have presented and defended your PhD thesis, you present something that is interesting and that shows that you possess knowledge of and interest for things outside of your own area of expertise.
This can be basically anything, but it typically meets a few of the following criteria: - it involves research-level mathematics and/or is a famous mathematical problem (Hilbert's problems or Millennium problems are quite popular, for example) - its presentation has to be self-contained and doable in about 10 minutes on a blackboard - it should be "beautiful" in some sense, but as always this is highly subjective - it is great if there is at least a very slight relation to your own area, some similar ideas or concepts or the like.
So my question is now: do you know any nice results in your respective fields that meets the above criteria? An interesting problem that has recently been solved by an elegant method, a surprising twist to a long-standing conjecture, or just a very beautiful and self-contained paper that is accessible for a non-expert? :) I have about 2 weeks to prepare.
Here is some more background information: My own area is stochastic analysis and stochastic PDEs, so anything too close to that area does not work. One of the examiners is from algebraic/arithmetic geometry, and that person would be the natural target audience for the question (although it should also not be too close to their area of expertise, of course). My background: I've had courses on numerical analysis, PDEs, dynamical systems, mathematical physics, and of course lots of probability and analysis. I don't know much about algebra or geometry, although I am trying to teach myself a bit of abstract algebra at the moment.
I have thought about some things myself: I've been wondering what these "perfectoid spaces" are, and there is a relation between stochastic analysis and algebra via so-called Hopf algebras, but both of these topics seem too vast and complicated and probably too hard to understand for a general mathematical audience, at least if they are being explained by someone of my capabilities.
Any ideas are welcome!
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