During the summer theory seminar for my research group, I like to cover a topic that is mathematically challenging, but not something that any of us would normally learn about in the course of our day-to-day researchThis is a great idea ! Usually, during the semester, I'm in market-driven mode, choosing topics that are more accessible, and are likely to draw a larger audience. But summer is a good time for harder material since you have smaller self-selected group and they're motivated.
Last summer I ran a "why can't we solve P vs NP" seminar with three students - we went through the standard obstacles, and along the way learnt a fair amount of complexity theory - it was a lot of fun. This summer we're doing lattice theory - I have selfish reasons for choosing this topic (some of my recent work needs it), and it's a very accessible (and relevant!) area, while still being nontrivial enough to get students to think.
We considered a number of other topics as well before settling on this one - we might even return to some of them later. They were, in no particular order.
- algorithmic game theory (from this book)
- concentration inequalities for random variables]
- quantum computing (possibly from these notes)
- complexity theory fundamentals (almost surely from Luca Trevisan's ongoing class)
- geometric flip theory (mulmuley's work on P vs NP). I suspect we'd go with this paper and these videos (and the associated reading material)
- A complexity series: the unique games conjecture, quantum computations as shortest paths and classical and quantum lower bounds + QIP=PSPACE)
- Fourier methods for boolean function analysis
- Geometric Approximations (full book)