Summer Seminars

Jared Saia at UNM and Machinations is working through Wilf's generatingfunctionology over the summer with his students. He says,

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 research

This 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) 

Especially for a summer seminar, it's important to have material that's generally available for free, since students might not want to purchase a book for a topic that's off the beaten track.