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I’ve been reading a lot of mathematics lately, but I don’t feel capable of explaining most of what I’ve been reading about, so I’m not sure what to blog about these days. Fortunately, SPUR will be starting soon, so I’ll start focusing on relevant material for my project eventually. Until then, here are some more random updates.

• Martin Gardner and Walter Rudin both recently passed away. They will be sorely missed by the mathematical community, although I can’t say I’m particularly qualified to eulogize about either.
• For my number theory seminar with Scott Carnahan I wrote a paper describing an important corollary of the Eichler-Shimura relation in the theory of modular forms. The actual relation is somewhat difficult to state, but the important corollary relates the number of points on certain elliptic curves $E$ over finite fields to the Fourier coefficients of certain modular forms of weight $2$. You can find the paper here. Although the class is over, corrections and comments are of course welcome. (Though I hope Scott doesn’t change my grade if someone spots a mistake he missed!)
• If you’re at all interested in the kind of mathematics where planar diagrams are used instead of traditional algebraic notation for computation, you should read Joachim Kock’s excellent book Frobenius Algebras and 2D Topological Quantum Field Theories. The book is much less intimidating than its title might suggest, and it is full of enlightening pictures and discussions. You might also be interested in a related MO question.

### 4 Responses

1. Had you already studied regular quantum mechanics before looking at this book?

• Nope. There’s no physics in the book; the author himself freely admits to being ignorant of the physics behind the subject. But there is some beautiful mathematics. If you’re interested in the physics, there are some good links in the MO question above.

2. What’s your research going to be on? (My two undergrad research projects were some of the most gratifying experience of my time at college.)

re: graphical representations, you may be interested in my graph-theoretical attack on the EDP. In a “pure” sense it’s messing with the number system at a primal level, redefining “counting” by bring it to a graphical representation level. (In practice I’m just wanting some discrepancy-larger-than-2 results that DON’T require an example out in the thousands.)

• I’m not sure; I’m still tossing around ideas with my mentor.