Currently the Riemann-Roch theorem is my nemesis, and I stumbled on Matt Baker’s math blog while I was looking for some help understanding figuring out how to use it. The post I came across, Riemann-Roch for Graphs and Applications, was not what I was looking for, but I’m glad I found it! Baker, a math professor at Georgia Tech, describes the Riemann-Roch theorem for graphs in fairly straightforward language and also gives some background about how he and his coauthor Serguei Norine discovered it. At the beginning it was a theorem in search of a precise formulation: “I stumbled upon the idea that there ought to be a graph-theoretic avatar of the Riemann-Roch Theorem while investigating ‘p-adic Riemann surfaces’ (for the experts: Berkovich curves). At the time I didn’t know precisely how to formulate the combinatorial Riemann-Roch theorem, but I knew that the following should be a special case…” I like seeing the incremental development of the idea, and it’s nice to see how many undergraduates were involved at different points in the process. His explanation of the theorem involves a game you can play on a graph, and he includes an applet for the game created by REU student Adam Tart.
Another post that caught my eye, probably because of the pictures, was Quadratic Reciprocity and Zolotarev’s Lemma. Who knew quadratic reciprocity could be described with a deck of cards? Baker writes, “Some time ago I reformulated Zolotarev’s argument (as presented here) in terms of dealing cards and I posted a little note about it on my web page. After reading my write-up (which was unfortunately opaque in a couple of spots), Jerry Shurman was inspired to rework the argument and he came up with this elegant formulation which I think may be a ‘proof from the book’. The following exposition is my own take on Jerry’s argument.” I’m not going to try to explain how quadratic reciprocity is like dealing cards. You should just go read his post.
Baker’s blog has several other posts that give background information and exposition for his research papers. I definitely appreciate reading about the motivation and false starts that usually get hidden away in the formal presentation of research. If that sounds like your thing, maybe you’d like to head on over and check it out.