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Archive for August, 2010

Hiatus

I will be unable to blog for the next two weeks. Two weeks after that, I’ll be moving into Christ’s College at the University of Cambridge! That is, if I can send my visa application in on time…

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The 2010 Fields Medalists

The 2010 Fields Medalists have just been announced! The moment I found their names on Twitter, they were already on the Wikipedia article. Go figure.

I am not familiar with any of their names, except that Ngô Bảo Châu’s name was tossed around on the blogosphere awhile back because of his proof of the fundamental lemma.

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I finally learned the solution to a little puzzle that’s been bothering me for awhile.

The setup of the puzzle is as follows. Let G be a weighted undirected graph, e.g. to each edge i \leftrightarrow j is associated a non-negative real number a_{ij}, and let A be the corresponding weighted adjacency matrix. If A is stochastic, one can interpret the weights a_{ij} as transition probabilities between the vertices which describe a Markov chain. (The undirected condition then means that the transition probability between two states doesn’t depend on the order in which the transition occurs.) So one can talk about random walks on such a graph, and between any two vertices the most likely walk is the one which maximizes the product of the weights of the corresponding edges.

Suppose you don’t want to maximize a product associated to the edges, but a sum. For example, if the vertices of G are locations to which you want to travel, then maybe you want the most likely random walk to also be the shortest one. If E_{ij} is the distance between vertex i and vertex j, then a natural way to do this is to set

a_{ij} = e^{- \beta E_{ij}}

where \beta is some positive constant. Then the weight of a path is a monotonically decreasing function of its total length, and (fudging the stochastic constraint a bit) the most likely path between two vertices, at least if \beta is sufficiently large, is going to be the shortest one. In fact, the larger \beta is, the more likely you are to always be on the shortest path, since the contribution from any longer paths becomes vanishingly small. As \beta \to \infty, the ring in which the entries of the adjacency matrix lives stops being \mathbb{R} and becomes (a version of) the tropical semiring.

That’s pretty cool, but it’s not what’s been puzzling me. What’s been puzzling me is that matrix entries in powers of A look an awful lot like partition functions in statistical mechanics, with \beta playing the role of the inverse temperature and E_{ij} playing the role of energies. So, for awhile now, I’ve been wondering whether they actually are partition functions of systems I can construct starting from the matrix A. It turns out that the answer is yes: the corresponding systems are called one-dimensional vertex models, and in the literature the connection to matrix entries is called the transfer matrix method. I learned this from an expository article by Vaughan Jones, “In and around the origin of quantum groups,” and today I’d like to briefly explain how it works.

(more…)

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I won’t get around to substantive blog posts for at least a few more days, so here is a puzzle.

We usually thinks of groups that occur in nature as permutations of a set which preserve some structure on that set. For example, the general linear group \text{GL}_n(F) preserves the structure of being an F-vector space of dimension n.

What structure does the special linear group \text{SL}_n(F) preserve?

(I have an answer, but I’m curious if it can be stated in a more elementary way.)

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