The Schrödinger equation on a finite graph

One of the most important discoveries in the history of science is the structure of the periodic table. This structure is a consequence of how electrons cluster around atomic nuclei and is essentially quantum-mechanical in nature. Most of it (the part not having to do with spin) can be deduced by solving the Schrödinger equation [...]

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Nationals 2010

MIT is hosting the United States Rubik’s Cube Championships this summer, August 6-8. All ages welcome! Normally I wouldn’t post about such things, but I happen to be a member of the Rubik’s Cube club here, and Some people use the Rubik’s cube group to motivate group theory. I’m a fan of hands-on mathematics, and [...]

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Fractional linear transformations and elliptic curves

The following two lemmas might be encountered in a basic course in complex analysis (the first in a basic course in group theory, even). Lemma 1: Fix a field . The group of fractional linear transformations acts triple transitively on and the stabilizer of any triplet of distinct points is trivial. Lemma 2: The group [...]

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Newton’s sums, necklace congruences, and zeta functions

The goal of this post is to give a purely combinatorial proof of Newton’s sums which would have interrupted the flow of the previous post. Recall that, in the notation of the previous post, Newton’s sums (also known as the first Newton-Girard identity) state that . One way to motivate a combinatorial proof is to [...]

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GILA V: The Polya enumeration theorem and applications

I ended the last post by asking whether the proof of baby Polya extends to the multi-parameter setting where we want to keep track of how many of each color we use. In fact, it does. First, we should specify what exactly we’re trying to compute. Recall the setup: we have colors (represented by variables [...]

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GILA III: The orbit-counting lemma and baby Polya

The orbit-stabilizer theorem implies, very immediately, one of the most important counting results in group theory. The proof is easy enough to give in a paragraph now that we’ve set up the requisite machinery. Remember that we counted fixed points by looking at the size of the stabilizer subgroup. Let’s count them another way. Since [...]

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GILA II: Orbits, stabilizers, and classifying group actions

Now that we’ve discussed group actions a bit, it’s time to characterize them. In this post I’d like to take a leaf from Tim Gowers’ book and try to make each step taken in the post “obvious.” While the content of the proofs is not too difficult, its motivation is rarely discussed. First, it’s important [...]

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GILA I: Group actions and equivalence relations

Sometimes I worry that I should be more consistent or more lenient about the background I expect of my readers. (Readers, I have to admit that I still don’t really know who you are!) Considering how important I think it is that mathematicians value communicating their ideas to non-specialists (what John Armstrong calls the Generally [...]

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