Going beyond your comfort zone

When someone linked me to Ravi Vakil’s advice for potential graduate students, I was struck by the following passage: …[M]athematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you’ll never get anywhere. Instead, you’ll have tendrils [...]

Read Full Post »

I hate axioms

(A more appropriate title for this post would probably be “I hate Bourbaki,” but I like it as is.) I spend a lot of my free time reading research papers, usually in combinatorics; those tend to require the least background. Today I decided to read everything I could find written by one of the great [...]

Read Full Post »

GILA VI: The cycle index polynomials of the symmetric groups

In the previous post we used the Polya enumeration theorem to give a sneaky, underhanded proof that . If you’ve never seen the exponential function used like this, you might be wondering how it can be “explained.” To explore this question, I’d like to give three other proofs of this result, the last of which [...]

Read Full Post »

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 [...]

Read Full Post »

GILA IV: The unreasonable effectiveness of generating functions in the combinatorial sciences

We’ve got everything we need to prove the Polya enumeration theorem. To state the theorem, however, requires the language of generating functions, so I thought I’d take the time to establish some of the important ideas. It isn’t possible to do justice to the subject in one post, so I’ll start with some references. Many [...]

Read Full Post »

Self-referential jokes in popular culture

I can’t resist mentioning a joke I heard from an episode of American Dad. Stan Smith has this to say about his training as a negotiator: Hey, you’ve got one of the CIA’s top negotiators on your side. Y’know, I negotiated my way through negotiator training. I should’ve failed the hell out of that class. [...]

Read Full Post »

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 [...]

Read Full Post »

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 [...]

Read Full Post »

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 [...]

Read Full Post »

Young diagrams, q-analogues, and one of my favorite proofs

I’ve decided to start blogging a little more about the algebraic combinatorics I’ve learned over the past year. In particular, I’d like to present one of my favorite proofs from Stanley’s Enumerative Combinatorics I. The theory of Young tableaux is a great example of the richness of modern mathematics: although they can be defined in [...]

Read Full Post »