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## The definition of a function

Tran Chieu Minh recently asked on Math Overflow why the definition of a function is asymmetric: it treats the domain and the codomain differently. In other words, why privilege functions over relations? (Edit: Tran has since deleted the question.)

I think this is an interesting question, but I don’t think it’s appropriate for Math Overflow, so I’m redirecting discussion of it here. My own answer is below.

## Putnam 2009

Kent Merryfield continues his tradition of posting the Putnam results every year. Before I came to MIT I didn’t understand why this was necessary, but after taking it I learned that the results are only sent to the relevant officials at each school, who distribute the results in a manner of their choosing. This seems to me to be a pretty inefficient system, and I’ve never really understood why it’s done this way, so kudos to Dr. Merryfield for making this information more widely available.

## 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 $F$. The group of fractional linear transformations $PGL_2(F)$ acts triple transitively on $\mathbb{P}^1(F)$ and the stabilizer of any triplet of distinct points is trivial.

Lemma 2: The group of fractional linear transformations on $\mathbb{P}^1(\mathbb{C})$ preserving the upper half plane $\mathbb{H} = \{ z \in \mathbb{C} | \text{Im}(z) > 0 \}$ is $PSL_2(\mathbb{R})$.

I used to only know extremely boring computational proofs of both of these statements. However, I now know better! Today I’d like to give shorter and conceptual proofs of both of these, and then briefly discuss how they come about in the study of elliptic curves (a subject I’d like to talk about in more detail once this semester is over).

## Update and wisdom

I forgot to mention some things in the last update:

• This summer I will be at MIT for SPUR, which should be a lot of fun. I debated for awhile between SPUR and applying to be a PROMYS counselor, but after thinking about it I’m too excited about the opportunity for research to turn it down.
• Next year I will be studying abroad at Cambridge. I’m really looking forward to this. Maybe I’ll even meet Timothy Gowers!

Incidentally, when I went to talk to David Jerison about SPUR, the conversation shifted to RSI (which he is also involved with). I told him I’d always been a little embarrassed about my RSI paper because the bulk of the argument in the first part of the paper is a rediscovery of a standard lemma in linear algebra (that two quadratic forms can be simultaneously diagonalized). Jerison replied that this wasn’t something to be embarrassed about; to the contrary, discovering a result “in the wild” meant that I would better understand its value (or something to that effect). Wise words.

I’ve been reading about a lot of interesting stuff; hopefully I’ll get some time to post about it during spring break.

## Walks on graphs and tensor products

Recently I asked a question on MO about some computations I’d done with Catalan numbers awhile ago on this blog, and Scott Morrison gave a beautiful answer explaining them in terms of quantum groups. Now, I don’t exactly know how quantum groups work, but along the way he described a useful connection between walks on graphs and tensor products of representations which at least partially explains one of the results I’d been wondering about and also unites several other related computations that have been on my mind recently.

Let $G$ be a compact group and let $\text{Rep}(G)$ denote the category of finite-dimensional unitary representations of $G$. As in the finite case, due to the existence of Haar measure, $\text{Rep}(G)$ is semisimple (i.e. every unitary representation decomposes uniquely into a sum of irreducible representations), and via the diagonal action it comes equipped with a tensor product with the property that the character of the tensor product is the product of the characters of the factors.

Question: Fix a representation $V \in \text{Rep}(G)$. What is the multiplicity of the trivial representation in $V^{\otimes n}$?