Median Putnam contestants, willing to devote one of the last Saturdays before final exams to a math test, are likely to receive an advanced degree in the sciences. It is counterproductive on many levels to leave them feeling like total idiots.

— Bruce Reznick, “Some Thoughts on Writing for the Putnam”

Last February I made a big public apology for having caused one of the biggest scoring errors in HMMT history, causing a lot of changes to the list of top individual students. Pleasantly, I got some nice emails from coaches who reminded me that most students and teams do not place highly in the tournament, and at the end of the day the most important thing is that the contestants enjoyed the tournament.

So now I decided I have to apologize for 2016, too.

The story this time is that I inadvertently sent over 100 students home having solved two …

In a previous post I tried to make the point that math olympiads should not be judged by their relevance to research mathematics. In doing so I failed to actually explain why I think math olympiads are a valuable experience for high schoolers, so I want to make amends here.

1. Summary

In high school I used to think that math contests were primarily meant to encourage contestants to study some math that is (much) more interesting than what’s typically shown in high school. While I still think this is one goal, and maybe it still is the primary goal in some people’s minds, I no longer believe this is the primary benefit.

My current belief is that there are two major benefits from math competitions:

1. To build a social network for gifted high school students with similar interests.
2. To provide a challenging experience that lets gifted students …

I recently had a combinatorics paper appear in the EJC. In this post I want to brag a bit by telling the “story” of this paper: what motivated it, how I found the conjecture that I originally did, and the process that eventually led me to the proof, and so on.

This work was part of the Duluth REU 2017, and I thank Joe Gallian for suggesting the problem.

1. Background

Let me begin by formulating the problem as it was given to me. First, here is the definition and notation for a “block-ascending” permutation.

Definition 1. For nonnegative integers $a_1$, …, $a_n$ an $(a_1, \dots, a_n)$-ascending permutation is a permutation on $\{1, 2, \dots, a_1 + \dots + a_n\}$ whose descent set is …

This is a rare politics post; I’ll try to keep this short and emotion-free. If parts of this are wrong, please correct me. More verbose explanations here, here, here, here, longer discussion here.

Suppose you are a math PhD student at MIT. Officially, this “costs” $50K a year in tuition. Fortunately this number is meaningless, because math PhD students serve time as teaching assistants in exchange for having the nominal sticker price waived. MIT then provides a stipend of about $25K a year for these PhD student’s living expenses. This stipend is taxable, but it’s small and you’d pay only $1K-$2K in federal taxes (about 6%).

The new GOP tax proposal strikes 26 U.S. Code 117(d) which would cause the $50K tuition waiver to also become taxable income: the PhD student would pay taxes on an “income” of $75K, at tax brackets of …

I wanted to quickly write this proof up, complete with pictures, so that I won’t forget it again. In this post I’ll give a combinatorial proof (due to Joyal) of the following:

Theorem 1 (Cayley’s Formula)

The number of trees on $n$ labeled vertices is $n^{n-2}$.

Proof: We are going to construct a bijection between

• Functions $\{1, 2, \dots, n\} \rightarrow \{1, 2, \dots, n\}$ (of which there are $n^n$) and
• Trees on $\{1, 2, \dots, n\}$ with two distinguished nodes $A$ and $B$ (possibly $A=B$).

This will imply the answer.

Let’s look at the first piece of data. We can visualize it as $\mathrm{n\; \dots }$

I’m reading through Primes of the Form $x^2+ny^2$, by David Cox (it’s good!). Here are the high-level notes I took on the first chapter, which is about the theory of quadratic forms.

(Meta point re blog: I’m probably going to start posting more and more of these more high-level notes/sketches on this blog on topics that I’ve been just learning. Up til now I’ve been mostly only posting things that I understand well and for which I have a very polished exposition. But the perfect is the enemy of the good here; given that I’m taking these notes for my own sake, I may as well share them to help others.)

1. Overview

Definition 1. For us a quadratic form is a polynomial $Q=Q(x,y)=a{x}^{\mathrm{2\; \dots }}$

(This is a bit of a follow-up to the solution reading post last month. Spoiler warnings: USAMO 2014/6, USAMO 2012/2, TSTST 2016/4, and hints for ELMO 2013/1, IMO 2016/2.)

I want to say a little about the process which I use to design my olympiad handouts and classes these days (and thus by extension the way I personally think about problems). The short summary is that my teaching style is centered around showing connections and recurring themes between problems.

Now let me explain this in more detail.

1. Main ideas

Solutions to olympiad problems can look quite different from one another at a surface level, but typically they center around one or two main ideas, as I describe in my post on reading solutions. Because details are easy to work out once you have the main idea, as far as learning is concerned you can …

(Ed Note: This was earlier posted under the incorrect title “On Designing Olympiad Training”. How I managed to mess that up is a long story involving some incompetence with Python scripts, but this is fixed now.)

Spoiler warnings: USAMO 2014/1, and hints for Putnam 2014 A4 and B2. You may want to work on these problems yourself before reading this post.

1. An Apology

At last year’s USA IMO training camp, I prepared a handout on writing/style for the students at MOP. One of the things I talked about was the “ocean-crossing point”, which for our purposes you can think of as the discrete jump from a problem being “essentially not solved” ($0+$) to “essentially solved” ($7-$). The name comes from a Scott Aaronson post:

Suppose your friend in Boston blindfolded you, drove you around for twenty minutes, then took the blindfold off …

In this post we’ll make sense of a holomorphic square root and logarithm. Wrote this up because I was surprised how hard it was to find a decent complete explanation.

Let $f \colon U \rightarrow \mathbb C$ be a holomorphic function. A holomorphic $n$-th root of $f$ is a function $g \colon U \rightarrow \mathbb C$ such that $f(z) = g(z)^n$ for all $z \in U$. A logarithm of $f$ is a function $g \colon U \rightarrow \mathbb C$ such that $f(z) = e^{g(z)}$ for all $z \in U$.

The main question …

In Spring 2016 I was taking 18.757 Representations of Lie Algebras. Since I knew next to nothing about either Lie groups or algebras, I was forced to quickly learn about their basic facts and properties. These are the notes that I wrote up accordingly. Proofs of most of these facts can be found in standard textbooks, for example Kirillov.

1. Lie groups

Let $K = \mathbb R$ or $K = \mathbb C$, depending on taste.

Definition 1. A Lie group is a group $G$ which is also a $K$-manifold; the multiplication maps $G \times G \rightarrow G$ (by $(g_1, g_2) \mapsto g_1g_2$) and the inversion map $G \rightarrow G$ (by $g\mapsto {\mathrm{g\; \dots }}^{}$