Prerequisites for this post: definition of Dirichlet convolution, and big $O$-notation.

Normally I don’t like to blog about something until I’m pretty confident that I have a reasonably good understanding of what’s happening, but I desperately need to sort out my thoughts, so here I go…

1. Primes

One day, an alien explorer lands on Earth in a 3rd grade classroom. He hears the teacher talk about these things called primes. So he goes up to the teacher and asks “how many primes are there less than $x$?”.

Answer: “uh. . .”.

Maybe that’s too hard, so the alien instead asks “about how many primes are there less than $x$?”.

This is again greeted with silence. Confused, the alien asks a bunch of the teachers, who all respond similarly, but then someone mentions that in the last couple hundred years, someone …

(It appears to be May 7 – good luck to all the national MathCounts competitors tomorrow!)

1. An 8.044 Problem

Recently I saw a 8.044 physics problem set which contained the problem

Consider a system of $N$ almost independent harmonic oscillators whose energy in a microcanonical ensemble is given by $E = \frac 12 \hbar \omega N + \hbar \omega M$. Show that this energy can be obtained is $\frac{(M+N-1)!}{M!(N-1)!}$.

Once you remove the physics fluff, it immediately reduces to

Show the number of nonnegative integer solutions to $M = \sum_{i=1}^N n_i$ is $\frac{(M …$

This is a draft of an appendix chapter for my Napkin project.

In the world of olympiad math, there’s a famous functional equation that goes as follows: $$f : {\mathbb R} \rightarrow {\mathbb R} \qquad f(x+y) = f(x) + f(y).$$ Everyone knows what its solutions are! There’s an obvious family of solutions $f(x) = cx$. Then there’s also this family of… uh… noncontinuous solutions (mumble grumble) pathological (mumble mumble) Axiom of Choice (grumble).

There’s also this thing called Zorn’s Lemma. It sounds terrifying, because it’s equivalent to the Axiom of Choice, which is also terrifying because why not.

In this post I will try to de-terrify …

In high school, I hated English class and thought it was a waste of time. Now I’m in college, and I still hate English class and think it’s a waste of time. (Nothing on my teachers, they were all nice people, and I hope they’re not reading this.)

However, I no longer think writing itself is a waste of time. Otherwise, I wouldn’t be blogging, even about math. This post explains why I changed my mind.

1. Guts

My impression is that teachers in high school got it all wrong.

In high school, students are told to learn algebra because “we all use math every day”. This is obviously false, and somehow the students eventually are led to believe it.

You can’t actually be serious. Do people really think that knowing the Pythagorean Theorem will help in your daily life? I sure don’t, and …

In the last week of December I got a position as the morning instructor for the A* USAMO winter camp. Having long lost interest in coaching for short-answer contests, I’d been looking forward to an opportunity to teach an olympiad class for ages, and so I was absolutely psyched for that week. In this post I’ll talk about some of the thoughts I had while teaching, in no particular order.

1. Class format

Here were the constraints I was working with. After removing guest lectures, exams, and so on I had four days of teaching time, one for each of the four olympiad subjects (algebra, geometry, combinatorics, number theory). I taught the morning session, meaning I had a three-hour block each day (with a 15-minute break). I had a wonderfully small class – just five students.

Here’s the format I used for the class, which seemed to work …

Good luck to everyone taking the January TST for the IMO 2015 tomorrow!

Now that we have products of irreducibles under our belt, I’ll talk about the finite regular representation and use it to derive the following two results about irreducibles.

1. The number of (isomorphsim classes) of irreducibles $\rho_\alpha$ is equal to the number of conjugacy classes of $G$.
2. We have $\left\lvert G \right\rvert = \sum_\alpha \left( \dim \rho_\alpha \right)^2$.

These will actually follow as corollaries from the complete decomposition of the finite regular representation.

In what follows $k$ is an algebraically closed field, $G$ is a finite group, and the characteristic of $k$ does not divide $\left\lvert G \right\rvert$. As a reminder, here are the …

Happy New Year to all! A quick reminder that $2015 = 5 \cdot 13 \cdot 31$.

This post will set the stage by examining products of two representations. In particular, I’ll characterize all the irreducibles of $G_1 \times G_2$ in terms of those for $G_1$ and $G_2$. This will set the stage for our discussion of the finite regular representation in Part 4.

In what follows $k$ is an algebraically closed field, $G$ is a finite group, and the characteristic of $k$ does not divide $\left\lvert G \right\rvert$.

1. Products of representations

First, I need to tell you how to take the product of two representations.

Definition. Let $G_1$ and $G_{\mathrm{2\; \dots }}$

Merry Christmas!

In the previous post I introduced the idea of an irreducible representation and showed that except in fields of low characteristic, these representations decompose completely. In this post I’ll present Schur’s Lemma at talk about what Schur and Maschke tell us about homomorphisms of representations.

1. Motivation

Fix a group $G$ now, and consider all isomorphism classes of finite-dimensional representations of $G$. We’ll denote this set by $\operatorname{Irrep}(G)$. Maschke’s Theorem tells us that any finite-dimensional representation $\rho$ can be decomposed as $$\bigoplus_{\rho_\alpha \in \operatorname{Irrep}(G)} \rho_{\alpha}^{\oplus n_\alpha}$$ where $n_\alpha$ is some nonnegative integer. This begs the question: what is $n_\alpha$

Good luck to everyone taking the December TST tomorrow!

The goal of this post is to give the reader a taste of representation theory, a la Math 55a. In theory, this post should be accessible to anyone with a knowledge of group actions and abstract vector spaces.

Fix a ground field $k$ (for all vector spaces). In this post I will introduce the concept of representations and irreducible representations. Using these basic definitions I will establish Maschke’s Theorem, which tells us that irreducibles and indecomposables are the same thing.

1. Definition and examples

Let $G$ be a group.

Definition. A representation of $G$ consists of a pair $\rho = (V, \cdot_\rho)$ where $V$ is a vector space over $k$ and $\cdot_\rho$ is a (left) group action of $G$

In this post I’ll cover three properties of isogonal conjugates which were only recently made known to me. These properties are generalization of some well-known lemmas, such as the incenter/excenter lemma and the nine-point circle.

1. Definitions

Let $ABC$ be a triangle with incenter $I$, and let $P$ be any point in the interior of $ABC$. Then we obtain three lines $AP$, $BP$, $CP$. Then the reflections of these lines across lines $AI$, $BI$, $CI$ always concur at a point $Q$ which is called the isogonal conjugate of $P$. (The proof of this concurrence follows from readily from Trig Ceva.) When $P$