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 …

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 $P$

This is a continuation of my earlier set theory post. In this post, I’ll describe the next three axioms of ZF and construct the ordinal numbers.

1. The previous axioms

As review, here are the natural descriptions of the five axioms we covered in the previous post.

Axiom 1 (Extensionality). Two sets are equal if they have the same elements.

Axiom 2 (Empty Set Exists). There exists an empty set $\varnothing$ which contains no elements

Axiom 3 (Pairing). Given two elements $x$ and $y$, there exists a set $\{x,y\}$ containing only those two elements. (It is permissible to have $x=y$, meaning that if $x$ is a set then so is $\{x\}$.)

Axiom 4 (Union). Given a set $a$, we can create $\cup a$, the union of the …

Back in high school, I sometimes wondered what all the big deal about ZFC and the Axiom of Choice was, but I never really understood what I read in the corresponding Wikipedia page. In this post, I’ll try to explain what axiomatic set theory is trying to do in a way accessible to those with just a high school background.

1. Motivation

What we’re going to try to lay out something like a “machine code” for math: a way of making math completely rigorous, to the point where it can be verified by a machine. This would make sure that our foundation on which we do our high-level theorem proving is sound. As we’ll see in just a moment, this is actually a lot harder to do than it sounds – there are some traps if we try to play too loosely with our definitions.

First of all …

This is an expanded version of an answer I gave to a question that came up while I was assisting the 2014-2015 WOOT class. It struck me as an unusually good way to motivate higher math using stuff that people notice in high school but for some reason decide to not think about.

In high school precalculus, you’ll often be asked to find the roots of some polynomial with integer coefficients. For instance, $$x^3 - x^2 - x - 15 = (x-3)(x^2+2x+5)$$ has roots $3$, $1+2i$, $-1-2i$. Or as another example, $$x^3-3x^2-2x+2=(x+1)(x^2-4x+\mathrm{2\; \dots }$$

I always wondered whether I could generate olympiad geometry problems by simply drawing lines and circles at random until three lines looked concurrent, four points looked concyclic, et cetera. From extensive experience you certainly get the feeling that this ought to be the case – there are tons and tons of problems out there but most of them have relatively simple statements, not involving more than a handful of points. Often I think, “I bet I could have stumbled upon this result just by drawing things at random”.

So one night, I decided to join the tangency point of $A$-mixtilinear circle with the orthocenter of a triangle $ABC.$ You can guess about how well that went. Nothing came up after two hours of messing around randomly.

Surprisingly, though, I found almost by accident that the following modification has had significant success:

1. First …