(Standard post on cardinals, as a prerequisite for forthcoming theory model post.)

An ordinal measures a total ordering. However, it does not do a fantastic job at measuring size. For example, there is a bijection between the elements of $\omega$ and $\omega+1$:

$$\begin{array}{rccccccc} \omega+1 = & \{ & \omega & 0 & 1 & 2 & \dots & \} \\ \omega = & \{ & 0 & 1 & 2 & 3 & \dots & \}. \end{array}$$

In fact, as you likely already know, there is even a bijection between $\omega$ and $\omega^2$:

$$\begin{array}{l|cccccc} + & 0 & 1 & 2 & 3 & 4 & \dots …$$

This one confused me for a long time, so I figured I should write this down before I forgot again.

Let $M$ be an abstract smooth manifold. We want to define the notion of a tangent vector to $M$ at a point $p \in M$. With that, we can define the tangent space $T_p(M)$, which will just be the (real) vector space of tangent vectors at $p$.

Geometrically, we know what this should look like for our usual examples. For example, if $M = S^1$ is a circle embedded in $\mathbb R^2$, then the tangent vector at a point $p$ should just look like a vector running off tangent to the circle.

Similarly, given a …

There are some notes on valuations from the first lecture of Math 223a at Harvard.

1. Valuations

Let $k$ be a field.

Definition 1. A valuation $$\left\lvert - \right\rvert : k \rightarrow \mathbb R_{\ge 0}$$ is a function obeying the axioms

• $\left\lvert \alpha \right\rvert = 0 \iff \alpha = 0$.
• $\left\lvert \alpha\beta \right\rvert = \left\lvert \alpha \right\rvert \left\lvert \beta \right\rvert$.
• Most importantly: there should exist a real constant $C$, such that $\left\lvert 1+\alpha \right\rvert < C$ whenever $\left\lvert \alpha \right\rvert \le 1$.

The third property is the interesting one. Note in particular it can be rewritten as $\mid a+b\mid <C\max \{\mid a\mid ,\mid \mathrm{b\; \dots }$

This blog post corresponds to my newest olympiad handout on mixtilinear incircles.

My favorite circle associated to a triangle is the $A$-mixtilinear incircle. While it rarely shows up on olympiads, it is one of the richest configurations I have seen, with many unexpected coincidences showing up, and I would be overjoyed if they become fashionable within the coming years.

Here’s the picture:

The points $D$ and $E$ are the contact points of the incircle and $A$-excircle on the side $BC$. Points $M_A$, $M_B$, $M_C$ are the midpoints of the arcs.

As a challenge to my recent USAMO class (I taught at A* Summer Camp this year), I asked them to find as many “coincidences” in the picture as I could …

Here I talk about my first project at the Emory REU. Prerequisites for this post: some familiarity with number fields.

1. Motivation: Arithmetic Progressions

Given a property $P$ about primes, there’s two questions we can ask:

1. How many primes $\le x$ are there with this property?
2. What’s the least prime with this property?

As an example, consider an arithmetic progression $a$, $a+d$, …, with $a < d$ and $\gcd(a,d) = 1$. The strong form of Dirichlet’s Theorem tells us that basically, the number of primes $\equiv a \pmod d$ is $\frac 1d$ the total number of primes. Moreover, the celebrated Linnik’s Theorem tells us that the first prime is $O({\mathrm{d\; \dots }}^{}$

In this post I will sketch a proof Dirichlet Theorem’s in the following form:

Theorem 1 (Dirichlet’s Theorem on Arithmetic Progression)

Let $$\psi(x;q,a) = \sum_{\substack{n \le x \\ n \equiv a \mod q}} \Lambda(n).$$ Let $N$ be a positive constant. Then for some constant $C(N) > 0$ depending on $N$, we have for any $q$ such that $q \le (\log x)^N$ we have $$\psi(x;q,a) = \frac{1}{\phi(q)} x + O\left( x\exp\left(-C(N) \sqrt{\log …$$

Prerequisites for this post: previous post, and complex analysis. For this entire post, $s$ is a complex variable with $s = \sigma + it$.

1. The $\Gamma$ function

So there’s this thing called the Gamma function. Denoted $\Gamma(s)$, it is defined by $$\Gamma(s) = \int_0^{\infty} x^{s-1} e^{-x} dx$$ as long as $\sigma > 0$. Here are its values at the first few integers:

$$\begin{aligned} \Gamma(1) &= 1 \\ \Gamma(2) &= 1 \\ \Gamma(3) &= 2 \\ \Gamma(4) &= 6 \\ \Gamma(5) &= 24. \end{aligned}$$

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 …

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 …