The following is an excerpt from a current work of mine. I thought I’d share it here, as some people have told me they enjoyed it.

As I’ll stress repeatedly, a matrix represents a linear map between two vector spaces. Writing it in the form of an $m \times n$ matrix is merely a very convenient way to see the map concretely. But it obfuscates the fact that this map is, well, a map, not an array of numbers.

If you took high school precalculus, you’ll see everything done in terms of matrices. To any typical high school student, a matrix is an array of numbers. No one is sure what exactly these numbers represent, but they’re told how to magically multiply these arrays to get more arrays. They’re told that the matrix

$$(\begin{array}{cccc}1& 0& \dots & 0\\ 0& 1& \dots & 0\\ \mathrm{⋮}& \mathrm{⋮}& \ddots & \mathrm{⋮}\\ 0& \mathrm{0\; \dots }\end{array}$$

This post has now become a PDF handout on my website.

Let $V$ be a normed finite-dimensional real vector space and let $U \subseteq V$ be an open set. A vector field on $U$ is a function $\xi : U \rightarrow V$. (In the words of Gaitsgory: “you should imagine a vector field as a domain, and at every point there is a little vector growing out of it.”)

The idea of a differential equation is as follows. Imagine your vector field specifies a velocity at each point. So you initially place a particle somewhere in $U$, and then let it move freely, guided by the arrows in the vector field. (There are plenty of good pictures online.) Intuitively, for nice $\xi$ it should be the case that the trajectory resulting is unique. This is the main take-away; the proof itself is just for …

(EDIT: These solutions earned me a slot in N1, top 16.)

I solved eight problems on the Putnam last Saturday. Here were the solutions I found during the exam, plus my repaired solution to B3 (the solution to B3 I submitted originally had a mistake).

Some comments about the test. I thought that the A test had easy problems: problems A1, A3, A4 were all routine, and problem A5 is a little long-winded but nothing magical. Problem A2 was tricky, and took me well over half the A session. I don’t know anything about A6, but it seems to be very hard.

The B session, on the other hand, was completely bizarre. In my opinion, the difficulty of the problems I did attempt was $$B4 \ll B1 \ll B5 < B3 < B2.$$

Model theory is really meta, so you will have to pay attention here.

Roughly, a “model of $\mathsf{ZFC}$” is a set with a binary relation that satisfies the $\mathsf{ZFC}$ axioms, just as a group is a set with a binary operation that satisfies the group axioms. Unfortunately, unlike with groups, it is very hard for me to give interesting examples of models, for the simple reason that we are literally trying to model the entire universe.

1. Models

(Prototypical example for this section: $(\omega, \in)$ obeys $\mathrm{PowerSet}$, $V_\kappa$ is a model for $\kappa$ inaccessible (later).)

Definition 1. A model $\mathscr M$ consists of a set $M$ and a binary relation $E \subseteq M \times …$

(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 $\omega^2$:

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

For Git users:

I’ve recently discovered the joy that is git aliases, courtesy of this blog post. To return to the favor, I thought I’d share the ones that I came up with.

For those of you that don’t already know, Git allows you to make aliases – shortcuts for commands. Specifically, if you add the following lines to your .gitconfig:

[alias]
    cm = commit
    co = checkout
    br = branch

Then running git cm will expand as git commit, git co master is git checkout master, and so on. You can see how this might make you happy because it could save a few keystrokes. But I think it’s more useful than that – let me share what I did.

The first thing I did was add

pu = pull origin
psh = push origin

and permanently save myself the frustration of forgetting to type origin. Not bad. Even more helpful was …

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 …