Happy Pi Day! I have an economics midterm on Wednesday, so here is my attempt at studying.

1. Mechanisms

The idea is as follows.

• We have $N$ people and a seller who wants to auction off a power drill.
• The $i$-th person has a private value of at most $\$1000$ on the power drill. We denote it by $x_i \in [0,1000]$.
• However, everyone knows the $x_i$ are distributed according to some measure $\mu_i$ supported on $[0, 1000]$. (Let’s say a Radon measure, but I don’t especially care). Tacitly we assume $\mu_i([0,1000]) = 1$.

Definition 1. Consider a game $M$ played as follows:

• Each player $i=1,\dots ,\mathrm{N\; \dots }$

These notes are from the February 23, 2016 lecture of 18.757, Representations of Lie Algebras, taught by Laura Rider.

Fix a field $k$ and let $G$ be a finite group. In this post we will show that one can reconstruct the group $G$ from the monoidal category of $k[G]$-modules (i.e. its $G$-representations).

1. Hopf algebras

We won’t do anything with Hopf algebras per se, but it will be convenient to have the language.

Recall that an associative $k$-algebra is a $k$-vector space $A$ equipped with a map $m : A \otimes A \rightarrow A$ and $i : k \hookrightarrow A$ (unit), satisfying some certain axioms.

Then a $k$-coalgebra is …

[EDIT 2018/03/05: This description seems significantly less accurate to me now than it did a few years ago, both because my views/values have changed substantially, and because SPARC has changed direction substantially since I attended as a junior counselor in 2015. I’ll leave it here as a reference, but should be taken with a grain of salt.]

I often get asked about what I learned from the SPARC summer camp. This is hard to describe and I never manage to give as a good of an answer as I want, so I want to take the time to write down something concrete now. For context: I attended SPARC in 2013 and 2014 and again as a counselor in 2015, so this post is long overdue (but better late than never).

(For those of you still in high school: applications for 2016 are now open, due March …

Occasionally I am approached by parents who ask me if I am available to teach their child in olympiad math. This is flattering enough that I’ve even said yes a few times, but I’m always confused why the question is “can you tutor my child?” instead of “do you think tutoring would help, and if so, can you tutor my child?”.

Here are my thoughts on the latter question.

Charging by Salt

I’m going to start by clearing up the big misconception which inspired the title of this post.

The way tutoring works is very roughly like the following: I meet with the student once every week, with custom-made materials. Then I give them some practice problems to work on (“homework”), which I also grade. I throw in some mock olympiads. I strongly encourage my students to email me with questions as they come up. Rinse and …

I know some friends who are fantastic at synthetic geometry. I can give them any problem and they’ll come up with an incredibly impressive synthetic solution. I also have some friends who are very bad at synthetic geometry, but have such good fortitude at computations that they can get away with using Cartesian coordinates for everything.

I don’t consider myself either of these types; I don’t have much ingenuity when it comes to my solutions, and I’m actually quite clumsy when it comes to long calculations. But nonetheless I have a high success rate with olympiad geometry problems. Not only that, but my solutions are often very algorithmic, in the sense that any well-trained student should be able to come up with this solution.

In this article I try to describe how I come up which such solutions.

1. The Three Reductions

Very roughly, there are …

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 $E \subseteq M \times …$