I’ve recently come to believe that “deep conversations” are overrated. Here is why.

Memory

Human short term memory is pretty crummy. Here is an illustration from linguistics:

A man that a woman that a child that a bird that I heard saw knows loves

This is a well-formed English phrase. And yet parsing it is difficult, because you need a stack of size four. Four is a pretty big number.

And that’s after I’ve written the sentence down for you, so your eyes could scan it two or three times to try and parse it. Imagine if I instead said this sentence aloud.

Other examples include any object with some moderately complex structure:

Let ABC be a triangle and let AD, BE, CF be altitudes concurrent at the orthocenter H.

This is not a very complicated diagram, but it’s also very difficult to capture in your …

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 $O({\mathrm{d\; \dots }}^{}$

Apparently even people on Quora want to know why I transferred from Harvard to MIT. Since I’ve been asked this question way too many times, I guess I should give an answer, once and for all.

There were plenty of reasons (and anti-reasons). I should say some anti-reasons first to give due credit – the Harvard math department is fantastic, and Harvard gives you significantly more freedom than MIT to take whatever you want. These were the main reasons why transferring was a difficult decision, and in fact I’m only ~70% sure I might the right choice.

Ultimately, the main reason I transferred was due to the housing.

At MIT, you basically get to choose where you live. All the dorms, and even floors within dorms, are different: living on 3rd West versus living on 5th East might as well be going to different colleges. Even if for some …

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 …

(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 …