Model theory is really meta, so you will have to pay attention here. Roughly, a ``model of $latex {\mathsf{ZFC}}&fg=000000$'' is a set with a binary relation that satisfies the $latex {\mathsf{ZFC}}&fg=000000$ axioms, just as a group is a set with a binary operation that satisfies the group axioms. Unfortunately, unlike with groups, it is very… Continue reading Models of ZFC
Cardinals
(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 $latex {\omega}&fg=000000$ and $latex {\omega+1}&fg=000000$: $latex \displaystyle \begin{array}{rccccccc} \omega+1 = & \{ & \omega & 0… Continue reading Cardinals
Git Aliases
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… Continue reading Git Aliases
Constructing the Tangent and Cotangent Space
This one confused me for a long time, so I figured I should write this down before I forgot again. Let $latex {M}&fg=000000$ be an abstract smooth manifold. We want to define the notion of a tangent vector to $latex {M}&fg=000000$ at a point $latex {p \in M}&fg=000000$. With that, we can define the tangent… Continue reading Constructing the Tangent and Cotangent Space
Some Notes on Valuations
There are some notes on valuations from the first lecture of Math 223a at Harvard. 1. Valuations Let $latex {k}&fg=000000$ be a field. Definition 1 A valuation $latex \displaystyle \left\lvert - \right\rvert : k \rightarrow \mathbb R_{\ge 0} &fg=000000$ is a function obeying the axioms $latex {\left\lvert \alpha \right\rvert = 0 \iff \alpha = 0}&fg=000000$.… Continue reading Some Notes on Valuations
The Mixtilinear Incircle
This blog post corresponds to my newest olympiad handout on mixtilinear incircles. My favorite circle associated to a triangle is the $latex {A}&fg=000000$-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… Continue reading The Mixtilinear Incircle
Conversations
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… Continue reading Conversations
Linnik’s Theorem for Sato-Tate Laws on CM Elliptic Curves
\title{A Variant of Linnik for Elliptic Curves} \maketitle Here I talk about my first project at the Emory REU. Prerequisites for this post: some familiarity with number fields. 1. Motivation: Arithemtic Progressions Given a property $latex {P}&fg=000000$ about primes, there's two questions we can ask: How many primes $latex {\le x}&fg=000000$ are there with this… Continue reading Linnik’s Theorem for Sato-Tate Laws on CM Elliptic Curves
Transferring
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… Continue reading Transferring
Proof of Dirichlet’s Theorem on Arithmetic Progressions
In this post I will sketch a proof Dirichlet Theorem's in the following form: Theorem 1 (Dirichlet's Theorem on Arithmetic Progression) Let $latex \displaystyle \psi(x;q,a) = \sum_{\substack{n \le x \\ n \equiv a \mod q}} \Lambda(n). &fg=000000$ Let $latex {N}&fg=000000$ be a positive constant. Then for some constant $latex {C(N) > 0}&fg=000000$ depending on $latex… Continue reading Proof of Dirichlet’s Theorem on Arithmetic Progressions
Power Overwhelming 