• February 21, 2021
• 6 minutes
Fractals on The Web

Last week, I made a little web application for visualizing some fractals, and I thought I’d write up a few thoughts about how it works.

• February 14, 2021
• 11 minutes
Spaced Repetition for Mathematics

Recently, I’ve been experimenting with using spaced repetition for self-studying “advanced” mathematics. This post goes through my motivations for adopting this system, as well as a few techniques I’ve used in adapting it to mathematics.

• February 02, 2021
• 17 minutes
Tychonoff's Theorem and Zorn's Lemma

Tychonoff’s theorem proves that the product (even infinite) of compact spaces is also compact. The proof makes judicious use of Zorn’s lemma. In fact, it uses it so well, that I gained an appreciation for how fun the lemma can be.

• December 13, 2020
• 12 minutes
Chinese Remainder Theorem for Programmers

This is a quick post about the Chinese Remainder Theorem. Specifically, how to use it to solve a system of system of simple modular equations.

• October 14, 2020
• 8 minutes
Monty Hall and Counterfactuals

This is about some shower thoughts I had recently about the infamous Monty Hall problem. Namely, how to make sense of the counter-intuitive results involved. We’ll see how reasoning counterfactually can make the best strategy seem a lot clearer.

• October 02, 2020
• 9 minutes
Categorical Graphs

This post is a basic introduction to the idea of Categorical Graphs, or just the standard theory of Graphs, developed through the lens of Category Theory, and generalized in the obvious ways that follow through that lens. This is just an introduction mainly because this theory doesn’t seem to have been developed very far yet, and I haven’t been able to develop it that much independently so far. I think I have a good grasp on some very basic ideas here, and wanted to present them while they’re still fresh in my head.

• September 09, 2020
• 10 minutes
Recursive Types as Initial Algebras

Recently (well, more like a month ago), I came across this interesting observation: In my head, I immediately jumped to the notion of Algebras in Category Theory. I had recently studied that notion, found it quite interesting, and was very happy to see this observation, because it was actually quite obvious to me thanks to what I’d recently learned. The goal of this post is to unpack that tweet, and then explain why that observation is true.

• August 30, 2020
• 17 minutes
Encoding the Naturals

In this post, we’ll cover 3 ways I know of encoding the natural numbers $\mathbb{N}$ in your standard functional language with recursive types and polymorphism. At least, these are the 3 most generalized ways of doing it. As we’ll see, some common encodings are just specific cases of a more general encoding. Prerequisites Some familiarity with defining data types in a functional-esque language might be helpful, but shouldn’t be strictly necessary.

• August 17, 2020
• 14 minutes
Empty vs NonEmpty Groups

The usual definition of a Group excludes empty groups by definition. There are alternate definitions of a Group that allow us to include the empty set, and which are quivalent to the normal definition in all other cases. This post explores this alternate definition, and the resulting differences with the normal concept of a Group. What is a Group? A Group is one of the most important structures in abstract algebra.

• June 18, 2020
• 7 minutes
Monomorphisms vs Epimorphisms

The concepts of monomorphism and epimorphism are very important in Category Theory. I always had a hard time remembering which one was which until I thought about a good mnemonic.