Exposition

This video, The Tale of Three Triangles, is a long form video discussing the convergent behavior of the random Chaos Game, the deterministic Sierpinski’s triangle, and the combinatorial Pascal’s triangle. This video took me months and over 5,000 lines of animation code in manim. It was eventually recognized by 3Blue1Brown as an honorable mention for storytelling in the first Summer of Math Exposition competition. It is available in both English and Mandarin and has over 100,000 views across both languages.

This video proves the closed-form expression \(C_n = {2n \choose n}/(n+1)\) for the Catalan numbers using a reflection proof. It was one of my first projects using manim and helped me develop the skills I would later use to create The Tale of Three Triangles.

Ramsey Theory studies the existence of monochromatic subgraphs in multicolored graphs (for more, see my notes on Graph Theory). In this video, I provide a visual proof that \(R(3,3) = 6\). This was my first project using manim.

This post discusses the use of probability theory in combinatorics, specifically to search for objects with particular properties (or prove no such objects exist). This is aimed at anyone interested in combinatorics who has basic exposure with probability theory (indicator variables, expected values). It introduces the use of probabilistic methods (notably expected value) in proving the existence of certain discrete objects, including graphs, families of sets, and sum-free sets. It also provides the background for Sum-Free Subsets of Finite Abelian Groups.

This post discusses applications of group theory to combinatorial problems. It introduces the ubiquitous concept of a group action before moving on to the Orbit-Stabilizer Theorem and its corollary, Burnside’s Lemma. Then, it applies Burnside's Lemma to three natural counting problems about necklaces, die, and graphs. This article is aimed at students finishing a first course in algebra who want to see how the subject is used in other, more “natural” areas of math.

This post, which roughly covers the first week of the Stanford course Algebraic Number Theory, discusses applications of group theory to number theory problems. It starts with ring theory (UFDs, PIDs, and Euclidean Domains) and moves on to sums of squares, a related Diophantine equation, and an example elliptic curve that can be solved using the technique of Fermat Factorization. This article is aimed at students finishing a first course in algebra who want to see how the subject is used in other, more “natural” areas of math.

This post discusses applications of group theory to problems in topology. It starts by defining paths and homotopies between them, then works towards a full description of the fundamental group - not just as a construction, but as a functor, and the power of that perspective. The post has plenty of drawings and examples, which is critical for this type of math. This article is aimed at students in the middle of a first course in algebra who want to see how the subject is used in other, more “natural” areas of math.

This post consists of lecture notes from a talk I gave in Stanford's MATH 159, Discrete Probabilistic Methods. The talk was on the application of the probabilistic method t problems in algebra; more precisely, the problem of being given a subset \(S\) of a finite abelian group \(A\), and trying to find the largest subset \(T\) of \(S\) which is “sum-free”. Here, we give a bound on the size of \(T\) in terms of \(S\), and show that it is tight. The proof, while not original, corrects inaccuracies and improves on various details of existing presentations.

This post focuses on a construction in category theory that I personally found particularly confusing; the notion of adjoint functors. Part of the confusion stems from the fact that there are 3 seemingly different definitions for what an adjoint is, and the process of grappling with these definitions and showing they are equivalent is often skipped in classes which use them, like MATH 210 or 216 at Stanford. This post discusses in full detail all three definitions and examples showing why choosing the right definition can substantially shorten proofs.

This post discusses two equivalent ways to define the same object: a quotient of a vector space. In doing so, it gives an idea of the difference between focusing on elements and focusing on maps, showing some elementary diagram-chasing. This article is aimed at anyone with sufficient linear algebra experience who is curious about the "categorical perspective" or wants to understand the philosophy of category theory. Why, for example, do algebraists often eschew bases, which are so central in a first course in linear algebra? This article hopes to hint at an answer.