This page contains short descriptions and links to some of the expository mathematics videos I've created over the years.
Most of them are posted on Spectral Collective, a channel which I help to run,
but I have also created a guest video for 3Blue1Brown.
Simulating and Analyzing Phase Change (2025)
This is a two-part series describing the basics of phase transitions from a mathematical perspective.
In the first video (on 3Blue1Brown), we give an intuitive proof of the Boltzmann formula and see how that leads to different
phases of matter, and we also explore some interesting behavior of the liquid–vapor model (also known as the Ising model).
In the second video (on Spectral Collective), we give a concise analysis of the mean-field version of this model (the Curie–Weiss model).
Card Shuffling (2024)
In this video, we explore what people mean when they say that "7 shuffles suffice" and we give an elegant
proof of a slightly weaker bound, showing that "12 shuffles suffice" using only basic probability theory.
This video received an honorable mention in the \(\pi\)th Summer of Math Exposition (#SoMEpi).
The Longest Increasing Subsequence (2023)
In a random permutation of the numbers \(1\) through \(n\), how long is the longest increasing subsequence?
This video explores the surprisingly deep and beautiful math that goes into answering this question.
This video received an honorable mention in the third Summer of Math Exposition (#SoME3).
Percolation (2022)
This video provides an introduction to the subject of percolation on the two-dimensional lattice, and presents the Peierls argument,
which proves that the critical parameter is bounded away from 0 and 1. Created with help from Caio Alves and Aranka Hrušková.
This video was selected as a winner in the second Summer of Math Exposition (#SoME2).
Sparse Graph Limits (2021)
This is a two-part series on Benjamini-Schramm convergence of sparse graphs to unimodular random rooted graphs.
In the first video (joint work with Caio Alves and Aranka Hrušková), we motivate the definition and go through a few examples.
In the second video, we work out a particularly enlightening example, the canopy tree.
The first video in this series received honorable mention in the first Summer of Math Exposition (#SoME1).