Research
A curiously slowly mixing Markov chain (with Persi Diaconis and Arun Ram) studies a particularly simple instance of the Burnside process, which is a Markov chain which allows one to sample uniformly from the orbits of a group action. The Burnside process has useful applications (for example for sampling large uniform partitions or contingency tables), but proving convergence to stationarity has been very difficult in almost all cases. We find that this particular "binary Burnside" chain can be explicitly diagonalized, and in particular we show that the mixing time looks very different when measured in l^1 (total variation) versus l^2 (chi-square) distance to stationarity.
Schur--Weyl duality for diagonalizing a Markov chain on the hypercube (with Persi Diaconis and Arun Ram) is a companion paper to the one above, which uses representation theory and algebraic combinatorics to find an explicit orthonormal basis of eigenvectors for the binary Burnside chain. This basis allows for more refined study of convergence to stationarity, and it also sheds light on certain nice symmetry properties of the chain in a way that can be generalized. Keep an eye out for more!
High-dimensional permutons: theory and applications (with Jacopo Borga) generalizes the notion of permuton convergence to higher-dimensional permutations, while providing the first examples of some (universal) permuton limits coming from natural combinatorial structures. It has some nice pictures and interesting constructions, and it would be nice to extend this work to other natural classes of permutations too! In Summer 2021, I participated in the Caltech SURF program under the mentorship of Professor Leonard Schulman. Here's the final report that I produced. In Summer 2020, I participated in the Texas A&M Probability and Algebra REU; here are the presentation slides and final report that I produced. Our group's work resulted in the paper Interacting particle systems with type D symmetry and duality.
|
|
Teaching
I am currently a Teaching Assistant for Math 61CM (Modern Mathematics: Continuous Methods), primarily teaching the "introduction to proofs" sections (for the second time in a row). Previously, I was a Teaching Assistant for Math 21 (Calculus) in Autumn 2024, Winter 2024, and Autumn 2023, and a Course Assistant for Math 20 (Calculus) in Autumn 2022 and Math 108 (Introduction to Combinatorics and Its Applications) in Spring 2025. I also taught a section of SOAR Mathematics / Summer Bridge Mathematics (a precalculus readiness crash course for incoming first-years) in the summers of 2023, 2024, 2025, and 2026.
During the 2024-2025 academic year, I was a CTL LIT fellow, working with Alexandra Stavrianidi to run quarterly TA discussion workshops and develop a department teaching handbook. I am also coordinating the TA mentorship program in the department; if you're teaching or want to chat about your philosophies or ideas, please don't hesitate to reach out! I will almost certainly have thoughts and want to hear your thoughts too.
At MIT, I was an Undergraduate Assistant for 18.100B (Real Analysis), as well as a Teaching Assistant for 18.600 (Probability and Random Variables) for two semesters. Those were some of my academic highlights during my time as an undergrad, and I'm happy to talk about my experiences! Other than that:
- I was one of the lecturers for 18.S097, an undergrad-led proof-writing workshop, during IAP 2021. Here are the lecture notes that I produced for the last two lectures of the class.
- I was a JC at Canada/USA Mathcamp in Summer 2019 and taught two short classes based on material I had learned a few months ago in 18.212: an evening talk on the matrix-tree theorem, as well as a Week 5 class cotaught with Shiyue Li on a proof of the hook-length formula.
Contact
Email me at lindrew@stanford.edu if you'd like to chat about anything!
|
|