I am a Ph.D. candidate in the MIT Department of Mathematics. My work spans several different fields, with an emphasis on applied mathematics and fluid dynamics. I have a few current projects centered around these topics. On one front, I am working to develop a new sub-grid-scale model for stratified turbulence, aiming to improve the efficiency and accuracy of ocean modelling. On another front, I am working to understand hydrodynamic analogues of quantum mechanics, and more specifically, to investigate which quantum phenomena can be closely approximated with classical models.
I graduated from MIT in 2022 with a B.S. in Mathematics, with a minor in German Studies. Afterwards, I studied in Cambridge University (Churchill College) under a Churchill Scholarship, and I graduated with an M.Phil. in Scientific Computing. My work in Cambridge was primarily oceanographic in nature, developing a new sub-grid-scale model for highly stratified flows in nature. There, I studied under the supervision of Professor Colm Caulfield, with external supervision by Professor Gregory Chini (University of New Hampshire).
A central focus of my work at MIT is the investigation of hydrodynamic quantum analogues, and particularly the "walking" oil droplet system of Couder and Fort. These walking droplets exhibit a wide variety of quantum-like features: stable spin states and quantised orbits, Bohm-like "surreal trajectories", and Fraunhofer-like diffraction patterns, among others.
My work lies on the theoretical end of this project; though these droplets exhibit qualitative similarities to quantum mechanics, they often fall short in achieving quantitative agreement. Through analytical and numerical approaches, I am investigating which quantum phenomena can be closely approximated with techniques of classical field theory.
Many key fluid flows in nature—including the atmosphere and the deep ocean—are stratified, meaning that their density varies significantly in the vertical direction. However, stratified turbulence gives rise to a separation of scales that precludes traditional methods of simulation, and so large-scale, accurate models of these flows remain outside our reach, even with current technology.
However, Chini et al. recently developed a "multi-timescale quasi-linear" (MTQL) algorithm to model such flows in the stratified limit; MTQL reduces and splits the Boussinesq system into a mean and fluctuating flow, drastically reducing the computation time needed to model a stratified flow. In my Master's dissertation at Cambridge—and in ongoing work—I have developed an improved, "non-linear cascading quasi-linear" (NCQL) approach to model these flows. Using weakly nonlinear theory, NCQL directly models the forward energy cascade of stratified flows, achieving significant improvements over MTQL in the moderately and highly stratified regimes (pictured above).
MIT PRIMES and MIT PRIMES USA are free, year-long programs organised by MIT to engage highly-talented high school students in novel mathematical research, under the guidance of faculty, post-docs, and graduate students. I have been involved in this program both as a student (in 2017) and as a mentor (since 2021), and have advised two research programs so far.
Eric Chen and Alex Zitzewitz are MIT PRIMES 2022 students, investigating the "accessory parameter problem" for the complex-analytic Heun equation. For large classes of Heun equations (generalising the Lamé equation, for which similar results were proved by Beukers), they proved a strong connection between unitarity of the underlying monodromy group and traces of certain monodromy matrices; this allows them to fully classify the spectrum of the accompanying real-analytic Heun operators. They further developed a numerical technique for identifying the accessory parameters that give rise to these conditions.
Andrew Du is an MIT PRIMES 2021 student, adapting recent techniques in inverse dynamics to build a dynamical model of the human arm. Specifically, combining a quaternion screw algebra formulism developed by Dumas et al. with novel models of human musculature, he developed an efficient, simple way to calculate kinematic and dynamic quantities associated with arm motion.
January 13, 2022 • Alli Armijo | MIT News correspondent
Senior David Darrow’s love of math fuels other passions such as mentoring and learning new languages...
January 12, 2022 • Julia Mongo | Office of Distinguished Fellowships
Seniors David Darrow and Tara Venkatadri and HST student James Diao will pursue master’s programs at Cambridge University....