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Current active project: I'm currently working on my master's thesis with Gautam Iyer, in the intersection of PDE, functional analysis, and dynamical systems. We study classes of double-exponentially mixing flows on the torus and (in)stability of their mixing properties under small perturbations, which is related to the formation of the Batchelor scale – the proposed length scale at which advection and diffusion perfectly balance one another out.
As this is active work, I can't say much more than that. :^)
(Image from Coti-Zelati, Cripper, Iyer, and Mazzucato's 2024 survey paper on mixing.)
Spring 2024: Seminar / Reading Group in Abstract Harmonic Analysis.
During my junior spring semester, I along with a few other friends took an undergraduate research seminar course taught by my good friend, CMU alum and current PhD student Subhasish Mukherjee. CMU doesn't generally have an undergraduate or graduate harmonic analysis course; this was simply because we were very interested in the subject. We started at locally compact Hausdorff topological groups and made it all the way to Pontryagin duality for locally compact Hausdorff abelian groups and the Peter-Weyl theorem on irreducible unitary representations of compact Hausdorff groups.
Summer 2023: The Muskat problem with surface tension.
With Ian Tice, using Folland's Introduction to Partial Differential Equations as a guide, I learned classical and modern PDE theory such as the theory of distributions, L^2-based Sobolev spaces using the Fourier transform, trace theory, and implicit function arguments in efforts to prove existence and uniqueness of solutions to the free-boundary Muskat problem with surface tension, a physical system which arises as a scaling limit of flow through porous media.
Summer 2022: Self-propulsion dynamics of an active particle with oscillatory surface activity.
My first summer at CMU, I conducted applied mathematics research in fluid dynamics in the Chemical Engineering department, with Aditya Khair and Kailasham Ramalingam. I performed numerical simulations using the Khair group's supercomputing cluster and explored analytic techniques for eigenvalue problems in order to classify instabilities leading to spontaneous self-propulsion of a chemically active particle in an unbounded fluid. The "run-and-halt" behavior I found in the simulations matched experimental results about drops whose activity was dependent on light, which was fantastic to see. I presented this work in a poster at the 2022 John Berg Chemical Engineering Undergraduate Research Symposium Poster Session, as well as CMU's 2023 Meeting of the Minds poster session, as this research was conducted under a Summer Undergraduate Research Fellowship.
Image from Khair and Kailasham, Dynamics of Forced and Unforced Autophoretic Particles, 2023, which was the paper (at the time, preprint) which served as the initial jumping-off point for my research.
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