Here are some of the projects I’m working on:
Formalization and mathlib
In the Fall of 2021 I began my journey in using Lean. Since then I have learned a ton, and am actively working on a couple projects in the area:
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I ran a Learning Lean Seminar at Northeastern University in Spring 2022. The focus of the seminar was to get undergraduates (and anyone else who is interested) up to speed with using Lean to formalize aspects of the undergraduate math curriculum, and maybe eventually get it up on mathlib. We got part of the way there.
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My first nontrivial (and in progress) formalization project is contained here wherein I formalize the proof of the equivalence of a few definitions for flat modules in Lean. The intention is that this will eventually get merged with mathlib.
Mathematics
A couple math things that I have been stewing over:
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Calculating the explicit image of of a \(\mathcal D\)-module on \(\mathbb A^1\) in the Sato Grassmannian under the Cannings-Holland correspondence.
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Realizing the Lax matrix for the 2D double elliptic integrable system as a Higgs field for something like an “elliptic Hitchin system”.
Projects nearer to completion will have preprints available. Unavailable preprints can be made available upon request.
My Papers
Some of my old papers on the arXiv:
Spectral Description of the Spin Ruijsenaars-Schneider System
In this paper I show that the classical spin Ruijsenaars-Schneider phase space can be given a moduli-theoretic interpretation in terms of twisted Higgs fields on cubic curves. The moduli-theoretic interpretation provides a completion for this phase space, and the RS flows are identified with the natural flows that exist on twisted Hitchin systems of this kind.
Langlands parameters of quivers in the Sato Grassmannian
In this paper, Martin Luu and I show that associated to the combinatorial data of a quiver in the Sato Grassmannian, one can construct a “local Langlands parameter” on \(\mathbb A^1\). In fact, there are a number of different ways of constructing this local Langlands parameter, and we prove some equivalences between them.
Old Notes
- Some notes on the class field theory. These are lecture notes I prepared for a seminar on the Langlands program. The hope is that I will find the time to continue preparing these notes, and eventually touch on subjects like Tate’s thesis, geometric class field theory, and finally the Langlands program.
This website
As you can tell, this website is a work in progress. My hope is that in tweaking the website I can get a little bit of experience and appreciation for Ruby, and web development more broadly. All that being said, I am still an amateur learning my way around this space so if you notice any issues or have any suggestions feel free to submit a PR at the page’s repo, and I’ll get on making the fixes!
Boats
Also I got a model ship building kit for Christmas, and I’ll be posting updates of my progress in building the HMS Bounty