Post 1: Introduction

22 Dec 2023

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Hi! This post is going to introduce myself and what I do.

Boring academic life story

First, let me recap a bit of my academic life.

I’ve liked math as long as I can remember. Naturally, what “math” meant to me changed over time. I excelled at the sort of naïve, calculation based mathematics, from when I was a toddler to when I was a slightly-more-grown-up-toddler (AKA, a middle schooler).

By the time I was in high school (the USA education system’s last 4 years before university), I was starting to learn about rudimentary mathematical logic and proofs. My first bouts of what I or other mathematicians might call “real math” were with group theory and graph theory at a summer program (opens in new tab).

Then, I attended the Louisiana School for Math, Science, and the Arts (or LSMSA (opens in new tab) for short). This is a residential high school¹. Essentially, it is a mini-university. All of the classes are university-level, and it is as easy as checking off some boxes on a form to actually get college credit for your courses. Most of the teachers, especially outside of the math department, have a Ph.D. in their area of expertise.

While at LSMSA, I also got into physics. Indeed, I completed probably a full physics major worth of content, from classical to quantum mechanics, and special to general relativity. I was initially going to double major in math and physics during university, but due to some annoyances², I did not.

Another great thing about LSMSA is that it was situated inside the campus of an actual university, Northwestern State University (or NSU, or NSULA). It is also relatively painless to say “hey, I want to take courses there” and do so. Since I quickly exhausted, or wasn’t satisfied with, LSMSA’s math offerings, I took quite a few courses at NSU: PDEs, probability theory, topology, complex analysis³, and advanced calculus⁴.

The top universities I applied to were UChicago⁵, Princeton, UCLA, and UMichigan. I got rejected from the first two, and accepted to the last two. Unfortunately, university is expensive in the USA, especially if you don’t live in the state in which the university is situated. For this reason, and some other reasons, I went to Louisiana State University (LSU).

Because of my studies in high school, I had a lot of credit going into university, and I also took placement exams for several courses that I didn’t have credit for. As a result, I really only took one undergraduate level math course as an undergraduate. I had to take a mandatory “lab” course, which was a primer in using Mathematica. I also took a “vertically integrated research” (VIR) courses, which both undergraduate and graduate students take. Besides those, I took graduate-level math courses, and got graduate credit for (most of) them as well. I also had to take some “general education” requirement courses, such as foreign language⁶, art, and an interdisciplinary honors course. Despite these annoyances, which usually make students at my level stay undergrads for 3 or 4 years, I was able to finish undergrad in just 2 years. In that time, I also completed an honors thesis, which became a formal paper published in the Journal of Geometry and Physics. You can find this paper on the arXiv here (opens in new tab).

Since I finished my undergrad so fast, and even though I was already doing graduate level courses and research, I didn’t feel prepared to go off and start a Ph.D. program. Instead, I stayed at LSU to do a masters (MS) degree. I chose to do the non-thesis track, which requires a certain number of courses and passing the qualifying exams (opens in new tab). As of this blog post, I am in between my penultimate and ultimate semesters of this program.

I applied to 15 Ph.D. programs, all situated in the USA. I won’t know results until some time in March. I don’t know exactly what I want to do for my thesis, but I would like to do something in the intersections of algebraic geometry, representation theory, and number theory. I also want to explore homotopy theory and algebraic K-theory, but I know relatively little about these.

What I’ve been doing as of late

As of this post, I am on winter break. My courses in the previous semester were:

• Geometric methods in representation theory, mainly focused on outlining the proof of the Springer correspondence.
• Algebraic topology, in particular chapters 3 and 4 in Hatcher.
• VIR on cluster algebras, going through the first 5-or-so chapters of the in-develolpment textbook by Fomin, Williams, and Zelevinsky.
• A reading course on perverse sheaves, using the book written by my advisor Pramod Achar, “Perverse Sheaves and Applications to Representation Theory”. I got through the first 4 chapters, and I am skipping chapter 5.

Here’s what I’ll be taking next semester:

• Modular forms.
• Functional analysis.
• Reading course on perverse sheaves, especially the “applications to representation theory” part.
• Reading course on intersection theory and enumerative geometry, with hopes of learning just enough to be powerful.

During my winter break, I’ve been doing the following things, which I will expand upon in future posts:

• Continue reading “Perverse Sheaves…”; I’m about half-way through chapter 6.
• Prepare for another reading course next semester, on intersection theory and enumerative geometry. Preparation consists of reading Fulton’s “Introduction to Intersection Theory in Algebraic Geometry”.
• Work on a quick reference/review sheet for algebra.⁷
• Work on research paper. See Post 2 for a description of my research.
• Learn the proof assistant Lean (opens in new tab), with hopes of contributing stuff on representation theory. See Post 3 for a discussion of Lean and what I want to do with it.

I think this is a good place to stop this post. See you next time!

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