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My Work

I haven’t done that much math yet. I don’t have the capacity to contribute to new math, so so far, my work takes the form of seminar talks and class projects.

Talks and Presentations

• Improvements on Lower Bounds of Sphere Packing Densities in High Dimensions

Class Projects

The following class projects are all relatively short (about 10 to 20 pages) research projects focusing on various topics in math. These were assigned in various graduate courses, and I’ll be updating these as I go. I’ll list them in reverse chronological order.

The LaTeX source is provided alongside the finished pdf product in a zip file, which also includes the BibLaTeX resource file. I have no clue why anyone would want this, I just think this is a good place to store it.

• Sharifi’s Conjecture (pdf | tex)
  Math 207C, Topics in Number Theory, Spring 2024
  This was a project focusing on a specific case of Sharifi’s conjecture, which is (in a sense) a generalisation of the Iwasawa Main Conjecture. I should say that modern algebraic number theory eludes big-picture understanding, and I feel like I don’t have a clear idea of the bigger context of this conjecture and the significance of the paper I based this project on. Nonetheless, this project and this yearlong sequence were fun ventures into the world of number theory.

• Rigid Analytic Spaces (pdf | tex)
  Math 207B, Topics in Number Theory, Winter 2024
  This was a very light introduction to Tate’s theory of rigid analytic geometry. It was definitely far less ambitious than my project from the previous quarter, but on the flip side, I felt like I had a much clearer picture of what was going on. Fortunately, I have taken algebraic geometry classes before, and that background helped quite a bit in putting all of this stuff into context.

• The Ramanujan-Petersson Conjecture (pdf | tex)
  Math 207A, Topics in Number Theory, Fall 2023
  I’m not quite as proud of this project as some of my other work here. I selected the topic of the Ramanujan-Petersson Conjecture because it seemed like an interesting motivation for the development of Hecke operators, but it quickly unraveled into a deep rabbit hole down the Langlands Program. I wish I understood more of what was going on, and I’ll be on the lookout for a topic more suited to my intelligence next quarter.

• Counting Solutions to Quadratic Forms (pdf | tex)
  Math 207B, Topics in Analytic Number Theory, Spring 2023
  This was a very eye-opening topic to explore. During this course, we had been discussing some problems related to quadratic forms, an ancient aspect of number theory. In fact, the instructor Professor Duke has made considerable contributions to this field. My interest stemmed from the Smith-Minkowski-Siegel mass formula, where I had first seen it in the context of the sphere packing problem. My intention was first to just outline a proof of this formula, but that was far too ambitious; the direction Prof. Duke advised me to take it in turned out to be very satisfactory to me.

• The First Hardy-Littlewood Conjecture (pdf | tex)
  Math 205B, Analytic Number Theory, Winter 2023
  This is really more of an exploration of the circle method than anything else, though it focuses on applications to the titular conjecture and similar problems.

• Thurston’s Geometrisation Conjecture (pdf | tex | slides | tex)
  Math 120B, Differential Geometry, Spring 2022
  I had an incredible amount of fun with this project. It does not delve too deeply into the math, but I think it’s a fun (and hopefully engaging) survey of this classification problem. The report was accompanied by a digital presentation, which I had way too much fun with to omit.

• Gromov’s Nonsqueezing Theroem (pdf | tex)
  Math 246C, Complex Analysis, Spring 2022
  This was a really rough project to scrape together. I think I pulled at least two all-nighters trying to understand what was going on. I still don’t understand anything about symplectic camels or needles or whatever Gromov was going on about.

• Stochastic Calculus and Brownian Motion (pdf | tex)
  Math 246B, Complex Analysis, Winter 2022
  The biggest mistake with this project was agreeing to do something adjacent to probability theory when I didn’t know anything about it. Learning all the notation and fundamentals alongside doing all the research is something I hope I never do again.

• Elliptic Functions (pdf | tex)
  Math 246A, Complex Analysis, Fall 2021
  Baby’s first research project! At that time, I was still majoring in Bioengineering and had no clue I wanted to pursue mathematics. Looking back, I think I have learned an incredible amount; content-wise, this project is almost way too elementary and could probably be covered in a single lecture (given a graduate audience).