math research senior year

Pre-College Issues High School Life
1

Hello all - I’m currently a junior with an interest in possibly attempting some math (or physics) research at some point prior to admissions. My academic focus is STEM, but my transcript isn’t reflective of that. Rather than entail my circumstance (as it’s full of shenanigans), I’ll just supply you my background:

Freshman Year – Geometry ©/Geometry (B)
Sophomore Summer: self-studied college algebra, qualified for trigonometry through ACCUPLACER
Sophomore Year - Trigonometry ©/nothing
from 4/1-9/1 I completed Calculus BC, Linear Algebra, Differential Equations, and Multivariable Calculus through John Hopkins’ CTY. My instructor told me she’s never covered so much content in a six-month timeframe. All As. With the remaining month of my enrollment I moved on to their Intro to Abstract Mathematics Course.
Junior Fall – finished Intro to Abstract Mathematics, took a course in PDEs at a nearby university
And currently this Spring, I’m taking Real and Complex Analysis concurrently, again through CTY. I also took up a graduate course in Calculus of Variations at the same university by my PDE professor’s advice.

Next year, I plan on taking topology (undergrad) and a math for physicists’ class (“theoretical physics”, but covers Poincaré expansions, integral transforms, Hilbert spaces, special functions and integral equations). My question is: what area of math would I benefit most from doing research in? To my knowledge, most high schoolers who do math research do combinatorics bc you don’t need a bunch of college classes for it, but I don’t want to teach myself any more math than I have to (I’m busy enough as is!). I have relatively good connections with the math faculty and have the option to take a graduate seminar on differential geometry as a senior, too. Should I learn how to use LaTex over the summer? And how much time does a research project usually take to complete? I’m also an IB student with EEs and IAs around the corner, do you think I could kill two birds with one published paper?

Also – I’ll be taking QM as two semester courses next year as well. How does physics research differ from math, and should I choose one over the other?

2

You will not be able to publish a paper in math in time for admissions. A publication-worthy research project in math can take many years, depending on the project, and more experience than you have. You may be able to collaborate on an applied math or physics project if you can code. I would talk to the professors in areas that interest you and find out who is willing to be your mentor over the summer. You also should definitely learn LaTeX.

3

Plenty of time to put together a nice math research project to enter in Regeneron. Kids at RSI put together winning projects in about 5 weeks every summer. No reason you can’t do the same with one of the professors you’ve developed a relationship with.

4

I think you need around 2 months to put together a good project that is competition ready. This is a great opportunity: take advantage of it.