Core Skills – Mathematics
Philosophy
Mathematics is a language that is unique in its precision. Certain fields require too much clarity to be expressed in colloquial, or even academic, terms. A deep understanding of computer science, physics, or even epistemology all require a level of rigor currently attainable only through mathematics.
For this reason, understanding the patterns of mathematical thinking is a unique strength. Difficult questions become obvious, and intractable ones approachable. Are there two people in the world who share the same number of hairs on their heads? Why are spam emails littered with misspellings? Why does it seem my friends all have more friends than me?
Background
I've been interested in math for as long as I can remember, but what the word "math" meant has evolved with more exposure.
In middle school, I skipped two grades worth of math, placing me in BC Calculus as a freshman. That meant the following two years would require that I enroll at Northern Virginia Community College. When I ran out of classes there, I had to apply to George Washington University and commute to D.C. It was then I learned just how deep mathematics can go.
I have since then had the privilege of studying and teaching university-level mathematics at Carnegie Mellon, where I could begin to see the iceberg underneath my feet. Below I've listed the most important fields I've delved into in considerable depth (although my background certainly isn't limited to these topics).
Notable Topics
Then I took it again as a freshman in college, from a more theoretical lens. (21-242)
Then I taught a linear algebra course to incoming freshmen. (21-241)
Then I took it again, but even more abstract, this time with fields and infinite dimensional vector spaces. (21-341)
Then I took it again from a data science perspective. (21-670)
Then I took it again with an emphasis on modern computational methods. (21-671)
Linear Algebra is my favorite topic in mathematics. It emerges endlessly in other domains, from vector calculus to graph theory. Its intuitions are introverted and secretive, but reveal themselves
with more careful study. It's essential to high-level physics and is geometrically beautiful.
My first exposure to real probability theory was in an introductory mathematics course in college. Here I learned that probability does not involve hand-waving. I would later teach this course for two semesters, which insisted I understand this axiomatic formulation of probability from the foundations. Between my mathematics and computer science majors, I was exposed to rigorous probability in the classroom through dedicated probability theory courses (21-325), through graduate coursework in Markov chains (36-610) and probability modeling (38-615), through mathematical finance (21-270), through advanced algorithms coursework (15-251, 15-210, 15-451), and even through coursework in the related field of combinatorics (21-301).
I first touched on analysis when studying calculus, perplexed by epsilon-delta proofs. I wish I had dug in deeper at that point.
Years later, I took vector analysis, which approached vector calculus with the rigor it deserves.
In later semesters, I would eventually take both Real Analysis I and II, which collectively stepped through constructing the reals, to deriving calculus from scratch,
all the way to Jacobians and Stroke's theorem.
I've found analysis most practical in my life in the computer science domain, not yet pure mathematics. Constants and bounds that seem arbitrary
and impossibly opaque become transparent when viewed with the tools and intuitions built up through analysis.
One of the best pieces of advice ever given to me (well, for computer science at least) was when stuck on a problem,
always start by turning it into a graph problem. Graph theory is always near in the computer science world.
My first exposure to graph theory was in Foundations of Theoretical Computer Science (15-251), but it struck a chord with me.
Enough so that I eventually took Graph Theory (21-484), where we focused solely on the theory and how vital it can be in mathematical research.
This bridged the gap wonderfully into Combinatorics (21-301), where graphs are the subject of countless queries. But graphs kept showing up,
as promised, in each of my subsequent algorithms courses (15-210, 15-451), including advanced flow algorithms that I would later employ in industry
at Qualcomm.
Teaching Experience
In my four years at Carnegie Mellon, I had the privilege of teaching mathematics during three different semesters.
My first course was Matrices and Linear Transformations (21-241), which delved into matrices, vector spaces, and the relation between them.
Despite having taken linear algebra courses before, I hadn't taken this course, and so much of the material I had to learn in my spare time well enough to teach it.
The following semester, I applied to be a teaching assistant for Professor John Mackey's Mathematical Concepts and Proofs (21-128), with its twin course Mathematical Foundations of Computer Science (15-151).
I taught one section of each during the Falls of 2020 and 2021, each time tailoring the content more towards mathematics or computer science given the section.
I received wonderful feedback from my students, often heartwarming in their praise for my ability to boil down concepts and translate them effectively.
I had the pleasure of working with Professor Po-Shen Loh, founder of Expii and coach of the United States International Math Olympiad team. During my time at Expii, I worked on his intensive coursework for mathematically motivated students at poshenloh.com, during which I created challenging problems tailored to different abilities, edited hours of production-level content, and even provided virtual expertise to students struggling with problems.
Qualcomm's Thinkabit Lab at Virginia Tech is a lab that serves the dual purpose of education and exploration. In my final years in high school, while working on numerous engineering projects, I also engaged heavily with the community through education. I worked on an “Augmented Reality Sandbox” for a local educator’s thesis, and I taught Arduino basics to underprivileged students to encourage STEM involvement. The experience left a lasting impact on me, and would later push me toward my role as a teaching assistant.
I have been tutoring privately since middle school, on topics ranging from basic algebra to multivariable calculus. Even years later, I'm still reminded through chance encounters with former students of the impact that I had when they were eager to learn but lacking good teachers.