I grew up reading Martin Gardner’s columns as a child, and was heavily inspired by Dr. Sun, who runs 九章 (Chiuchang Mathematics Education Foundation). I also did a lot of math competitions in middle and high school. One of the weirder things that blossomed out of this was me presenting about Triangles of Absolute Differences at the Gathering 4 Gardener 9. Anyway, recreational math is great but now that I’ve actually gone to college there’s a lot more math to talk about.
Combinatorics was easily my favorite area of mathematics throughout middle and high school (although I was mostly only comparing against other school and olympiad subjects).
Even today I feel like Cayley’s formula for the numer of trees is one of my favorite theorems. Although the “standard” proof is by Prüfer codes, there are many other proofs out there that are just as cool, if not cooler. This might be one of the reasons I like this theorem so much:
- Joyal’s bijective proof;
- Pitman’s proof by double counting;
- A surprisingly elegant proof by generating functions.
I took 18.217 with Prof. Richard Stanley at MIT.
A handful of illustrative combinatorial problems (possibly liberally defined):
- Given n distinct points on a circle, what is the maximum number of acute triangles that can be formed from three of these points?
- Given n distinct points on the plane, mark the midpoint of every pair of distinct points. Prove that at least 2n − 3 distinct points are marked.
- In an n × n matrix of integers, every pair of adjacent integers differ by at most 1. Prove that some integer appears at least n times.
Theoretical Computer Science
One of my earliest ventures into this space was in middle school where I submitted a paper on simple variations on the secretary problem. It won a silver medal for some reason.
I took 6.841/18.405 Advanced Complexity Theory with Prof. Dana Moshkovitz. For my final project I co-wrote a survey on communication complexity, which is pretty cool.
Later I took 6.854/18.??? Advanced Algorithms. For my final project I wrote a survey of the prophet inequality, perhaps a sibling of the secretary problem.
Finally during Summer 2017 I participated in SPUR where I learned about tree reconstruction. Afterwards I continued the research under a new direction with Prof. Elchanan Mossel.