This will be old news to anyone who does algebraic topology, but oddly enough I can't seem to find it all written in one place anywhere, and in particular I can't find the bit about $latex {\mathsf{hPairTop}}&fg=000000$ at all. In algebraic topology you (for example) associate every topological space $latex {X}&fg=000000$ with a group, like… Continue reading Algebraic Topology Functors
Tag: math
A Sketchy Overview of Green-Tao
These are the notes of my last lecture in the 18.099 discrete analysis seminar. It is a very high-level overview of the Green-Tao theorem. It is a subset of this paper. 1. Synopsis This post as in overview of the proof of: Theorem 1 (Green-Tao) The prime numbers contain arbitrarily long arithmetic progressions. Here, Szemerédi's… Continue reading A Sketchy Overview of Green-Tao
Approximating E3-LIN is NP-Hard
This lecture, which I gave for my 18.434 seminar, focuses on the MAX-E3LIN problem. We prove that approximating it is NP-hard by a reduction from LABEL-COVER. 1. Introducing MAX-E3LIN In the MAX-E3LIN problem, our input is a series of linear equations $latex {\pmod 2}&fg=000000$ in $latex {n}&fg=000000$ binary variables, each with three terms. Equivalently, one… Continue reading Approximating E3-LIN is NP-Hard
First drafts of Napkin up!
EDIT: Here's a July 19 draft that fixes some of the glaring issues that were pointed out. This morning I finally uploaded the first drafts of my Napkin project, which I've been working on since December 2014. See the Napkin tab above for a listing of all drafts. Napkin is my personal exposition project, which… Continue reading First drafts of Napkin up!
The Structure Theorem over PID’s
In this post I'll describe the structure theorem over PID's which generalizes the following results: Finite dimensional vector fields over $latex {k}&fg=000000$ are all of the form $latex {k^{\oplus n}}&fg=000000$, The classification theorem for finitely generated abelian groups, The Frobenius normal form of a matrix, The Jordan decomposition of a matrix. 1. Some ring theory… Continue reading The Structure Theorem over PID’s
Miller-Rabin (for MIT 18.434)
This is a transcript of a talk I gave as part of MIT's 18.434 class, the ``Seminar in Theoretical Computer Science'' as part of MIT's communication requirement. (Insert snarky comment about MIT's CI-* requirements here.) It probably would have made a nice math circle talk for high schoolers but I felt somewhat awkward having to… Continue reading Miller-Rabin (for MIT 18.434)
Things Fourier
For some reason several classes at MIT this year involve Fourier analysis. I was always confused about this as a high schooler, because no one ever gave me the ``orthonormal basis'' explanation, so here goes. As a bonus, I also prove a form of Arrow's Impossibility Theorem using binary Fourier analysis, and then talk about… Continue reading Things Fourier
18.099 Transcript: Bourgain’s Theorem
As part of the 18.099 Discrete Analysis reading group at MIT, I presented section 4.7 of Tao-Vu's Additive Combinatorics textbook. Here were the notes I used for the second half of my presentation. 1. Synopsis We aim to prove the following result. Theorem 1 (Bourgain) Assume $latex {N \ge 2}&fg=000000$ is prime and $latex {A,… Continue reading 18.099 Transcript: Bourgain’s Theorem
18.099 Transcript: Chang’s Theorem
As part of the 18.099 discrete analysis reading group at MIT, I presented section 4.7 of Tao-Vu's Additive Combinatorics textbook. Here were the notes I used for the first part of my presentation. 1. Synopsis In the previous few lectures we've worked hard at developing the notion of characters, Bohr sets, spectrums. Today we put… Continue reading 18.099 Transcript: Chang’s Theorem
Stop Paying Me Per Hour
Occasionally I am approached by parents who ask me if I am available to teach their child in olympiad math. This is flattering enough that I've even said yes a few times, but I'm always confused why the question is "can you tutor my child?" instead of "do you think tutoring would help, and if… Continue reading Stop Paying Me Per Hour
Power Overwhelming 