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
___ Students Have to Suffer
This will be old news to most of the readership of this blog, but I realize I've never written it down, so time to fix that. Fill in the blank Let's begin by playing a game of "fill in the blank". Suppose that today, the director of secondary education at your high school says: "___… Continue reading ___ Students Have to Suffer
Artin Reciprocity
I will tell you a story about the Reciprocity Law. After my thesis, I had the idea to define $latex {L}&fg=000000$-series for non-abelian extensions. But for them to agree with the $latex {L}&fg=000000$-series for abelian extensions, a certain isomorphism had to be true. I could show it implied all the standard reciprocity laws. So I… Continue reading Artin Reciprocity
Against Perfect Scores
One of the pieces of advice I constantly give to young students preparing for math contests is that they should probably do harder problems. But perhaps I don't preach this zealously enough for them to listen, so here's a concrete reason (with actual math!) why I give this advice. 1. The AIME and USAMO In… Continue reading Against Perfect Scores
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
Shifting PDF’s using gs
Some time ago I was reading the 18.785 analytic NT notes to try and figure out some sections of Davenport that I couldn't understand. These notes looked nice enough that I decided I should probably print them out, But much to my annoyance, I found that almost all the top margins were too tiny, and… Continue reading Shifting PDF’s using gs
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
Mechanism Design and Revenue Equivalence
Happy Pi Day! I have an economics midterm on Wednesday, so here is my attempt at studying. 1. Mechanisms The idea is as follows. We have $latex {N}&fg=000000$ people and a seller who wants to auction off a power drill. The $latex {i}&fg=000000$th person has a private value of at most $latex {\$1000}&fg=000000$ on the… Continue reading Mechanism Design and Revenue Equivalence
Power Overwhelming 