I'm reading through Primes of the Form $latex {x^2+ny^2}&fg=000000$, by David Cox (link; it's good!). Here are the high-level notes I took on the first chapter, which is about the theory of quadratic forms. (Meta point re blog: I'm probably going to start posting more and more of these more high-level notes/sketches on this blog… Continue reading Positive Definite Quadratic Forms
Category: Linear and Abstract Algebra
Facts about Lie Groups and Algebras
In Spring 2016 I was taking 18.757 Representations of Lie Algebras. Since I knew next to nothing about either Lie groups or algebras, I was forced to quickly learn about their basic facts and properties. These are the notes that I wrote up accordingly. Proofs of most of these facts can be found in standard… Continue reading Facts about Lie Groups and Algebras
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
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
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
Tannakian Reconstruction
These notes are from the February 23, 2016 lecture of 18.757, Representations of Lie Algebras, taught by Laura Rider. Fix a field $latex {k}&fg=000000$ and let $latex {G}&fg=000000$ be a finite group. In this post we will show that one can reconstruct the group $latex {G}&fg=000000$ from the monoidal category of $latex {k[G]}&fg=000000$-modules (i.e. its… Continue reading Tannakian Reconstruction
Rant: Matrices Are Not Arrays of Numbers
The following is an excerpt from a current work of mine. I thought I'd share it here, as some people have told me they enjoyed it. As I'll stress repeatedly, a matrix represents a linear map between two vector spaces. Writing it in the form of an $latex {m \times n}&fg=000000$ matrix is merely a… Continue reading Rant: Matrices Are Not Arrays of Numbers
Representation Theory, Part 4: The Finite Regular Representation
Good luck to everyone taking the January TST for the IMO 2015 tomorrow! Now that we have products of irreducibles under our belt, I'll talk about the finite regular representation and use it to derive the following two results about irreducibles. The number of (isomorphsim classes) of irreducibles $latex {\rho_\alpha}&fg=000000$ is equal to the number… Continue reading Representation Theory, Part 4: The Finite Regular Representation
Represenation Theory, Part 3: Products of Representations
Happy New Year to all! A quick reminder that $latex {2015 = 5 \cdot 13 \cdot 31}&fg=000000$. This post will set the stage by examining products of two representations. In particular, I'll characterize all the irreducibles of $latex {G_1 \times G_2}&fg=000000$ in terms of those for $latex {G_1}&fg=000000$ and $latex {G_2}&fg=000000$. This will set the… Continue reading Represenation Theory, Part 3: Products of Representations
Representation Theory, Part 2: Schur’s Lemma
Merry Christmas! In the previous post I introduced the idea of an irreducible representation and showed that except in fields of low characteristic, these representations decompose completely. In this post I'll present Schur's Lemma at talk about what Schur and Maschke tell us about homomorphisms of representations. 1. Motivation Fix a group $latex {G}&fg=000000$ now,… Continue reading Representation Theory, Part 2: Schur’s Lemma
Power Overwhelming 