As is well-known, the lattice of submodules of a module is modular. What I did not know is that the converse is not true, and that lattices of submodules must satisfy a stronger property, the arguesian law.
The Arguesian law is a lattice-theoretic analogue of Desargues’ theorem in projective geometry. I read the statement of the theorem several times and I have no intuition about what it means.
There is a kind of converse to this result: a complemented lattice can be embedded into the lattice of submodules of a module if and only if it is arguesian. (I found the result in Gratzer’s book on lattice theory, which is viewable in Google Books.)
Chandan Singh Dalawat has a nice survey article about K2. It just gives the highlights of the theory, without proofs, so it’s closer to a teaser trailer than it is to full-length movie. But sometimes you just want a teaser trailer to tell you if you want to invest the time in the movie.
I came across this terrific article that describes a sequence of results beginning with Pappas’ theorem through the Cayley-Bacharach theorem to modern formulations in terms of the Gorenstein (!) condition.
The connection between classical topics in algebraic geometry and modern techniques is fascinating.
I periodically feel like I should learn more about nonassociative algebra. (I’ve studied Lie algebras, and technically Lie algebras are non-associative, but they’re pretty atypical of nonassociative algebras.) There’s a mysterious circle of “exceptional” examples that are all related — the octonions, the five exceptional Lie algebras, the exceptional Jordan algebra — that I would like to understand better. John Baez has an article about the direct connection that I post about before, but what I don’t understand about the general theory is how relaxing assocativity gives you so few new examples.
Project Gutenberg has a book by Schafer on the general theory of nonassociative algebras. Kevin McCrimmon has an unpublished draft of a book on the structure theory of alternative algebras.
I previously linked to an article classifying the simple Moufang loops. The only examples that are not groups are again related to the octonions.
One of my ambitions in life is to understand projective determinacy. Fortunately, Tim Gowers has written a series of posts to explain Martin’s proof that Borel sets are determined.
The main source of interest in determinacy is that results suggest that it is the strongest regularity property that a set can have, in that it it tends to imply other nice properties such as Lebesgue measurability. Here is a short proof by Martin that determinacy implies Lebesgue measurability. Justin Palumbo has a nice set of lecture notes that relate determinacy to other regularity properties.
(One nuance is that determinacy for a single set usually doesn’t imply strong regularity properties — the proofs typically require several auxiliary games for a single set. The Martin and Palumbo links use the setting of the axiom of determinacy, which is the axiom that all sets are determined. This is actually false in ZFC: it contradicts the axiom of choice. There are analogous results that hold in ZFC where you keep track of which sets you need to have determined.)
I came across a number theory paper Twists of X(7) and Primitive Solutions of x2 + y3 = z7 that I find completely fascinating. I find it fascinating because a) the question is so easy, b) the answer is so hard, and yet c) someone was able to answer it.
An earlier expository paper, Faltings plus epsilon, Wiles plus epsilon, and the Generalized Fermat Equation, talks about the general question of finding solutions to the “generalized Fermat eqution”, xp + yq = zr.
Ugh, I suck at this blogging thing. I periodically get ambitious, and make big plans. That doesn’t actually lead to any completed posts, just many long half-finished posts, and hundreds of open tabs in Firefox. I think I’ll start with some short posts.
Linear types are one of those things that I’d always wanted to learn more about. The idea seems somewhat natural &emdash; practically speaking the amount of resources an object uses is part of its signature &emdash; but the details are sufficiently complex that I’ve never quite mastered it. This presentation by Francois Pottier seems like a nice place to start.
Via Lambda the Ultimate.
Justin Curry has written an excellent introduction to cosheaves. Cosheaves are the dual notion to sheaves, but many specific properties of sheaves of sets do not dualize, so they have a somewhat different flavor. The introduction includes some applications of cosheaves in networks.
Simon Willerton has thoughts on PERT charts as copresheaves.
As part of my new program of bringing you 2009’s internet to you today, I was fiddling around with Translation Party, which repeatedly translates a sentence in English into Japanese and back, until it finds a fixed point. Once I got tired of song lyrics, I tried various mathematical statements paraphrased into plain English. Most of the time, it converges right away on something close to the original sentence, or on gibberish, but I did find one example where it converted a true mathematical statement into an intelligible, but false, mathematical statement: that a matrix group is a group algebra.
I might be the last person in the world to find out, but I just found a website, Detexify, that clearly works on magic. You draw a symbol, and it looks for the closest Latex symbol. It works surprisingly well. I just draw a terrible approximation of the Weierstrass p symbol, and the actual Weierstrass p symbol was the 4th hit.