Greg Friedman has written one of the greatest expository articles of advanced mathematics that I’ve ever read: An elementary illustrated introduction to simplicial sets. The combinatorial notion of simplicial set completely captures the homotopy of simple spaces (such as CW complexes). I never made any headway in understanding this until I read Friedman’s article.
Here’s something I didn’t know. There exists nonlinear (but algebraic) ordinary differential equations such that solutions to that differential equation are dense in the space of continuous functions. These are known as universal differential equations. An explicit construction of one is given in this preprint by Keith Briggs. If I understand the construction correctly, the trick seems to be that the nonlinearity gives you branch points where you have a choice for the direction in the solution. This allows you to paste together solutions in enough ways that you can achieve density.
This thread at Math Overflow has the feel of advanced alien technology. Forcing is a technique for constructing models of set theory where various hypotheses fail. For example, forcing can be used to construct a model of set theory where the continuum hypothesis is violated.
There are some statements whose value cannot be affected by forcing. These statements are known as absolute. Forcing is useless for establishing such a statement is independent, but this can be a virtue. If you can create a model using forcing such that you can prove that an absolute statement is true in that model, then it must already be true in the universe of ordinary sets. The thread gives several specific examples of theorems you can prove this way.
Eugene Lim went to Haiti after the earthquake to teach calculus after the earthquake. He has posted a first-hand account at Cosmic Variance.
Grothendieck categories are a generalization of categories of modules. Sheaves of abelian groups over a topological space also form a Grothendieck category. Grothendieck categories are a special case of abelian categories, but the extra structure allows additional theorems to be proved.
Grigory Garkusha has a thorough introduction to the subject on arXiv.
snarXiv is a site the generates parody abstracts for high-energy physics theory papers, a la arXiv. While the abstracts don’t quite make sense, they eerily resemble the real thing.
snarXiv versus arXiv is another site that gives you a random snarXiv and arXiv paper title, and asks you to tell the fake from the real thing. The fake titles are much harder to recognize than the fake abstracts. Initially, I got the first 5 right, but after about 25 I was down to random chance.
Via Not Even Wrong.
When I first encountered the subject of Lie algebras, I thought it was pointless and unmotivated. I also had the impression from high school physics that classical mechanics was built out of a bunch of random facts that were true for no reason, like the conservation of angular momentum. Also, I thought that potential energy was a sort-of a con — that if you can simply declare that a body has potential energy that you can make the law of conservation of energy tautologically true. Reading Arnold’s Mathematical Methods in Classical Mechanics changed all that. Arnold starts with one-dimensional systems like the inverse-square law and harmonic oscillator, and then to three-dimensional systems where he explains how symmetries in the equations of motion lead to conservation laws. Along the way, he explains how Lie groups lead to Lie algebras, and how in particular how rotational symmetries in 3d lead to the Lie algebra of so(3), which physicists use in the guise of the cross-product of vector calculus. He also introduces the Lagrangian and Hamiltonian formulations of classical mechanics. Most importantly, (since you can learn the equivalent from a physics text like Goldstein’s Classical Mechanics), he puts in the language of mathematicians rather than the language of physicists.
Years after I studied the subject of ODEs, I almost bought Arnold’s (expensive) Ordinary Differential Equations just because it was such a beautiful introduction to the subject. Lots of textbooks allude to the dynamical systems viewpoint for ODEs, but his book really communicates that viewpoint.
The main challenger to the incumbent party in Columbia the former mayor of Bogota, Antanas Mockus. As this profile make clear, Mockus is a man with a flair for the dramatic. According to the profile, he apparently once mooned an auditorium full of students. While mayor, he would occasionally dress up as a superhero named “Supercitizen”.
Intriguingly, the profiles list his job description as a “mathematician”, but they don’t really make clear what this means.