Here is a draft of a book by Aldous and Fill on the theory of reversible Markov chains.

Start with a σ-algebra of sets, as usual. Instead of associating a probability with each set, associate a member of the set {Not Possible, Possible}. The empty set is assigned the value Not Possible, while the whole space is assigned the value Possible. A countable disjoint union of sets is Possible if and only if at least one of the individual sets is Possible.

A measure takes values in the semigroup of the nonnegative real numbers closed under addition. Here, we’ve replaced that semigroup with the semigroup of {Not Possible, Possible} under the commutative binary operation +, with multiplication table:

I’ll explain the relationship with topological Markov chains in a future post.

Grigory Garkusha has a thorough introduction to the subject on arXiv.

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.

Intriguingly, the profiles list his job description as a “mathematician”, but they don’t really make clear what this means.

One of his essays convinced me back in high school that trying to visualize the fourth dimension was dangerous. Charles Hinton invented a system of cubes to teach you to visualize the fourth dimension. Gardner printed a letter (copies at banubula and waggish) from someone who said that the cubes were bad for your mental health. It wasn’t until sometime after taking linear algebra that the feeling dissipated.

More on the cubes can be found at The Fairyland of Geometry.