The connection between classical topics in algebraic geometry and modern techniques is fascinating.

]]>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 Kevin McCrimmon has an unpublished draft of a book on the structure theory of alternative algebras.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.

]]>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.)

]]>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”, x^{p} + y^{q} = z^{r}.

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.

]]>Simon Willerton has thoughts on PERT charts as copresheaves.

]]>Over the past year, several posts convey palpable excitement over a series of preprints that prove two conjectures: the virtually Haken conjecture and its generalization the virtually fibered conjecture. These were apparently the outstanding open conjectures after the proof of geometrization. This post in particular describes the techniques involved in the proof. To see how fast things changed over the past year, this post on the Wise conjecture (an important ingredient of the proof) makes it clear that from the perspective of March of this year it was very much an open question which way the result would most likely turn out.

I’d been meaning to learn more about the subject, just to have a better idea of what happened. (For example, I still don’t really understand what a Haken manifold is, even though I’ve read the definition. Fortuitously, Erica Klarreich has written a long

general-audience article that gives at least some of the flavor of what’s going on.