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 Kevin McCrimmon has an unpublished draft of a book on the structure theory of alternative algebras.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

What’s interesting about the clip is that it’s clear to a math audience that the student who keeps interrupting is a blowhard who has no idea what he’s talking about. While it would be clear to any audience that the student is arrogant, I don’t know if it would be clear that the student doesn’t know what he’s talking about.

Sociologists Donald McKenzie and Taylor Spears have a new paperon the history of the Gaussian copula model, based on detailed interviews with quants before and after the crisis. They find that the limitations of the Li model were well understood by financial modelers on Wall Street, and that none of them took the model as literally true. The formula became widely used for institutional reasons outside of the quant community — for example, once the industry had a standard model, then the model could be used in evaluating profit and loss. This is further evidence for the thesis that mathematics is only dangerous when it falls into the wrong hands.

Cathy O’Neal highlights the parts of the paper that address the questions of institutional politics and blame.

Finite simple groups have a complete classification. I was wondering if there were any weakenings of the axioms of group that also allowed a complete classification of the simple objects. (Here, I mean no nontrivial quotients.) Surprisingly, there’s a classification for semigroups. In the theory of semigropus the term “Simple& is used for a weaker notion. Semigroups with no nontrivial quotients are known as “congruence-free”. The classification of finite congruence-free semigroups splits into two cases: for semigroups with a zero (an element 0 such that 0x = 0) there’s an explicit construction, while a congruence-free semigroup without a zero must be a simple group.

Another direction to generalize is weaken the form of associativity. The most-studied weakening is the Moufang property, which includes the octonions as a non-trivial example. Here, the complete classification is also known: a finite simple Moufang loop is either a group or a Paige loop, which is a non-associative construction closely related to the octonions, but defined over a finite field. It’s interesting that in this case, the one non-associative family resembles simple groups of Lie type, in that it’s parameterized by the finite fields. This classification relies non-trivially on the classification of simple groups, in that the explicit classification is used to rule out any other non-associative examples.

The paper Octonions, simple Moufang loops and triality by Gábor Nagy and Petr Vojtechovský, explains Moufang loops, and how the classification of non-associative Moufang Loops reduces to a question about finite simple groups.