George Lowther at Almost Sure has written a terrific series of posts explaining stochastic processes and the stochastic calculus. Stochastic calculus is widely used in physics and finance, so there are many informal introductions that get across the main ideas in a form sufficient for applications. Most of the formal presentations of the subject seem very far away from the informal ones, to an unusual extent. For example, for the important technical notion of semimartingale the Wikipedia definition is the usual one, which has a very different flavor from the naive picture useful in applications. Lowther introduces it directly in terms of the stochastic integral, and the stochastic integral itself is introduced as a limiting process of random sums of a particularly simple form. The random sums are pretty much the same things you would write down in a naive presentation.
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.
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 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.
I was never really sure I believed Weibel’s famous footnote that a proof of the Snake Lemma appeared in a 1980 romantic comedy, It’s My Turn, but Oliver Knill has put together a gallery of math clips from movies and TV shows, and it’s there.
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.
David Li’s Gaussian copula formula has been called the formula that killed Wall Street, because of how it was used in pricing mortgage-backed securities.
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.
I had a question that I was going to ask on Math Overflow, but after some research I managed to find the answer.
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 GÃ¡bor Nagy and Petr VojtechovskÃ½, explains Moufang loops, and how the classification of non-associative Moufang Loops reduces to a question about finite simple groups.
I’ve been trying to learn about stacks, something that is much easier in the Internet age. The Stacks Project is a collaborative textbook that introduces the subject from the ground up, including all of the machinery necessary. The book is already up to 3000(!) pages.
Project Gutenberg has David Hilbert’s Foundations of Geometry available. It is a translation of Hilbert’s Grundlagen der Geometrie, which is famous as the first modern axiomization of Euclidean geometry. The difference between Hilbert’s approach and that of Euclid is that Hilbert fills in all of the fiddly little details required to meet modern standards of rigor.
The book is elementary, and (as translated by Townsend) is a pleasant read. Much of the book centers around constructing the field of real numbers in terms of the axiomized geometrical constructions. This in turn allows Hilbert to show that the set of axioms is complete. The topic leads naturally to one of the main themes of research in plane geometry in the early part of the last century, which is to consider different algebraic objects and how they can serve as coordinates for different notions of affine or projective planes. The reals can be replaced with an arbitrary division ring, for example. For a projective plane, the most general object is a planar ternary ring, with has a ternary operation that serves as a hybrid of addition and multiplication. Determining the projective planes with a finite number of points is still an open question.