Peter Woit reports that the entire editorial board of the journal Topology has resigned to protest the high cost of subscription to the journal. John Baez has posted the resignation letter here.
John has long been an advocate of boycotting expensive journals, and has written a call to action. You can find a list of math journals and their costs on this webpage by Ulf Rehmann.
There’s a new preprint on arXiv, Counterexample to the Hodge Conjecture, by Kim and Roush, that proposes a counterexample to the Hodge conjecture, one of the Clay Mathematics Institute’s 7 Millenium Problems. It’s a bold claim, and the 12-page paper looks like it would be easy for an expert (which I am not) to follow, so we’ll probably hear a definite verdict relatively soon. Interestingly, part of the argument involves a computer calculation.
Via Peter Woit.
Update. The authors have withdrawn their claim of a counterexample.
This thread about famous women mathematicians on Cocktail Party Physics, reminded me of an interesting figure in history that I came across while doing researching for a Wikipedia article: Grete Hermann. (The Wikipedia article is a skeleton that I created; it could use a lot of work.)
Hermann was a student of Emmy Noether. Noether was one of the iconic figures of twentieth-century mathematics, a key figure in the century’s trend toward abstraction. A typical example is her proof of the Lasker-Noether theorem. The theorem, that every ideal has a primary decomposition, was originally proven for polynomial rings by Emanuel Lasker, using a difficult computational argument. Noether identified the key abstract condition behind the result — the ascending chain condition on ideals — and used it to give a shorter proof of a much more general theorem. Rings that satisfy the ascending chain condition on ideals are now known as Noetherian rings in her honor.
While Hermann was Noether’s student, her thesis was a throwback to the nineteenth century’s computational approach. Hermann showed that Lasker’s approach could be turned into an effective procedure for computing primary decompositions. Hermann did this before the invention of the computer, or even before the notion of an effective procedure had been formalized. (As her definition, Hermann used the existence of an explicit upper bound on time complexity, and gave such a bound for primary decomposition, and other questions in commutative ring theory.)
Hermann went on to work in philosophy and the foundations of physics. John Von Neumann had proposed a proof that a hidden variable theory of quantum mechanics could not exist. (A hidden variable theory is one that explains the random behavior of quantum mechanical systems in terms of unobserved deterministic variables.) Hermann discovered and published the flaw in Von Neumann’s proof back in 1935, a result that has no impact until it was rediscovered by John Bell some thirty years later.
(The thread on Cocktail Party Physics is instructive for just how unfamous mathematicians really are. For physicists, Karl Weierstrauss is an obscure historical figure. For mathematicians of course, Weierstrauss is five times as famous as Madonna and Britney Spears combined. It was interesting to learn that Sofia Kovalevskaya is not particularly well-known among physicists, even though part of her research was in classical mechanics.)
David Bachman has written a text, available online that introduces differential forms at the level of second-year calculus. The subject is a little too familiar to me for me to judge how well he succeeds, but he actually uses the notes as the text when he teaches Calculus III.
Week 236 of John Baez’ This Week’s Finds in Mathematical Physics is up. The bulk of this week’s entry is about large countable ordinals. (Something I’ve always wanted to understand is in what sense the Feferman-Schütte ordinal captures the idea of an impredicative definition.
John explains how the spaces between interesting ordinals grows large in terms of driving through South Dakota. If you ever drive I-90 the length of South Dakota, you’ll see prairie occasionally interrupted by billboards. Unfortunately, there’s not that much worth advertising, but since there’s a law of conservation of the number of billboards, fake tourist attractions have sprung up simply to catch bored travellers. After seeing billboards for Wall Drug for 400 miles (and the first one travelling west really says “Wall Drug — 400 miles”), you’ll be tempted to stop too.
The August Notices of the AMS is out. David Ruelle contributes What is… a Strange Attractor. Samuel Maratek’s The Differential Geometry and Physical Basis for the Applications of Feynman Diagrams summarizes the basics of gauge theory.
The issue also features a debate between David Eisenbud and Ronald Stern about whether the AMS should establish an AMS Fellow program.
I was looking at the description of espace étalé (see Wikipedia’s article on sheaves) in J.S. Milne’s lecture notes on etale cohomology where I saw this scary sentence: “It is possible to avoid using these spaces — in fact Grothendieck has banished them from mathematics — but they are quite useful, for example, for defining the inverse image of a sheaf”. Well, I hope for Milne’s sake that Grothendieck never finds out.
In a glimpse of humanity’s future, which will be a grim dystopia for me and a paradise for everyone else, John Iskra is writing an undergraduate algebra text written purely from a categorical point of view, called Really Modern Algebra. It’s far from complete, but at 70 pages you can see where he’s going. He’s currently teaching a course out of the book, and also provides his slides from lectures.
If anyone is interested in some more synthetic differential geometric goodness, the point of view of the book Natural Operations in Differential Geometry by Ivan Kolar, Jan Slovak and Peter W. Michor, while couched in a more traditional language, is quite close to that of synthetic differential geometry. In Natural Operations, the authors are trying to classify functors on the category of differentiable manifolds (this is what they call a natural operation). Synthetic differential geometry tries to define a larger category so that those functors become representable.
Sean at Cosmic Variance has a very interesting post on Boltzmann and entropy. Given that entropy is generally increasing, why was the universe ever in a low-entropy state? One idea proposed by Boltzmann himself is that we are living in a small low-entropy fluctuation in a much larger universe. (According to statistical mechanics, entropy can decrease, but is just very unlikely to do so.) If we take Boltzmann’s idea seriously, then we would expect to be living in the most likely fluctuation compatible with our existence, which does not seem to be the case. Sean has much more on this.