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.