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.

I was curious if the statement of Sard’s Theorem was the best possible. Curiously, the best answer I could find was on Everything 2, which describes an improvement in terms of Hausdorff measure zero sets.

I see via Peter Woit that a new preprint by John Morgan and Gang Tian has appeared on arXiv, Ricci Flow and the Poincare Conjecture, which claims to offer a complete proof of the Poincare Conjecture, based on Perelman’s sketch. Huai-Dong Cao and Xi-Ping Zhu’s proof of the complete Geometrization Conjecture has been published in the Asian Journal of Mathematics, and is now available online as A Complete Proof of the Poincaré and Geometrization Conjectures – Application of the Hamilton-Perelman theory of the Ricci flow.

I found some nice lecture notes on local cohomology (in commutative algebra) by Craig Huneke, with an appendix by Amelia Taylor.

Synthetic differential geometry is an attempt to reformulate differential geometry to allow infinitesimals. Unlike nonstandard analysis, these infinitesimals are nilpotent, and the operation of taking the derivative of a function at a point becomes just evaluating the function at a nilpotent infinitesimal near that point. The idea was used heuristically in the nineteenth century, but the inspiration from the modern reformulation comes from commutative algebra, where the idea is unproblematic.

Anders Kock has made his book on the subject, Synthetic Differential Geometry, available for download on his website. The book is being reprinted, so he asks readers not to circulate printed copies.

I was recently reminded of the invariant subspace problem in Hilbert spaces: the question of whether every bounded operator on a Hilbert space has a closed invariant subspace. Of famous open problems in mathematics, this one is perhaps the most surprising. It sounds like it should be exercise 7 of chapter 2 of a book on Hilbert spaces; yet the answer is still unknown. (I have no idea what the answer should be; I’m just surprised that it’s so hard to figure out one way or the other. What makes it particularly surprising is the answer is known for Banach spaces.)

B. F. Yadav has a survey article on the subject The Invariant Subspace Problem. It appears in Nieuw Archief voor Wiskunde, a publication of the Royal Dutch Mathematical Society, which prints the occasional article in English.

I’ve just had the finest accomplishment of my mathematical career. I was trying to remember the definition of Jategaonkar’s second layer condition, which arises in noncommutative ring theory when studying the analogue of localizing at a prime ideal. So I typed Jategaonkar into Google, and lo and behold, I spelled it right the first time. (I didn’t say this was much of an accomplishment; just my finest.)

(As for the actual definition, I had less luck. The best I could find was this Ph. D. thesis, by Paul Chong-Hyun Kim.)

Kenny Easwaran and David Corfield are discussing whether mathematical concepts have essences. Kenny’s example is that of a normal subgroup of a group, that while it can be defined several different ways, the essence of the notion is that it is the kernel of a group homomorphism. David relates the question to his larger program to redirect the philosophy of mathematics away from its traditional concerns towards elucidating the meaning of mathematical concepts.