The latest issue of the Bulletin of the AMS is out. The feature article is Expander graphs and their applications by Hoory, Linial, and Wigderson. Expander graphs are a kind of graph that are important in computational complexity theory; we discussed them once before. Y.S. Sinai, a mathematician who works on topics quite close to physics, has an interesting article on the cultural differences called Mathematicians and physicists = cats and dogs?.

I was reading Hilton and Stammbach’s A Course in Homological Algebra, when I spotted this rather forlorn passage:

Of course, if A is free, Ext(A,Z) = 0, but it is still an open question whether, for all abelian groups A, Ext(A,Z) = 0 implies A free.

It is forlorn because we now know that we’ll never know: this is the Whitehead problem, and in 1973 Saharon Shelah proved that it is independent of the axioms of set theory.

David Corfield discusses some speculation originally from Israel Gelfand:

Sporadic simple groups are not groups, they are objects from a still unknown infinite family, some number of which happened to be groups, just by chance.

(In David’s terminology, that means that sporadic finite simple groups are not a natural kind.)

I used to believe this very same thing, so I find it interesting that others have speculated the same thing. A couple of years ago, though, I came across a remark by Michael Aschbacher that made me rethink my view: the classification of finite simple groups is primarily an asymptotic result. Every sufficiently large finite simple group is either cyclic, alternating, or a group of Lie type.

Results that are true only for large enough parameter values are common enough that the existence of small-value counterexamples does not require special explanation. For example, the classification of simple modular Lie algebras looks completely different over small characteristics than it does over large characteristics. The best known results for number theoretic results such as Waring’s problem and Goldbach’s conjecture are asymptotic. Small numbers are just bad news.

Did anyone else recieve a press release from Shing-Tung Yau’s lawyer? With no explanation, I was sent this press release from Howard Cooper, Yau’s lawyer, denying the version of events described in Nasar and Gruber’s New Yorker article, Manifold Destiny. There’s nothing in the e-mail, other than press release, so as far as I know they either a) sent it to me because I linked to the New Yorker article, b) sent it to everyone with an e-mail address on this site, or c) everyone in the world. (In fact, I almost deleted the mail as spam without reading it.)

The web version of the press release links to this letter from Cooper to the article’s authors, detailing their specific charges. The letter is careful to make it sound like they could sue, but they haven’t made up their mind to do so yet.

The October Notices of the AMS are also out. Not that much grabbed me from this issue. The feature article, Mathematics in Facial Surgery, describes how PDEs are used in modelling the results of facial reconstructive surgery. This month’s What is…, What is… a bad end describes a concept in complex manifold theory.

I finally had a chance to take a look at the September Notices of the AMS. Allyn Jackson’s Conjectures No More summarizes the conventional wisdom that the Poincaré and Geometrization conjectures are now theorems. What is… a quasicrystal?, by Marjorie Senechal, describes quasicrystals (crystals whose diffraction pattern implies they have symmetries that cannot be explained by a cyrstallographic group) and Penrose tilings.

The feature article, Notes on the Deuring-Heilbronn Phenomenon, by Jeffrey Stopple, discusses some results on Dirichlet L-functions.

I missed these a couple of months ago when they came out. But, if you are looking for something to do this Labor day (in the US anyway, I’m currently in the UK and finding it difficult not to write “Labour”), here are the problems for the 47^th International Mathematical Olympiad (IMO): day 1 and day 2. Have fun!