Ars Mathematica

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.

Ariel Rubinstein has made his book, Modeling Bounded Rationality, available online. Economic models tend to assume that humans are mistake-free calculating machines; economists have tried to introduce more realistic assumptions under the banner of bounded rationality. This is a far-from-settled problem, mainly because there are more ways to be wrong than there are to be right.

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.

Either Murray Gell-Mann or Richard Feynman said “Physics is to mathematics as sex is masturbation.” Discuss.

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.

A sunspot equilibrium is a market equilibrium in which prices depend on otherwise irrelevant random variable. The name is inspired by a theory of nineteenth century economist William Jevons that sunspots affected the stock market. (The theory, while wrong, isn’t quite as rediculous as it sounds. Jevons thought, by looking at the data he had on sunspots and agricultural prices, that he detected a pattern that indicated that sunspots caused crops to fail, which in turn caused recessions.)

A sunspot equilibrium would then be a self-fulfilling prophecy. If everyone expected that sunspots caused recessions, that in principle could be sufficient to cause a recession, even if the cause-and-effect existed entirely in people’s heads. Note that this outcome, while not optimal, would still be individually rational: even if you knew that sunspots didn’t really cause recessions, you would know that everyone else was expecting a recession, so you would act accordingly.

Karl Shell, one of the inventors of the concept, has a list of links to his papers on sunspot equilibria. In particular, he links to a short survey article he co-authored with Bruce Smith.

Interestingly, known public-key cryptosystems all seem to depend on the difficulty of the hidden subgroup problem. Suppose you have a group that can observe, and a subgroup that you cannot observe. Instead, you have a function that is constant along cosets of the group and different for different subgroups. The hidden subgroup problem is to compute a generating set for the subgroup just by evaluating the function. The problem generalizes integer factorization, the graph isomorphism problem and the problem of finding the shortest vector in a lattice. An efficient algorithm would apparently crack all known public-key cryptosystems.

Chris Lamont has a survey paper on the hidden subgroup problem in quantum computing, one that does not assume any background in quantum mechanics. Dave Bacon has some thoughts on an alternate approach.

I was under the impression that the uncracked public-key cryptosystems were all based on number theory, which made them vulnerable to variants of Shor’s algorithm. Yesterday I learned via Dave Bacon that there are cryptosystems based on the hardness of finding the shortest vector on a lattice. Here is a survey paper on the subject by Oded Regev. There is also the McEliece cryptosystem, which is based on coding theory.

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.