The 2006 Nobel Prize in Chemistry was awarded to Roger Kornberg for eukaryotic transcription, something that most people would regard as biology. This touched off some agonizing about the Meaning of it All at Uncertain Principles, In the Pipeline, and Adventures in Ethics and Science. Paul Bracher even went so far as to suggest that chemists move in on the physics prize.
Hey, at least they have a prize.
Backreaction has a long post on applications of the AdS/CFT correspondence in heavy ion physics. The topic is interesting because it is a potential experimental prediction derived from string theory. Interestingly, it doesn’t involve string theory as a theory of everything; instead it uses string theory ideas to make calculations about quark-gluon plasma.
I just came across a reference to the work of the philosopher of mathematics Hartry Field. This is what Wikipedia has to say:
Fictionalism was introduced in 1980 when Hartry Field published Science Without Numbers, which rejected and in fact reversed Quine’s indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as true, Field suggested that mathematics was dispensable, and therefore should be rejected as false. He did this by giving a complete axiomatization of Newtonian mechanics that didn’t reference numbers or functions at all. He started with the “betweenness” axioms of Hilbert geometry to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields.
Does anyone know if that is an accurate summary of Field’s argument? It seems obviously wrong to me.
Via Crooked Timber.
November Notices of the AMS are out. The entire issue is devoted to Alan Turing.
Penny Smith has posted a preprint to arXiv, Immortal Smooth Solution of the Three Space Dimensional Navier-Stokes System that, if correct, would solve one of the Clay Institute’s Millenium Problems. Christina Sormani has created detailed summary of Smith’s work on PDEs and Navier-Stokes.
The Navier-Stokes equation is a set of equations that describe fluid flow in Newtonian mechanics. The equations are notoriously difficult to analyze. The existence of smooth solutions for all time (the meaning of “immortal†in the paper title) has long been an open question. One now perhaps closed.
Via Peter Woit.
Update. The paper has been withdrawn. (Via John Baez in the comments.)
I’d like to point out that the 2006 Nobel Prize in Physics was awarded for this diagram:
No word on whether the original diagram included the same caption. More information available from Backreaction. Graphic via XKCD.
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.
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.