The questions for the 46th International Mathematical Olympiad are now available.
There are six problems and the competition was over two days. Each day the contestants got 4 hours 30 minutes to solve three problems. So, close your office door, download the first set of problems and see how you go.
Oh, and by the way, the contestants were high school students.
Project Euclid is a project of Cornell University to mathematical journals online. The site hosts a mix of free and for-pay journals, but several journals are available completely free:
So for example, if you’re interesting in seeing Feit and Thompson’s proof of the odd order theorem in the Pacific Journal of Mathematics, you can find it here.
Last week, when Michael asked for a list of fundamental theorems in different branches of mathematics, Juan de Mairena suggested the Lax Equivalence Theorem as a candidate. Today on ArXiv I spotted a paper that makes the rather dramatic claim that the theorem is “wrong” — not that it is wrong in the strict mathematical sense, but that its conditions are not realistic for real-world problems. I’m not in a position to evaluate the claim (I never even heard of the result until Juan’s comment), but I thought it was interesting to see a paper on the subject so soon after we discussed it.
Astronomers have discovered a planet that has three suns, something that was long thought to be impossible. (Astronomers argued that the orbits would be too unstable.) Can a planet in a figure eight orbit be far behind?
Deformation theory is the study of how mathematical structures vary with respect to parameters. Pavel Etingof has written an introduction to the deformation theory of associative algebras. Marco Manetti has provided extensive lecture notes on deformations of complex structures.
Deformation theory for associative algebras can be related to both algebraic topology and quantum field theory. Alexander Voronov has some lecture notes from a course he taught on the connections.
You knew it was only a matter of time before Flash was used for pure evil. John Tantalo, who could be using his talents to cure cancer or something, has written the Planarity Flash Game. The game generates random planar graphs and draws them to hide the fact that they are planar. You the player (or rather victim) must move the vertices around to show that it really is planar. The game is hard, but moving the little dots around is incredibly hypnotizing.
Via Eszter at Crooked Timber.
M. Lothaire is a pseudonym for a group of authors who wrote the book Combinatorics on Words. The study of words — strings of letters drawn from a fixed alphabet — is surprisingly fruitful in mathematics. For example, finite words form a basis of the free algebra. Sets of infinite words closed under shifts form dynamical systems known as symbolic dynamical systems. Many apparently more-complicated dynamical systems can be reduced to symbolic systems.
A more complicated application is that of Lyndon words. The property of being an aperiodic word is preserved under cyclic permutations. Let two aperiodic words which are cyclic permutations of each other be considered equivalent. Lyndon words are a particular method of choosing one member of each equivalence class. Surprisingly Lyndon words can be used to write down a vector space basis for the free Lie algebra.
M. Lothaire is back with a sequel, Algebraic Combinatorics on Words. The best part? It’s available online.
MOND, the acronym for MOdified Newtonian Dynamics is a theory put forth by Moti Milgrom in 1983 to resolve problems with galaxy kinematics without resorting to dark matter. The theory decouples inertial and gravitational mass (breaking the equivalence principle) positing that at very small accelerations (those below an observationally determined constant a0), the gravitation force felt by a body is actually smaller than the force predicted by the famous Newtonian equation F = ma. Aside from the bizzare nature of the change, the theory has several things going for it, not the least of which is that it made predictions that were later verified (namely the existence of low surface brightness galaxies).
Another point in its favor is that it can explain the Pioneer anomoly. When I first heard of the theory and the problems it was meant to solve, I thought that the theory was crazy, but that it might just be crazy enough to be true. I even bandied about the idea of writing a popular science book about it which itself would have several things going for it:
Some final food for thought; I haven’t done the calculation myself, but apparently constant acceleration at a0 for our best guess at the age of the universe produces a velocity of – wait for it – the speed of light. I don’t know if this new world would be cool enough for Walt to have to put on shades before he glanced at it, but quite a few textbooks would need to be rewritten 
And some MOND links:
Wikipedia now has everything. I was looking up Green’s relations on Google, and there it was. Green’s relations are one of the key concepts in the theory of semigroups, but semigroups themselves are one of the less-fashionable areas of mathematics. So I think it’s safe to say that Wikipedia’s work is done.
Reinhard Diestel has a new edition of his graduate text, Graph Theory available on his website. The book is a dense but broad introduction to the field.