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
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.
Pure Planar Evil
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.
Algebraic Combinatorics on Words
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.
MONDieu!!
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:
- MOND could actually be correct – such a book would be early to the game
- While there are many science popularization books written, there certainly aren’t many speculative hard science popularizations written – unless you count all the string theory stuff – it could jump start a whole category!
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:
Green’s relations
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.
Diestel on Graph Theory
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.
Pioneer Anomaly
The Pioneer 10 and 11 spacecraft have experienced an unexplained drag that has caused them to travel slower than predicted, a phenomenon known as the Pioneer anomaly. Attempts to explain it using normal physics have failed, which leads people to speculate that it will require brand-new physical theories to explain. I personally hope that this is like the discovery of radioactivity by Roentgen in 1895 — a first initial glimpse into a new world.
Water Mechanics
Despite the fact that in theory it is entirely reducible to quantum mechanics, chemists do not have a mathematical model of water molecules that completely explains its behavior.
Update. Sigfpe has more thoughts at his blog.
Crawley-Boevey on Quivers et al
If you’re interested in recent developments in abstract algebra, an excellent place to look is the homepage of William Crawley-Boevey. He provides lecture notes covering quivers (which we’ve discussed before), the cohomological approach to central simple algebras, and invariant theory.