At the new group physics weblog Cosmic Variance, Sean Carroll has posted a defense of string theory, Two Cheers for String Theory. Chad Orzel and Peter Woit have also weighed in.
Author Archives: Walt
Expander Graphs
It’s basically impossible to know all of the important concepts and results in mathematics. It’s impossible to even have heard of all of the important concepts and results in mathematics. For example, I’d never heard of expander graphs, which apparently have widespread applications in combinatorics and computer science, and even have an interpretation in terms of group representations.
Michael Nielsen has a series of posts on expander graphs beginning here. For more background, he links to lecture notes on the subject by Linial and Wigderson.
Igor Dolgachev
Igor Dolgachev, a mathematician at the University of Michigan, has made available lecture notes on topics in algebraic geometry and physics. The lecture notes in algebraic geometry include invariant theory and what he calls “classical algebraic geometry”. He also provides an introduction to theoretical physics for mathematicians, and as well as one on string theory.
Project Euclid
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:
- Annals of Mathematics
- Communications in Mathematical Physics
- Nagoya Mathematical Journal
- Pacific Journal of Mathematics
- Probability Surveys
- Proceedings of the Japan Academy, Series A
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.
Lax Attack
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.
Figure Eight Revisited
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.
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.