Lieven Le Bruyn has posted his vacation reading. I was planning on eventually writing something about Victor Ginzburg’s Lectures on Noncommutative Geometry, which is a survey of noncommutative geometry. Berest and Chalykh’s A∞ modules and Calogero-Moser Spaces. A∞ algebras are a generalization of algebras where the multiplication goes horribly wrong. Calogero-Moser space is a space that parametrizes right ideals in the Weyl algebra. These are two topics that I’d like to learn, so it looks interesting.
Smolin on Background Independence
Lee Smolin has posted a philosophical article on alternative approaches to quantum gravity: The case for background independence.
Topology Atlas
Topology Atlas is a portal site for topologists. It’s most intriguing feature is Topology Q+A, a collection of discussion forums for mathematics questions. The possibilities include Ask a Topologist and Ask an Analyst. For example, here’s Abhijit Dasgupta’s answer to a question about non-Borel measurable sets.
Probability lecture notes online
I was hunting for lecture notes on (measure-theoretic) probability, and I found a couple of nice links:
- Rich Bass has posted succinct introduction to probability as well as lecture notes on other topics in probability.
- Ivan Wilde has a series of lecture notes in real analysis, including one set on Measure, Integration, and Probability. He also covers C* algebras and von Neumann algebras, among other topics.
Two Cheers for String Theory
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.
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.
International Mathematical Olympiad
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
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.