For anyone who finds the Connes-Marcolli paper we recently linked to heavy going, Lieven Le Bruyn recommends some lighter reading: an interview with Alain Connes himself from the conference in Tehran where the paper was presented. In the interview Connes talks about physics and noncommutative geometry. He also has some sharp criticisms with how mathematics research works in the United States in comparison to France (where he lives and works).

Peter Woit also discussed this interview a couple of weeks back.

Oddly enough, an important part of recent set theory is two-person zero-sum infinite games. The question of solvability of games defined on certain types of sets turn out to depend on delicate set-theoretical questions (most of which turn out to be independent of ZFC). Solvability of particular types of games on a set can be thought of a refinement of measurability for that set.

Jindrich Zapletal has some lecture notes on the subject.

We’ve discussed semirings before. One interesting application is tropical geometry, which studies the analogue of algebraic varieties over the max-plus semiring (sometimes known as the tropical semiring).
Grigory Mikhalkin has posted a survey article on the subject, Tropical geometry and its applications, to arXiv. (The “applications” of the title are applications to ordinary algebraic geometry.)

Update. Commenter ansobol has compiled an online bibliography of recent works in tropical geometry . For pre-1996 works, there is another bibliography by Stephane Gaubert.

My calender reads January 9, but the February Notices of the AMS are up.

Alain Connes and Mathilde Marcolli have posted a new survey paper on Arxiv, A walk in the noncommutative garden. There are many contenders for the title of noncommutative geometry, but Connes’ flavor is the most successful. Most of Connes’ examples are variants of the same basic idea: when a group acts nicely on a space, you can define a new space by collapsing each orbit of the group action to a single point (this construction is known as the quotient space of the action). Unfortunately, most group actions are not nice.

Connes and Marcolli describe an alternative construction. By a theorem of Gelfand, you can study spaces by instead studying its ring of continuous functions (see this Wikipedia article for precise details). Gelfand’s result puts the commutative in commutative geometry. For group actions that have badly behaved quotients, Connes introduced a noncommutative ring that functions as the analogue of the quotient space.

I found two interesting online resources on the Foundations of Mathematics:

• What is Mathematics: Gödel’s Theorem and Around by Karlis Podnieks
• Foundations of Mathematics by Alexander Sakharov