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.

Bjorn’s Maths Blog has collected various mathematical methods for catching a lion in the Saharan desert.

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