I realize that 99% of mathematicians are in the “the new decade doesn’t begin until 2011″ camp, but I started wondering, how does mathematics look different now than it did in 2000? The big news of the decade was the solution of the Poincaré conjecture and more generally the geometrization conjecture. At the time, I remember hearing it widely predicted that this spelled doom for the topic of 3-manifolds. Is that what really happened?

Also, while progress in certain areas, such as algebraic geometry, algebraic topology, and number theory are high profile, what’s happened in the rest of mathematics? Graph theory saw the proof of the graph minor theorem (which I remember being earlier, but Wikipedia claims was only completed in 2004), but I don’t know what else happened in the area. Were there any major new breakthroughs, or changes in perspective in group theory? Logic? Universal algebra? Game theory?

In a related note, the proof of a conjecture known as the Fundamental Lemma made Time magazine’s list of the top scientific discoveries of 2009.

Dan Piponi of Neighborhood of Infinity has written a paper about how he has successfully used automatic differentiation in the movie industry. Unfortunately, in the course of the article he gives the best argument against the use of automatic differentiation I’ve heard:

In this paper we will present one approach to automatic
differentiation and describe one application that was used with considerable success during the post-production of Matrix Reloaded and Revolutions.

I rest my case.

Two series of monographs on logic are now available (for free) on Project Euclid:

• Perspectives on Logic.
• Lecture Notes on Logic.

I think several of the monographs are well-known in their areas, such as Model Theory of Fields.

CMU students conduct a mock protest at the G20. Slogans include “Safer Data Mining” and “MapReduce, MapReuse, MapRecycle: Green Data Processing”.

The Bohr-Einstein debates, now in convenient puppet form.