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 PoincarÃ© 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?