The 2008 Abel Prize has been announced. This year’s winners are John Thompson and Jacques Tits.

Thompson is most famous for his work on the Feit-Thompson theorem, that every group of odd order is solvable. Solvable groups resemble upper triangular matrices: a solvable group is constructed in layers out of abelian groups.

Tits invented the notion of BN pairs and buildings. The opposite of a solvable group is a simple group, which cannot be split up into layers. Simple groups tend to resemble the set of all invertible n-by-n matrices over a field (which itself is not simple, but is pretty close to it). Tits identified the key property that makes the resemblence work: the existence of special subgroups B and N. For the group of invertible matrices, B is the set of upper-triangular matrices, while N is the set of permutation matrices. Buildings are a geometric explanation of BN pairs.

What makes a well-written proof? Who writes proofs well? Discussions of mathematical exposition usually revolve around larger-scale questions, such as how to organize the material, what kinds of examples to use, or how much background is necessary; by and large, around the question of what to put between the proofs. There is the art of choosing a proof, which is a subjective measure of taste. But once you’ve chosen the proof, what’s the best way to lay it out?

The style of my own proofs tends towards alternating sentences that begin with “Let” and sentences that begin with “Thus”.

A reader sent me two news articles (here and here) announcing a generalization of the Schwarz-Christoffel mapping in complex analysis. The paper itself is not freely available, but I found this summary from SIAM news that fills out many of the details.

The Schwartz-Christoffel mapping an explicit mapping from the inside of a polygon to the unit disk that is conformal: it preserves angles (it does usually preserve straight lines). The recent work extends this to give conformal mappings from polygonal regions with polygonal holes to circular regions with circular holes. It was known before this that you couldn’t necessarily map any polygon region with holes to any circular region with holes while preserving angles. The two regions must share the same moduli, which are a sets of numbers you can associate with a region. (These moduli are related to the moduli that arise in the theory of Riemann surfaces.)

The breakthrough is not showing that a conformal map exists when the moduli agree, but giving an explicit means of calculating it. The result is not as explicit as the original Schwartz-Christoffel result, but can be calculated numerically.