There is a buzz of expectancy waiting for the announcement of this year’s Fields Medal for mathematics, always assuming that they award one this year. All the smart money, indeed all the money full stop, is on Grigory ‘Grisha’ Perelman for outlining a proof of the Poincare conjecture. There is plenty of chatter about Perelman’s supposed eccentricities, whether he will accept the million dollar prize from the Clay Mathematics Institute if his proof is correct and generally why the Poincare conjecture is important in the first place, but I want to highlight the way that Perelman chose to share his discovery.
In 2002 Perelman posted a paper on arXiv, “The Entropy Formula for the Ricci Flow and its Geometric Applications“, claiming to give “a sketch of an eclectic proof of this conjecture“. Two more papers followed in 2003: “Ricci Flow with Surgery on Three-Manifolds” and “Finite Extinction Time for the Solumtions to the Ricci Flow on Certain Three-Manifolds“. Since then other mathematicians have taken up the challenge of explicitly proving that Perelman’s approach provides a proof for Poincare, and indeed surpasses it by proving a more general conjecture of which Poincare’s is a special case.
This is a superb example of how peer review in the open works. Perelman put his work out in the public domain and the mathematical community has collaborated to bring out its full impact. Indeed it is highly debateable whether Perelman’s paper could have been published in a ‘conventional’ journal with a ‘conventional’ impact factor.
This isn’t unusual with papers published on arXiv, it is just a particularly high profile example. If Grisha Perelman is awarded the Fields on Tuesday it perhaps it should be viewed not just as the acknowledging a specific Russian genius but also as a victory for collaborative approaches to science.