Ricci Flow and the Poincaré Conjecture

The eld of Topology was born out of the realisation that in some fundamentalsense, a sphere and an ellipsoid resemble each other but di er from a torus { thesurface of a rubber tube (or a doughnut). A striking instance of this can be seenby imagining water owing smoothly on these. On the surface of a sphere or anellipsoid (or an egg), the water must (at any given instant of time) be stationarysomewhere. This is not so in the case of the torus.In topology, we regardthe sphere and the ellipsoid as having the same topologicaltype, which we make precise later. Topology is the study of properties that areshared by objects of the same topological type. These are generally the globalproperties. Understanding the di erent topological types of spaces, the so calledclassi cation problem, is thus a fundamental question in topology.In the case of surfaces (more precisely closed surfaces), there are two in nitesequences of topological types. The rst sequence, consisting of the so called ori-entable surfaces, consist of the sphere, the torus, the 2-holed torus, the 3-holedtorus and so on (see gure 1). One would like to have a similar classi cation inall dimensions. However, due to fundamental algorithmic issues, it is impossible tohave such a list in dimensions four and above.There is a simple way to characterise the sphere among surfaces. If we take anycurve on the sphere, we can shrink it to a point while remaining on the sphere. Aspacewith this propertyis called simply-connected. A torusis not simply-connectedas a curve that goes around the torus cannot be shrunk to a point while remainingon the torus. In fact, the sphere is the only simply-connected surface.In 1904, Poincare raised the question as to whether a similar characterisation ofthe (3-dimensional) sphere holds in dimension 3. That this is so has come to beknown as the Poincare conjecture. As topology exploded in the twentieth century,severalattempts weremade to provethis (and someto disproveit). However,at theturnofthemillennium thisremainedunsolved. Surprisingly,the higherdimensionalanalogue of this statement turned out to be easier and has been solved.In 2002-2003, three preprints ([8], [9] and [10]) rich in ideas but frugal with de-tails, were posted by the Russian mathematician Grisha Perelman, who had beenworking on this in in solitude for seven years at the Steklov Institute. These werebasedonthe Ricci ow, whichwasintroducedbyRichardHamiltonin 1982. Hamil-ton had developed the theory of Ricci ow through the 1980’s and 1990’s, provingmany important results and developing a programme[4] which, if completed, wouldlead to the Poincare conjecture and much more. Perelman introduced a series ofhighly original ideas and powerful techniques to complete Hamilton’s programme.It has taken two years for the mathematical community to assimilate Perelman’sideas and expand his preprints into complete proofs. Very recently, an article [1]and a book [7] containing complete and mostly self-contained proofs of the Poincare