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Fields Medals at Madrid

Grigori Perelman

Perelman was born in the then Soviet Union in 1966. He has worked in the USA and at the Steklov Institute in St Petersburg, but currently holds no specific academic position.

The Fields medal citation for Perelman centres on his work on the Ricci flow, a systematic technique for deforming a manifold—the multidimensional generalisation of a surface—in a way that tends to simplify its geometry. The most important potential consequences of his work are proofs of two fundamental conjectures in topology: The Geometrization Conjecture of William Thurston and a crucial special case, the Poincaré Conjecture.

Special circumstances surrounding these applications mean that Perelman’s work on them has not yet been published in a refereed journal, so its status remains uncertain. A consensus appears to be emerging that Perelman has proved both conjectures, but his methods involve several difficult areas of mathematics and his outlines of the proofs are complicated and condensed. Perelman’s increasingly reclusive nature has added an unusual human dimension to the mathematical story. It is perhaps a pity that this story reinforces the stereotype of the eccentric mathematician.

Both conjectures are about the topology of three-dimensional manifolds, and they attempt to extend to three dimensions methods and theorems that have been successfully applied to two-dimensional manifolds—surfaces. More than a century ago mathematicians succeeded in listing all possible topological types of surfaces, on the assumption that there is no boundary, the surface is orientable (two-sided), and the surface is of finite extent. (The non-orientable case, which involves surfaces such as the Klein bottle, with only one side, can also be solved.) Every surface of this kind is topologically equivalent to a sphere, a torus, a torus with two holes, a torus with three holes, and so on.

How could a creature living on such a surface, ignorant of any surrounding space, work out which surface it inhabits? That is, how can we characterise such surfaces intrinsically? By 1900 it was understood that the way to answer such questions is to consider closed loops in the surface, and how these loops can be deformed. For example, on a sphere (by which we mean just the surface, not the solid interior) any closed loop can be continuously deformed to a point—‘shrunk’. For example, the circle running round the equator can be gradually moved towards the north pole, becoming smaller and smaller until it coincides with the north pole itself. In contrast, every surface that is not equivalent to a sphere contains loops that cannot be deformed to points. Such loops ‘pass through a hole’ and the hole prevents them being shrunk. So the sphere can be characterised as the only surface in which any closed loop can be shrunk to a point.

In 1904 Henri Poincaré, one of the all-time greats in mathematics, was trying to understand three-dimensional manifolds, and for a time he assumed that the characterisation in terms of shrinking loops generalised to that case. Later he realised that one plausible version of this statement is actually wrong, while another closely related formulation seemed difficult to prove but might well be true. He posed a question, later reinterpreted as the ‘Poincaré conjecture’—if a three-dimensional manifold (without boundary, of finite extent, and so on) had the property that any loop in it can be shrunk to a point, that manifold must be topologically equivalent to the 3-sphere (a natural three-dimensional analogue of a sphere). Subsequent attempts to prove or disprove this conjecture met with zero success, but versions in dimensions 4 and higher were all proved to be true.

The Geometrization Conjecture goes further, and applies to all three-manifolds, not just those in which every loop can be shrunk. Its starting-point is an interpretation of the classification of surfaces in terms of non-Euclidean geometry. The torus can be obtained by taking a square in the Euclidean plane, and identifying opposite edges. As such, it is flat—has zero curvature. The sphere has constant positive curvature. A torus with two or more holes can be represented as a surface of constant negative curvature. So the topology of surfaces can be reinterpreted in terms of three types of geometry, one Euclidean and two non-Euclidean. Namely, Euclidean geometry itself, elliptic geometry (positive curvature), and hyperbolic geometry (negative curvature).

Might something similar hold in three dimensions? In the 1980s William Thurston pointed out some complications: there are eight types of geometry to consider, not three. And it is no longer possible to use a single geometry for a given manifold: instead, the manifold must be cut into several pieces, using one geometry for each. He formulated his Geometrization Conjecture: there is always a systematic way to cut up a three-dimensional manifold into pieces, each corresponding to one of the eight geometries. The Poincaré conjecture would be an immediate consequence, because the condition that all loops shrink rules out seven geometries, leaving just the geometry of constant positive curvature—that of the 3-sphere.

In 1982 Richard Hamilton introduced a new technique into the area, based on the mathematical ideas used by Albert Einstein in General relativity. According to Einstein, space-time can be considered as curved, and the curvature describes the force of gravity. Curvature is measured by the so-called curvature tensor, and this has a simpler relative known as the Ricci tensor after its inventor Gregorio Ricci-Curbastro.

Changes in the geometry of the universe over time are governed by the Einstein Equations, which say that the stress tensor is proportional to the curvature. In effect, the gravitational bending of the universe tries to smooth itself out as time passes, and the Einstein Equations quantify that idea. The same game can be played using the Ricci version of curvature, and it leads to the same kind of behaviour: a surface that obeys the equations for the ‘Ricci flow’ will naturally tend to simplify its own geometry by redistributing its curvature more equitably. Hamilton showed that the two-dimensional Poincaré conjecture can be proved using the Ricci flow—basically, a surface in which all loops shrink simplifies itself so much as it follows the Ricci flow that it ends up as a perfect sphere. Hamilton also suggested generalising this approach to three dimensions, and made some progress along those lines, but hit some difficult obstacles.

In 2002 Perelman caused a sensation by placing several papers on the arXiv, a website for physics and mathematics research that lets researchers provide public access to unrefereed, often ongoing, work. The aim of the website is to avoid long delays that occur while papers are being refereed for official publication, and all users are made aware that the material has not (yet) been subjected to expert scrutiny—other than by readers of the arXiv. Previously, this role had been played by informal ‘preprints’. These papers ostensibly were about various properties of the Ricci flow, but it became clear that if the work was correct, it implied the Geometrization Conjecture, hence that of Poincaré. The basic idea is the one suggested by Hamilton. Start with an arbitrary three-dimensional manifold, equip it with a notion of distance so that the Ricci flow makes sense, and let the manifold follow the flow, simplifying itself. The main complication is that ‘singualrities’ can develop, where the manifold pinches together and ceases to be smooth. The new idea is to cut it apart near such a singularity, cap off the resulting holes, and let the flow continue. If the manifold manages to simplify itself completely after only finitely many singularities have arisen, each piece will support just one of the eight geometries, and reversing the cutting operations (‘surgery’) tells us how to glue those pieces back together to reconstruct the manifold.

The Poincaré Conjecture is famous for another reason: it is one of the eight Millennium Mathematics Problems selected by the Clay Institute, and as such its solution—suitably verified—attracts a million-dollar prize. However, Perelman had his own reasons not to want the prize—indeed, any reward save the solution itself—and therefore had no strong reason to expand his often-cryptic papers on the arXiv into something suited for publication. Experts in the area therefore developed their own versions of his ideas, trying to fill in any apparent gaps in the logic, and generally tidy up the work into something acceptable as a genuine proof. Two such attempts have already been published, but neither has been fully accepted, for various reasons. Progress continues on another redevelopment of Perelman’s ideas, and a comprehensive and definitive version appears close to acceptance by the mathematical community. However, the ideas are so difficult, and the potential pitfalls so subtle, that the jury is currently still out. However, it is widely agreed that many parts of Perelman’s work are definitely correct, and those parts are so important in their own right that he fully deserves a Fields medal for them.

As it happens, he has no more interest in a Fields medal than he does in a Clay Prize. Initially, he lectured on his work and corresponded widely with interested mathematicians, explaining how to fill in missing detail. He has since become more reclusive, and appears somewhat disillusioned. This may be a natural reaction to publicity, and a sense of anticlimax—how can you top that kind of achievement? It may stem from disappointment that the solution has not yet been recognised as correct—but for a problem of this importance, recognition comes only after the severest scrutiny. Extraordinary claims require extraordinary evidence. Or it may be just a facet of Perelman’s complex personality. Not everyone craves worldly success.