This week’s That’s Maths column in The Irish Times (TM061, or search for “thatsmaths” at irishtimes.com) is about the remarkable mathematician Grisha Perelman and his proof of a one-hundred year old conjecture.
During the twentieth century topology emerged as one of the pillars of mathematics, alongside algebra and analysis. Geometers consider lengths, angles and other details. Topologists are more interested in the overall, global aspects, properties that survive when a shape is distorted or stretched, as long as it is not torn.
Mathematicians talk of manifolds rather than shapes. Manifolds exist in all spatial dimensions: curves, surfaces, solids and so on. Topology is concerned with properties of manifolds that remain unchanged under continuous deformations. Two shapes, like a circle and square, that can be transformed into each other by deformation without cutting or tearing, are treated as topologically equivalent. They are said to be homeomorphic.
But if distortions are allowed, how are different manifolds distinguished? One way is to count the holes in a manifold. For a topologist, a sphere is the same as a cube: they are both 3-dimensional shapes without any holes. A doughnut has a hole, so it is different from a sphere. A tea-cup also has a hole through the handle, so it is similar (homeomorphic) to a doughnut. Spectacle frames have two holes, as have hand-cuffs; and so on.
Shapes with holes
Imagine stretching a rubber band around the Equator. If it is moved northwards, it will gradually shrink until, reaching the North Pole, it converges to a single point. A surface for which any loop can be shrunk to a point within the surface without tearing is called a simply connected surface. So the 2-sphere is simply connected. On the other hand, if a rubber band is looped around the hole in a doughnut, there is no way to shrink it to a point without breaking the rubber band (or the doughnut). So the surface of the doughnut is not simply connected.
Two-dimensional surfaces have been completely classified; this was one of the triumphs of nineteenth-century mathematics. The torus has a hole, the sphere has none. Generally, manifolds that are simply connected have no holes. The number n of holes is a topological characteristic of a manifold. For (oriented) surfaces, the genus (g = 2–2n) determines the surface up to homeomorphism.
The French mathematician Henri Poincaré knew that a 2-sphere is essentially characterized by the property of simple connectivity, or by having no holes. In 1904 he posed the corresponding question for the 3-sphere (the set of points in four dimensional space at unit distance from the origin).
Poincaré’s question became known as the Poincaré Conjecture:
Every simply connected closed 3-manifold is homeomorphic to the 3-sphere.
This question is fundamental to achieving an understanding of three-dimensional shapes (compact manifolds in three dimensions). Numerous claims of a proof were made but, until recently, all were shown to be flawed. Through the twentieth century, false proofs of the conjecture came and went; the problem seemed unassailable.
Poincaré’s conjecture was considered to be one of the most important outstanding problems in mathematics. The hypersphere or 3-sphere is a 3-D shape with no holes. The question that Poincaré posed is whether the 3-sphere is, topologically speaking, the only 3-dimensional shape with no holes. But he could not answer the question.
Surprisingly, the equivalent questions for dimensions 4 and higher were proved during the 20th century, but Poincaré’s original problem on the 3-sphere remained unsolved.
When mathematicians speak of a sphere, they mean the two-dimensional surface of a spherical ball, like the surface of the Earth, not the solid Earth itself. A sphere is the set of points that are all the same distance from a central point. A sphere has two dimensions: any position on the Earth’s surface can be pinned down by giving two numbers, the latitude and longitude. So, to be precise, we call it the 2-sphere. But the 2-sphere “lives” in three-dimensional space.
Now we move up a notch, and consider the 3-sphere or hypersphere, the surface of a 4-ball in 4-space. The 3-sphere is the set of points that are equidistant from a central point in 4-space. It is much more difficult to visualise than the 2-sphere. Just as we depict three-dimensional scenes on a two-dimensional canvas by using perspective, we can represent the 3-sphere (which lives in 4-space) in three dimensions. Locally, the 3-sphere looks like ordinary 3-space, but it is finite in size and yet lacks any boundary.
In 1982, the American mathematician Richard Hamilton came up with an ingenious plan of attack. He would treat curvature as something fluid, that could change over time. His idea was to use ideas from physics. Heat always flows from hotter to a colder places, smoothing out differences. Heat flow is described by a partial differential equation called the diffusion equation.
What characterizes a sphere is its constant positive curvature. More general manifolds have curvature that varies with position. What if we tried to diffuse the curvature of a manifold, in the same way that heat diffuses, causing it to become more uniform? With Hamilton’s approach, we smooth out curvature instead of heat. This is much more complicated, because, in three dimensions, curvature is not a single number like temperature, but a tensor with several components. Worse still, the equation that governs the smoothing is non-linear and much harder to solve than the heat equation.
Hamilton’s idea was to start with an arbitrary shape without holes, and simplify it by smoothing out the curvature, allowing it to flow around the shape. He was hoping that this would always lead to a 3-sphere, proving Poincaré’s conjecture. Sadly, there was a major hitch: the flow sometimes caused the curvature to grow without limit, forming a singularity that caused the solution to blow up.
Hamilton imposed a Riemannian metric on the manifold and then systematically changed the metric properties using a PDE so that it became simpler, equivalent or similar to a known manifold. The distortion is called a Ricci flow and the equation is
where gij is the metric tensor and Rij is the Ricci tensor, a contracted form of Riemann’s curvature tensor (these objects, while not known to most of us, are very familiar to students of general relativity). Ricci flow expands the negative-curvature parts of the manifold and contracts the positive-curvature parts. The difficulty with Hamilton’s strategy is that, during the process of distortion, singularities can develop and these can raise a barrier against further progress.
In 2002 the Russian mathematician Grigori (Grisha) Perelman posted the first of three preprints on arXiv, a scientific repository on the internet. Perelman used the strategy outlined by Richard Hamilton. He managed to overcome the difficulties posed by singularities by performing “surgery” on the shape, removing the singularities and allowing the flow to proceed. He found that in this way every shape without holes would eventually become a 3-sphere.
Perelman’s preprints left several gaps and it was not immediately clear whether or not he had actually proved the conjecture. But over the following years his work was intensively studied by several high-profile teams of mathematicians, who concluded that all the gaps were minor and could be filled in using techniques well-known to Perelman himself.
Of the fifty or so recipients of the Fields Medal – comparable in prestige to a Nobel Prize – about one third have been honoured for their work in topology. Three were awarded for work relating to the Poincaré conjecture.
In 2006, Perelman was offered the award of a Fields Medal for his work. But he refused the prize saying he was not interested in money or fame. He is clearly uncomfortable with being in the public eye and is quoted as saying “I don’t want to be on display like an animal in a zoo”.
CMI Millennium Prize
To celebrate mathematics in the new millennium, the Clay Mathematics Institute, a privately funded organisation, offered seven $1 million prizes for solutions to any of seven of the deepest and most difficult problems in mathematics. The prizes were announced at a meeting in Paris in May, 2000. The problems were chosen by a Scientific Advisory Board that consulted leading experts worldwide.
In March 2010 the Clay Mathematics Institute announced the award of the first Millennium Prize to Perelman for his proof of the Poincaré conjecture. This would have made him a millionaire overnight. But Perelman turned down the prize, saying that his contribution to proving the conjecture was no greater than Hamilton’s. Perelman now lives reclusively in St Petersburg and it is unclear whether he has abandoned mathematics or continues to do research. Let us hope the latter is the case, for he is fearsomely brilliant and such people are very rare. Moreover, six Millennium Prizes still remain to be solved.
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