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Wrap a rubber band around an apple and you can slide it off to a point without tearing anything. Try the same around a doughnut, through the hole, and you cannot. In 1904 Henri Poincaré asked whether a property like this is enough to recognise the simplest three-dimensional shape. The question stood for nearly a century, until Grigori Perelman posted three papers online in 2002 and 2003. This post covers who he is, what the conjecture says, how his proof worked in outline, the prizes he declined, and a topology warm-up you can do with pencil and paper.
Who is Grigori Perelman?
His MacTutor biography gives his birth in Leningrad, now St Petersburg, on 13 June 1966. His mother taught mathematics and his father gave him logic and mathematics problems to think about. He trained in Sergei Rukshin's mathematics club, attended the city's School 239, and at the 1982 International Mathematical Olympiad in Budapest won a gold medal with a perfect score.
He graduated from Leningrad State University in 1987 and defended his thesis on saddle surfaces in 1990. Postdoctoral posts in the United States followed, including a Miller Research Fellowship at Berkeley, and his 1994 proof of the soul conjecture of Cheeger and Gromoll made his name. He turned down professorships at leading universities and returned to the Steklov Institute in St Petersburg in 1995.
What does the Poincaré conjecture say?
The Clay Mathematics Institute's page on the conjecture uses the rubber band picture. A surface on which every loop can be shrunk to a point is called simply connected. Poincaré knew that the two-dimensional sphere is essentially characterised by this property, and in 1904 he asked the same question for the three-dimensional sphere.
The Clay press release of 18 March 2010 explains what that is. Just as the ordinary sphere is the set of points at a fixed distance from a centre in three-dimensional space, the three-sphere is the set of such points in four-dimensional space. Since nobody can picture it directly, Poincaré wanted a test you could carry out from inside the shape.
How did Perelman prove it?
According to the press release, a new line of attack opened around 1982 with Richard Hamilton's Ricci flow, an equation related to the one Fourier introduced for the flow of heat. It smooths out a shape's geometry over time, but progress stalled because singularities formed that nobody could control. Perelman understood how those singularities form, and introduced a quantity he called entropy that, like its namesake in thermodynamics, increases as time passes.
His three arXiv papers were The entropy formula for the Ricci flow and its geometric applications on 11 November 2002, Ricci flow with surgery on three-manifolds on 10 March 2003, and Finite extinction time for the solutions to the Ricci flow on certain three-manifolds on 17 July 2003. The Clay page adds that he also proved William Thurston's geometrization conjecture: every three-dimensional manifold is built from standard pieces, each with one of eight well-understood geometries.
Why did he decline the prizes?
The International Mathematical Union's 2006 Fields Medal page lists Perelman among the medallists and states that he declined to accept the award. MacTutor notes that he was the first person to do so. In 2010 the Clay Institute awarded him its Millennium Prize, and MacTutor reports that in July 2010 he refused the one million dollars, saying he considered Hamilton's contribution to the solution no less than his own.
Worked example: a number that ignores stretching
Topologists look for numbers that stay the same however a shape is stretched. One is V − E + F, where V is the number of vertices, E edges and F faces.
Cube. 8 − 12 + 6 = 2.
Tetrahedron. 4 − 6 + 4 = 2.
Square pyramid. 5 − 8 + 5 = 2.
Torus. Draw a 3 × 3 grid of squares and glue the top edge to the bottom and the left to the right, which makes a doughnut surface. After gluing there are 9 vertices, 18 edges and 9 faces, so 9 − 18 + 9 = 0.
Every surface shaped like a sphere gives 2; the torus gives 0. Like the rubber band test, the number tells the apple from the doughnut. This is not Perelman's method, which works in three dimensions with far deeper tools, but it shows the kind of question topology asks.
What do students often get wrong?
Picturing a solid ball. The three-sphere is the three-dimensional boundary of a four-dimensional ball, not a filled ball.
Forgetting the gluing. On the torus grid, edges that are glued together count once.
Thinking one paper did it. MacTutor notes that experts took time to check the proof and to see that it settled geometrization too.
Euler's formula is part of our post on Leonhard Euler, and Poincaré's other great problem is in our post on the three-body problem. If faces, edges and vertices blur together, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing.
What can a student take from Perelman's way of working?
Build on others and say so. His proof grew from Hamilton's Ricci flow, and he insisted Hamilton deserved equal credit.
Ask questions. MacTutor records how much he valued Hamilton's openness when he asked him about his work.
Say a lot in few words. The Clay release praises the elegant brevity with which he described his results.
Next step: compute V − E + F for a triangular prism and a hexagonal prism, and explain why you get the same answer.
