As the end of the year approaches many publications are releasing their top 10 lists for the year and Science is no exception. Last year Science named evolution as its top breakthrough of the year, but was accused of pandering to the political/religious debates that were/are raging throughout the world, especially in the United States. This year, Science (open access) named a breakthrough that has no connections to politics or religion: the proof of the Poincaré Conjecture by Russian mathematician Grisha Perelman.
The Poincaré Conjecture was originally proposed by Henri Poincaré in 1904 and deals with the topology of everyday objects, namely what, in topological terms, defines a sphere. The Conjecture remained unsolved for almost 100 years, although not for lack of trying, and in the year 2000 the Clay Mathematics Institute (CMI) named the Poincaré Conjecture as one of its six Millennium Problems. These problems have solutions that have eluded mathematicians for years and carry a US $1,000,000 prize to anyone who solves them (either in a positive or negative manner). As stated in the CMI's official problem declaration, the Poincaré Conjecture asks
If a compact three-dimensional manifold M3 has the property that every simple closed curve within the manifold can be deformed continuously to a point, does it follow that M3 is homeomorphic to the sphere S3?
Clear, no? Well for those of us who do not hold advanced degrees in topology or geometry, it asks a fairly simple question, namely "Can an arbitrary closed surface be turned into a sphere by only stretching it appropriately?" This question has formed a cornerstone in the field of mathematics known as
topology, which studies the geometry of surfaces undergoing deformations. Poincaré suggested that ALL three-dimensional surfaces that had no holes could be turned into a three-dimensional sphere without needing to tear a section apart or stitch two sections together. He also suggested the converse to be true, that a surface with a hole could NEVER be turned into a sphere without tearing or sewing sections. A real world example from breakfast would be that a muffin can never be turned into a doughnut: the hole in the doughnut cannot be created in the muffin by simply stretching it—one would need to tear a hole in the muffin to make it ever resemble the doughnut.
Over the years, special solutions for specific dimensions were developed. For one and two dimensions, the proof was trivial; seven or more dimensions were handled by a proof from
Stephen Smale, developed in 1960. Smale then extended his proof to all dimensions greater than or equal to five, for which he was awarded the Fields Medal in 1966. More than 20 years later,
Micheal Friedman proved the Conjecture for the four dimensional case in 1980 and was awarded the Fields Medal in 1986. Poincaré had now been proven for ALL dimensions EXCEPT three, the original dimension for which Henri Poincaré first proposed the problem.
The final unproven dimension fell in a series of papers published to the web by Grisha Perelman, all available to the
curious for free from
arXiv.org. Dr. Perelman extended previous work done by Richard Hamilton, a mathematician who proposed that an arbitrarily lumpy space could "flow" towards a smooth space—imagine a Koosh ball morphing into a soccer ball—through equations akin to the heat equation, and named the process "Ricci flow." In Ricci flow, lumpy areas and areas that are highly curved tend to smooth themselves out until in entire surface has a constant curvature. In the absence of any major problems, this idea of Ricci flow could show how an arbitrary surface could be morphed by simple stretching into a smooth sphere, thus providing a method to prove or disprove Poincaré's Conjecture. However, problems arose; singularities, such as necks (thin areas between two larger areas, e.g. the bar in a dumbbell weight) would pinch and close themselves off, forming two separate objects with a uniform geometry, violating the rules of topological stretching. Perelman had to work for many years to find a way to overcome this problem. In November of 2002, he published his first paper on the subject with a new quantity added into the mix, what he called Ricci flow "entropy", borrowing from a term in statistical mechanics that tends to increase until a equilibrium is reached. This entropy idea proved that the problems in Hamilton's work—the singularities—could be overcome, yet Perelman still faced other road block to overcome before a full proof could be developed. In his subsequent articles, he showed that these problems areas would occur one at a time, as opposed to all at once. He then went on to show that it could be "pruned" through surgery before it would cause a problem with the Ricci Flow.
This series of papers was published in 2002 and 2003, yet it took the mathematics community at large three more years to accept this solution as a true proof to the Poincaré Conjecture. However, the story does not end there. In 2006 the International Mathematical Union announced that it had awarded the Fields Medal to Grisha Perelman, but he declined it. In a rare
interview with
The New Yorker, Dr. Perelman announced that he was retiring from mathematics, stating he was disheartened by what he viewed as ethical lapses by some of his colleagues. This
New Yorker article has caused quite a commotion in the mathematical world, with people claiming their words were distorted and threats of lawsuits traded. This has caused a black cloud to hang over what is the greatest mathematical breakthrough of the millennium (so far). Fortunately mathematical proofs are not affected by the feelings of those who create them. The proof of the Poincaré Conjecture remains atop the list of the greatest scientific breakthroughs of the year.
