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Too simply pure

Book Book Review, Title Perfect Rigor, Author Masha Gessen, Rating 3.5,

Perfect Rigor

Masha Gessen

Book Review

The Poincaré conjecture, one of the great unsolved topological problems, was finally proven at the end of the 20th century by Grigoriy Perelman, a Russian mathematician of genius. Topology might be described as distilled geometry. The historian Masha Gessen, who grew up herself in the Russian mathematical culture, invites us into the Aspergian milieu of world-class geometers to tease out the tale.

-CC-BY-SA 3.0, Salix alba at English Wikipedia

Fig 1: a loop around a sphere can be continually reduced to a singe point. Attrib: Salix alba at English Wikipedia, CC-BY-SA 3.0.


Henri Poincaré was a fin-de-siècle polymath, brilliant in both physics and mathematics. In physics, he came close to working out special relativity prior to Einstein’s seminal 1905 paper. He was also a leader in the foundation of topology. His Poincaré conjecture can be expressed succinctly: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. (note 1)

Succinctness is a hallmark of mathematical expression, but it is also obscure to those of us who have not been initiated. The Clay Mathematical Institute provides an more accessible introduction to the Poincaré conjecture: "If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is 'simply connected,' but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere."

While the author takes us on the journey of Perelman’s solution of the conjecture, we learn about the closed and edgy culture of elite mathematicians, those who see themselves as thinkers of pure thoughts, who see themselves as unironically direct and honest about all. During Perelman’s coming-of-age in the USSR, for a brief period the Soviet mathematical world became a safe harbor for those who needed a place to be eccentric. All about them the institutions of the Soviet Union systematically taught and enforced the conformity of its distorted version of communism, while the mathematical elite were trained in schools and math clubs that were as free as was possible of such constraints, a happy accident helped by the immense Cold War arms race, which required skilled mathematicians.

It is the mind of Perelman, a problem solver of the highest order, that Gessen seeks to place in its context, if not to explain. His world was an immense shelter, of which he seemed largely unaware, his gifts slightly larger than his vulnerabilities. Without the forbearance and protection of determined people, without the convergence of events, the gifted Perelman may not have sustained the effort required to complete the pursuit of the conjecture.

Perelman, who rivaled Candide in his unworldliness, swallowed whole the mathematics of entire fields, and in conquering the up-to-then unconquerable Poincaré conjecture, was swallowed in turn by his pure belief in meritocracy, unable to see that great things can be done in a world colored in the grey of human ambition. His own ambition was ferociously channeled into his mathematics, so much so that when he achieved the unachievable, he forsook the further pursuit of mathematics, at the same time refusing the ordinary rewards of position, money (one million dollars), and fame, turning down the Fields medal, the mathematical equivalent of the Nobel prize.

Gessen conjectured, in topological terms most ironical, that for Perelman, "like a rubber band slipping inexorably off a sphere, his world was about to shrink to a point" (page 175).




The idea of a homeomorphism in topology is roughly analogous to the idea of similarity in Euclidean geometry. To say that two triangles are similar is to say that they have the same geometric proportions: their angles are the same, and all corresponding sides are in proportion. One of the similar triangles can be transformed into the other by shrinking or stretching the sides simultaneously in the same proportion.
-GNU FDL 1.2, Nguyenthephuc

Fig 2: The small triangle is similar to the large triangle because the the large triangle can be created by proportionally doubling the length of each of the small triangle’s three sides. Attrib: Nguyenthephuc, GNU FDL 1.2.


Analogously in topology, to say that 2-manifolds, that is 2-dimensional surfaces, are homeomorphic is to say that both geometrical objects have the same topological properties, such that the shape of one surface can be transformed into the other by continuously stretching or bending it into the shape of the other surface.

-CC BY-NC-SA 2.0, jasohill cropped, pixabay

Fig 3: a dice is homeomorphic to a sphere because it's surface can be stretched continually to form a spherical surface. Attrib: jasohill cropped, pixabay, CC BY-NC-SA 2.0.


The two-dimensional equivalent to the conjecture is more easily visualized. In Figure 1 above, the loop around the 2-dimensional surface of the sphere, the 2-sphere, can be continually tightened until it is reduced to a point. In fact, any compact 2-dimensional surface without boundary, such as the dice in Figure 3 above, for which every loop around its surface can be continuously tightened to a point, is topologically the same, homeomorphic, as a 2-sphere. The Poincaré conjecture extends this to the 3-dimensional surface of a sphere, the 3-sphere, which is not easily visualized.

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