Prize for Resolution of the Poincare Conjecture Awarded to Dr. Grigoriy Perelman

Date: 2015-10-07; view: 349.

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The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincare conjecture. The citation for the award reads:

The Clay Mathematics Institute awards the Millennium Prize for resolution of the Poincare conjecture to Grigoriy Perelman.

The Poincare conjecture is one of the seven Millennium Prize Problems established by CMI in 2000. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude.

Formulated in 1904 by the French mathematician Henri Poincare, the conjecture is fundamental to achieving an understanding of three-dimensional shapes. The simplest of these shapes is the three-dimensional sphere. It is contained in four-dimensional space, and is defined as the set of points at a fixed distance from a given point, just as the two-dimensional sphere (skin of an orange or surface of the earth) is defined as the set of points in three-dimensional space at a fixed distance from a given point (the center).

We cannot directly visualize objects in n-dimensional space. The goal was to recognize all three-spheres even though they may be highly distorted. Poincare found the right test. In this work he was led to topology, a still new kind of mathematics related to geometry, and to the study of shapes of all dimensions.

The simplest shape was the circle, or distorted versions of it such as the ellipse. Both the circle and the two-sphere can be described in words or in equations as the set of points at a fixed distance from a given point (the center). Thus it makes sense to talk about the three-sphere, the four-sphere, etc. These shapes are hard to visualize, since they naturally are contained in four-dimensional space, five-dimensional space, and so on, whereas we live in three-dimensional space. Nonetheless, with mathematical training, shapes in higher-dimensional spaces can be studied just as well as shapes in dimensions two and three.

In topology, two shapes are considered the same if the points of one correspond to the points of another in a continuous way. Thus the circle, the ellipse, and the wild piece of string are considered the same. This is much like what happens in the geometry of Euclid. Suppose that one shape can be moved, without changing lengths or angles, onto another shape. Then the two shapes are considered the same A round, perfect two-sphere, like the surface of a ping-pong ball, is topologically the same as the surface of an egg. There remained the original conjecture of Poincare in dimension three. It seemed to be the most difficult of all, as the continuing series of failed efforts, both to prove and to disprove it. There came three developments that would play crucial roles in Perelman's solutions of the conjecture.

Geometrization. The first of these developments was William Thurston's geometrization conjecture. It laid out a program for understanding all three-dimensional shapes in a coherent way, much as had been done for two-dimensional shapes in the latter half of the nineteenth century. According to Thurston, three-dimensional shapes could be broken down into pieces governed by one of eight geometries, somewhat as a molecule can be broken into its constituent, much simpler atoms. This is the origin of the name, «geometrization conjecture».

A remarkable feature of the geometrization conjecture was that it implied the Poincare conjecture as a special case. Such a bold assertion was accordingly thought to be far, far out of reach – perhaps a subject of research for the twenty-second century. Nonetheless, Thurston was able to prove the geometrization conjecture for a wide class of shapes that have a sufficient degree of complexity. While these methods did not apply to the three-sphere, Thurston's work shed new light on the central role of Poincare's conjecture and placed it in a far broader mathematical context.

Limits of spaces.The second current of ideas did not appear to have a connection with the Poincare conjecture until much later. While technical in nature, the work, in which the names of Cheeger and Perelman figure prominently, has to do with how one can take limits of geometric shapes, just as we learned to take limits in beginning calculus class. Think of Zeno and his paradox: you walk half the distance from where you are standing to the wall of your living room. Then you walk half the remaining distance. And so on. With each step you get closer to the wall. The wall is your "limiting position," but you never reach it in a finite number of steps. Now imagine a shape changing with time. With each «step» it changes shape, but can nonetheless be a «nice» shape at each step-smooth, as the mathematicians say. For the limiting shape the situation is different. It may be nice and smooth, or it may have special points that are different from all the others, that is, singular points, or «singularities». Imagine a Y-shaped piece of tubing that is collapsing: as time increases, the diameter of the tube gets smaller and smaller. Imagine further that one second after the tube begins its collapse, the diameter has gone to zero. Now the shape is different: it is a Y shape of infinitely thin wire. The point where the arms of the Y meet is different from all the others. It is the singular point of this shape. The kinds of shapes that can occur as limits are called Aleksandrov spaces, named after the Russian mathematician A.D. Aleksandrov who initiated and developed their theory.

Differential equations. The third development concerns differential equations. These equations involve rates of change in the unknown quantities of the equation, e.g., the rate of change of the position of an apple as it falls from a tree towards the earth's center. Differential equations are expressed in the language of calculus, which Isaac Newton invented in the 1680s in order to explain how material bodies (apples, the moon, and so on) move under the influence of an external force. Nowadays physicists use differential equations to study a great range of phenomena: the motion of galaxies and the stars within them, the flow of air and water, the propagation of sound and light, the conduction of heat, and even the creation, interaction, and annihilation of elementary particles such as electrons, protons, and quarks.

Conduction of heat and change of temperature play a special role. This kind of physics was first treated mathematically by Joseph Fourier in his 1822 book, Theorie Analytique de la Chaleur. The differential equation that governs change of temperature is called the heat equation. It has the remarkable property that as time increases, irregularities in the distribution of temperature decrease.

Differential equations apply to geometric and topological problems as well as to physical ones. But one studies not the rate at which temperature changes, but rather the rate of change in some geometric quantity as it relates to other quantities such as curvature. A piece of paper lying on the table has curvature zero. A sphere has positive curvature. The curvature is a large number for a small sphere, but is a small number for a large sphere such as the surface of the earth. Indeed, the curvature of the earth is so small that its surface has sometimes mistakenly been thought to be flat. For an example of negative curvature, think of a point on the bell of a trumpet.

An early landmark in the application of differential equations to geometric problems was the 1963 paper of J. Eells and J. Sampson. The authors introduced the «harmonic map equation», a kind of nonlinear version of Fourier's heat equation. It proved to be a powerful tool for the solution of geometric and topological problems. There are now many important nonlinear heat equations – the equations for mean curvature flow, scalar curvature flow, and Ricci flow.

Also notable is the Yang-Mills equation, which came into mathematics from the physics of quantum fields. In 1983 this equation was used to establish very strong restrictions on the topology of four-dimensional shapes on which it was possible to do calculus. These results helped renew hopes of obtaining other strong geometric results from analytic arguments – that is, from calculus and differential equations. Optimism for such applications had been tempered to some extent by the examples of Rene Thorn and Milnor.

Perelman's proof of the Poincare and geometrization conjectures is a major mathematical advance. His ideas and methods have already found new applications in analysis and geometry; surely the future will bring many more.