| February 26, 2001 |
Abstract: 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. Poincare, 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 (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since. (This abstract is taken from the CMI website, see http://www.claymath.org). |
| March 7, 2001 |
Abstract: Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern, however the German mathematician G.F.B. Riemann (1826 - 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function z(s) called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation z(s) = 0 lie on a straight line. This has been checked for the first 1,500,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers. (This abstract is taken from the CMI website, see http://www.claymath.org). |
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| March 15, 2001 |
Abstract: Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations. (This abstract is taken from the CMI website, see http://www.claymath.org). |
| April 5, 2001 |
Abstract: In the twentieth century mathematicians discovered powerful ways to investigate the shapes of complicated objects. The basic idea is to ask to what extent we can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension. This technique turned out to be so useful that it got generalized in many different ways, eventually leading to powerful tools that enabled mathematicians to make great progress in cataloging the variety of objects they encountered in their investigations. Unfortunately, the geometric origins of the procedure became obscured in this generalization. In some sense it was necessary to add pieces that did not have any geometric interpretation. The Hodge conjecture asserts that for particularly nice types of spaces called projective algebraic varieties , the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles . (This abstract is taken from the CMI website, see http://www.claymath.org). |
| April 23, 2001 |
Abstract: Mathematicians have always been fascinated by the problem of describing all solutions in whole numbers x,y,z to algebraic equations like x2 + y2 = z2. Euclid gave the complete solution for that equation, but for more complicated equations this becomes extremely difficult. Indeed, in 1970 Yu. V. Matiyasevich showed that Hilbert's tenth problem is unsolvable, i.e., there is no general method for determining when such equations have a solution in whole numbers. But in special cases one can hope to say something. When the solutions are the points of an abelian variety, the Birch and Swinnerton-Dyer conjecture asserts that the size of the group of rational points is related to the behavior of an associated zeta function z(s) near the point s=1. In particular this amazing conjecture asserts that if z(1) is equal to 0, then there are an infinite number of rational points (solutions), and conversely, if z(1) is not equal to 0, then there is only a finite number of such points. (This abstract is taken from the CMI website, see http://www.claymath.org). |
| June 7, 2001 |
Abstract: The laws of quantum physics stand to the world of elementary particles in the way that Newton's laws of classical mechanics stand to the macroscopic world. Almost half a century ago, Yang and Mills discovered that quantum physics reveals a remarkable relationship between the physics of elementary particles and the mathematics of geometric objects. Predictions based on the Yang-Mills equations have been verified in high energy experiments performed at laboratories all over the world: Brookhaven, Stanford, CERN, and Tskuba. Nonetheless, there are no known solutions to their equations which both describe massive particles and are mathematically rigorous. In particular, the mass gap hypothesis, which most physicists take for granted and use in their explanation for the invisibility of "quarks", has never received a mathematically satisfactory justification. Progress on this problem will require the introduction of fundamental new ideas both in physics and in mathematics. (This abstract is taken from the CMI website, see http://www.claymath.org). |
| TBA |
Abstract: It is Saturday evening and you arrive at a big party. Feeling shy, you wonder whether you already know anyone in the room. Your host proposes that you must certainly know Rose, the lady in the corner next to the dessert tray. In a fraction of a second you are able to cast a glance and verify that your host is correct. However, in the absence of such a suggestion, you are obliged to make a tour of the whole room, checking out each person one by one, to see if there is anyone you recognize. This is an example of the general phenomenon that generating a solution to a problem often takes far longer than verifying that a given solution is correct. Similarly, if someone tells you that the number 13,717,421 can be written as the product of two smaller numbers, you might not know whether to believe him, but if he tells you that it can be factored as 3607 times 3803 then you can easily check that it is true using a hand calculator. One of the outstanding problems in logic and computer science is determining whether questions exist whose answer can be quickly checked (for example by computer), but which require a much longer time to solve from scratch (without knowing the answer). There certainly seem to be many such questions. But so far no one has proved that any of them really does require a long time to solve; it may be that we simply have not yet discovered how to solve them quickly. Stephen Cook formulated the P versus NP problem in 1971. (This abstract is taken from the CMI website, see http://www.claymath.org). |