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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 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 (the set of points in four dimensional space at unit distance from the origin). This question turned out be be extraordinarily difficult, and mathematicians have been struggling with it ever since.

...but if you can prove it, [or any of six other 'millenium prize problems'] the clay mathematics institute wants to line your pockets with $1M

...but if you can prove it, [or any of six other 'millenium prize problems'] the clay mathematics institute wants to line your pockets with $1M

Can I cut my teeth on the $32,000 and $64,000 questions, first? Can I use a lifeline?

posted by dhartung at 5:17 AM on May 25, 2000

posted by dhartung at 5:17 AM on May 25, 2000

Wow, the Riemann hypothesis. That's a good 'un. I had a professor in college who described it as the deepest unsolved problem in number theory.

A brief description: 19th-century complex analysis guy Riemann extended Euler's zeta function (1 + 1/2^n + 1/3^n + ...) to the complex plane (that is, the plane where the real numbers are one axis and the real numbers times the square root of negative one are the other). Unlike Euler's function, this new "Riemann zeta function" can have zeroes, as positive numbers raised to imaginary powers can be negative. Riemann noted that all non-trivial zeroes (there are an infinite number of "trivial" solutions that can be described quite simply) were symmetric around the line of points where the real component was 1/2 (the "critical line") and within a "critical strip" of distance 1/2 from this line. He then theorized that all non-trivial zeroes were actually on the critical line.

Proving the Riemann hypothesis is one of the Hilbert problems--the top unsolved questions of mathematics circa 1900. If the Riemann hypothesis is shown to be true (which most people believe it to be), we would know the precise formula that describes the frequency of prime number distribution, as opposed to now, when it allows for a possible error.

I'm completely incapable of giving a succinct summary of any of the other problems, though. This is what happens when you graduate and start working with computers instead of studying math.

posted by snarkout at 10:17 AM on May 25, 2000

A brief description: 19th-century complex analysis guy Riemann extended Euler's zeta function (1 + 1/2^n + 1/3^n + ...) to the complex plane (that is, the plane where the real numbers are one axis and the real numbers times the square root of negative one are the other). Unlike Euler's function, this new "Riemann zeta function" can have zeroes, as positive numbers raised to imaginary powers can be negative. Riemann noted that all non-trivial zeroes (there are an infinite number of "trivial" solutions that can be described quite simply) were symmetric around the line of points where the real component was 1/2 (the "critical line") and within a "critical strip" of distance 1/2 from this line. He then theorized that all non-trivial zeroes were actually on the critical line.

Proving the Riemann hypothesis is one of the Hilbert problems--the top unsolved questions of mathematics circa 1900. If the Riemann hypothesis is shown to be true (which most people believe it to be), we would know the precise formula that describes the frequency of prime number distribution, as opposed to now, when it allows for a possible error.

I'm completely incapable of giving a succinct summary of any of the other problems, though. This is what happens when you graduate and start working with computers instead of studying math.

posted by snarkout at 10:17 AM on May 25, 2000

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posted by wendell at 10:03 PM on May 24, 2000