Tuesday, August 08, 2006

Unsolved problems in mathematics

I just noticed this page in Eric Weisstein's excellent Mathworld site: Unsolved Problems. It's a nice list of 16 problems in mathematics -- some of which date back two hundred years or more -- that have defied the efforts of many mathematicians to find a solution. Everyone is familiar with the feeling, of course.

Curiously, however, most of the problems concern number theory, in the general sense that they deal primarily with questions involving integers. More precisely, 11 of the 16, by my count. No doubt this is because such problems can be explained to anyone who knows what an integer is, without reference to any "higher" mathematics.

I won't discuss them here, since the Mathworld site itself is an excellent encyclopedic reference to just about any topic in mathematics you might care to name. For that reason alone it's worth perusing the site if you aren't already familiar with it. Each problem, with related concepts, has its own page of explanation on the site.

However, there's a hell of a lot of mathematics out there that's not concerned primarily with integers, and a correspondingly large number of unsolved problems. Does the site have more to say about such? Yes, it does. A little poking around reveals an index page: Mathematical Problems, and beneath that five lower level index pages that deal either with unsolved problems, noteworthy unsolved problems of the past that have subsequently been solved, or important conjectures that have been found to be mistaken.

One of the lower level indexes, Unsolved Problems, lists no fewer than 223 individual pages (as of today) describing a great variety of unsolved problems. Aha. Now there's something to sink one's teeth into.

Another lower level index lists 17 "Prize Problems" -- problems considered important enough by someone to put up a substantial sum of money to be awarded to the first person who provides a valid solution. If you'd like to try your math skills, 16 of these are (as far as I'm aware) still unsolved. The 17th is Fermat's Last Theorem, which, as you know, was resolved in 1993-95 (primarily) by Andrew Wiles. I wrote quite a bit about this here.

Yet another index page covers Problem Collections -- lists of diverse unsolved problems compiled by various mathematicians. Oddly, one of the most well-publicized recent lists of this kind -- the Millennium Problems of the Clay Mathematics Institute -- is not included. This is especially odd because the Instutite offers a prize of US $1,000,000 for a solution (positive or negative) to each of its seven problems. Five of these problems are listed among Mathworld's Prize Problems -- but two (Navier-Stokes equations and Yang-Mills theory) are not.

There's one Millennium problem which deserves special comment here -- the Poincaré Conjecture. I won't even attempt to describe it in this post, though it's not too hard to explain what it's about, but less easy to explain why it's so important. The reason it's worth special comment here is that it has (apparently) just recently been resolved after kicking around for about 100 years since Henri Poincaré called attention to it. See this post from Peter Woit's blog for some of the latest news. See here and here for technical background.

What's going on with the Poincaré Conjecture definitely deserves a lot more discussion than a few sentences in this post. Looks like I'll have to make a project out of that -- as soon as I figure out what actually is going on...


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