Voevodsky also solved a major mathematical problem, called the Milnor conjecture. Milnor, one of the best mathematicians of the past half century and a close friend of John Nash, the mathematician portrayed in the movie "A Beautiful Mind," believed there was an equivalence between different ways of describing the properties of different kinds of surfaces. (This is a vast oversimplification, but I'm worried again that I'm losing you.) Voevodsky created new mathematical tools that, in 1996, enabled him to solve the problem.

Voevodsky is also a Clay Institute Prize fellow, and the institute sponsors his visits to Russia to lecture and inspire the current generation of young students there. Russia has a proud tradition of mathematicians and mathematical physicists (built in part, it has been speculated, because the country was unable to mount major efforts in the experimental sciences), which like other areas has suffered with the collapse of the Soviet Union.

How do mathematicians like Voevodsky work? This is, in the end, difficult to say. Hardy spent most days at the side of a cricket field, drinking tea. The great Grothendieck, who as much as anyone else is responsible for a vision of the unity of all mathematics, spent 18 hours a day creating mathematics that has astonished and inspired mathematicians ever since. (Grothendieck won his own Fields Medal in 1966, Milnor in 1962.) "There was far more imagination in the head of Archimedes than in that of Homer," Voltaire said, which English majors might doubt. But they would be wrong.

Your tax dollars pay for about $300 million dollars of mathematical research each year, and we are finally going to get to something you can understand.

As you read this article, you trust that the words your computer retrieved from Salon's Web servers are faithful to their original. But how do you know?

You know because mathematicians and engineers have thoughtfully included error-correcting codes in the computers that talk to one another across the Internet, because sometimes bits get scrambled, dropped, or mutilated. Madhu Sudan, winner of the 2002 Nevanlinna Prize, explained it to his daughter as follows: "When my 3-year-old daughter, Roshni, asked me what I do, I told her I correct errors. She asked me what is an error and what does it mean to correct them. So I wrote 'Rothni' on a piece of paper and asked her to circle any mistakes (without explaining what I intended to write). She circled the 't.' I told her that's what I do for a living. She understood what I did, but not why it was a big deal."

It's a big deal because Sudan has shown that certain codes can correct many more errors than was previously thought possible.

Sudan is a theoretical computer scientist. He does not, he said, use a computer in his work. He was recognized for his breakthroughs in error-correcting codes, probabilistically checkable proofs, and the non-approximability of optimization problems.

Given a proposed proof of a mathematical statement -- say, the statement that there is an infinite number of prime numbers (numbers, such as 7 or 17, evenly divisible only by 1 and themselves) -- the theory of probabilistically checkable proofs recasts the proof so that its fundamental logic is encoded as a sequence of bits that can be stored in a computer. Checking only some of these bits, Sudan and others have shown, can determine with high probability whether the proof is correct. Amazingly, the number of bits one must examine can be made extremely small.

How small? Let's just leave it as "small," because this article must soon come to an end, and because ... well, you know.

Consider two sets: all towns in your state, and all states whose names end in the letter "A." Given a finite collection of finite sets such as this, what is the largest size of a subcollection such that every two sets in the subcollection have no overlap? I forgot to ask Sudan about this particular problem, but he probably doesn't know anyway -- hey, it's a tough problem. You might propose a solution, which could be easily checked, but in general there may be no known algorithm that will easily produce a solution from scratch.

What Sudan and others showed is that, for many such problems, approximating an optimal solution is just as hard as finding an optimal solution. Now, this could obviously be useful to scientists and engineers. It also has implications for a fundamental mathematical problem called P=NP (if you solve it, the Clay Institute will give you a million dollars).

Why, in the end, does all of this mathematics matter? Why have Lafforgue, Voevodsky, and Sudan been culled from the set of all mathematicians for the highest honors?

Yes, mathematics is wonderfully useful for calculating the properties of superstrings and the path of the next comet that will collide with earth. Yada yada.

But, really, does a poem matter only because it can be read at a memorial? Isn't a busker's real worth the gleam in his eye when he performs? Is whatever art you may have created more important than the way you felt when you created it, or the way others felt when it was received?

"The case for my life, then," Hardy wrote in his "Apology," "or for that of any one else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any other artists, great or small, who have left some kind of memorial behind them."

This story has been corrected.

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