Stubborn Theorems

Seven math mysteries

It looks like a typical residential home in Cambridge, but inside you're likely to find a bustling television crew chatting with a prominent mathematics scholar. The Clay Mathematics Institute (CMI), a two-year-old independent foundation that seeks "to increase and disseminate mathematical knowledge," has piqued media interest. That's not due to its novelty, though board members say it's the only institute of its kind. Rather, the attention stems from an intriguing contest: CMI has offered $1 million rewards for the solutions to seven classic mathematical conundrums. It's the largest prize ever offered in the field.

The challenge, issued last May, is the brainchild of CMI president Arthur Jaffe, who is also Clay professor of mathematics and theoretical science at Harvard. Jaffe joined CMI at its inception, and many Harvard students, alumni, and professors have followed him into its ranks. Onetime mathematics concentrator William Randolph Hearst III '71, for example, sits on the board of directors.

Jaffe devised the "Millennium Prize Problems" last year with an advisory board that includes Andrew Wiles, who in 1995 published a proof of Fermat's last theorem, a problem that had stumped his fellow mathematicians for 350 years. None of the seven questions on CMI's list has gone unsolved for that long, though scholars have worked on each for at least three decades. The oldest conundrum on the list is the Riemann hypothesis, first proposed in 1859, which asks whether prime numbers occur randomly or in a pattern. Other stumpers, like the Yang-Mills theory in particle physics, are in relatively new fields. "We're not saying that there aren't other interesting problems at the frontiers of fast-moving areas--there are," says Jaffe. "But we tried to get a certain set of classic, old mathematics problems. There's no question that these are important."

Nonmathematicians may not immediately grasp this import. Just stating the Yang-Mills problem takes 15 pages. But the jumble of cryptic symbols and equations contains levers that can move the real world. The Yang-Mills theory, considered as important to physics as Einstein's theory of relativity, explains how subatomic particles (such as quarks) interact. The unsolved problem is to determine whether there are actual solutions to the Yang-Mills equations, and if these solutions support physicists' assumptions. The Navier-Stokes equations, another CMI problem, relate to fluid mechanics, the science dealing with how water and air respond to pressure exerted on them. Forms of these equations help design airplanes and ships and predict weather.

This work may be esoteric, but, as Jaffe notes, "[Mathematics] is the basis of every science. It's the enabling science. It's very glamorous to talk about the Human Genome Project, but without mathematics, you couldn't have gotten the Human Genome Project. There's no practical outcome from [the Yang-Mills] problem that I can imagine today. But I'm very shortsighted. I'm looking at it in terms of what I now know. A person 10 years from now might look at it very differently."

Of course, it could take much longer than that for a mathematician to solve the first Millennium Prize Problem. Even then, he or she can claim the $1 million reward only after the solution has been published in a well-established journal and has stood up to peer scrutiny for two years. But financial incentive is secondary, Jaffe asserts. "If you solve one of these problems, you become immortal. Your name will be in mathematics books 500 years from now. That's much more important to most mathematicians than the money." Indeed, Jaffe's inspiration for the contest was a prizeless challenge issued a century ago by the German mathematician David Hilbert, who presented 23 mathematical problems (including the Riemann hypothesis) to his colleagues. The resulting work, says Jaffe, "determined the course of twentieth-century mathematics."

The Clay Mathematics Institute doesn't have such lofty goals for its Millennium Prize Problems. There is much else to focus on. CMI supports research by a steady stream of talented scholars from around the world. (Mathematician Dennis Gaitsgory, a junior fellow of the Society of Fellows, is one of several CMI Long-Term Prize Fellows.) The institute also runs summer schools, workshops, and conferences for everyone from tenured professors to high-school students.

"We're not trying to lift the whole population," says founder Landon T. Clay '50, a former Boston businessman who underwrites the institute. "What we want to do is support the exceptional students." The next Fermat or Riemann might depend on that.

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