How Terence Tao's quest for elegance earned him a Fields Medal and a MacArthur Fellowship

Terence Tao   Courtesy of the International Congress of Mathematicians Press Office

If I told you this article was about a young superstar whose work involves needles, honeycombs, puzzles, and progressions, you might never guess that my subject, Terence Tao, is a mathematician—and not just any mathematician, but one of the biggest names of his generation. In July, at the International Congress of Mathematicians in Madrid, he was awarded a Fields Medal—the “Nobel Prize” of mathematics. Just this week he added a MacArthur Foundation “genius grant” to his trophy case.

I met Terry in 1987, in Havana, when we were high school students competing in the International Math Olympiad (Terry for Australia, me for the United States.) Terry stood out even then—mainly for being a 12-year-old among the mostly late-adolescent contestants. But he wasn’t overmatched, either mathematically or socially. He had a sprightly, engaging demeanor and was keen to leap into conversation about whatever mathematical topic might present itself. In other words, he was a lot like he is today. You might think a former prodigy who got his doctorate from Princeton at 20 would, like many leading mathematicians, choose only to express himself in highly technical monographs. But Terry, now 29, has expended a great deal of energy trying to make the main ideas of his work accessible to students and researchers in other areas. His Web page features dozens of brief write-ups, labeled “short stories,” that explicate a single theorem or idea in a small space, clearing away the technical underbrush to expose the idea beneath. (Terry warned me not to make too much of the literary metaphor—his explanation is that he didn’t know the more common term “exposé” when he first set up the site.) He has also written a book for high school students, Solving Mathematical Problems: A Personal Perspective, that provides insight into problems like the ones we competed to solve in the Math Olympiad—problems that are simple to state but conceal a deep and beautiful mathematical framework.

Perhaps it’s no coincidence that the research problems Terry made his name with have something of the same nature. His most recent spectacular result, proved in collaboration with Ben Green of the University of Bristol, concerns “long arithmetic progressions in prime numbers.” Exactly how prime numbers are distributed among the whole numbers has been a central mystery of number theory for a few thousand years, starting from Euclid’s proof circa 300 BC that there are infinitely many primes. One must then ask (and making this vague question precise is half the battle): Are the prime numbers sprinkled randomly throughout the whole numbers? Or are there hidden irregularities and structures that we can perceive if we look at numbers from the right perspective?

What’s remarkable about Terry’s contribution to this problem is that he’s not a number theorist at all, but a harmonic analyst. Practitioners of this mathematical specialty study functions as they break up into simpler ones—for example, the way a complex waveform (like that produced by a complicated chord) can be deconstructed into simpler waves (the single notes). Harmonic analysis is also good at teasing out a very slight pattern in an otherwise random data set—like stripping the ambient noise from a recording and leaving the music, even if the music is much quieter than the noise. The prime numbers do obey some simple patterns—for instance, all primes but 2 are odd numbers. The great achievement of the Green-Tao theorem is its use of subtle and novel methods of harmonic analysis (very much indebted to the work of 2002 Fields Medalist Timothy Gowers) to show that these simple patterns are essentially the only structure the primes possess. Beyond these basic structures, the primes look random—they are, in a sense, all noise and no music.

If this idea seems depressing, it’s only because I’ve given no indication of the beauty of the proof, or the satisfaction of knowing so much more than previously about the way the long-mysterious prime numbers really behave. Music in the primes would be great, but understanding the truth is better. Terry has posted on his website a line by Tom Robbins that captures this sentiment very well: “The scientist keeps the romantic honest, and the romantic keeps the scientist human.”

Originally published September 21, 2006

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