Friday, January 30, 2015

A Simple and Clear-Cut Constellation

The New Yorker, Feb 2, 2015

from Alec Wilkinson's profile of Yitang Zhang "The Pursuit of Beauty":

The British mathematician G. H. Hardy wrote in 1940 that mathematics is, of "all the arts and sciences, the most austere and the most remote." Bertrand Russell called it a refuge from "the dreary exile of the actual world." Hardy believed emphatically in the precise essence of math. A mathematical proof, such as Zhang produced, "should resemble a simple and clear-cut constellation," he wrote, "not a scattered cluster in the Milky Way." 
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The books on his shelves have titles such as "An Introduction to Hilbert Space" and "Elliptical Curves, Modular Forms, and Fermat's Last Theorem." There are also books on modern history and on Napoleon, who fascinates him, and copies of Shakespeare, which he reads in Chinese, because it's easier than Elizabethan English.
"Bounded Gaps Between Primes" is a back-door attack on the twin-prime conjecture, which was proposed in the nineteenth century, and says that, no matter how far you travel on the number line, even as the gap widens between primes you will always encounter a pair of primes that are separated by two. The twin-prime conjecture is still unsolved. Euclid's proof established that there will always be primes, but it says nothing about how far apart any two might be. Zhang established that there is a distance within which, on an infinite number of occasions, there will always be two primes.
"You have to imagine this coming from nothing," Eric Grinberg said. "We simply didn't know. It is like thinking that the universe is infinite, unbounded, and finding it has an end somewhere."... 
When we reached Zhang's office, I asked him how he had found the door into the problem. On a whiteboard, he wrote, "Goldston-Pintz-Yildirim" and "Bombieri-Friedlander-Iwaniec." He said, "The first paper is on bounded gaps, and the second is on the distribution of prime numbers in arithmatic progression. I compare these two together, plus my own innovations, based on the years of reading in the library." 
"Many people have tried that problem," Iwaniec said. "He's a private guy. Nothing is rushed. If it takes him another ten years, that's fine with him. Unless you tackle a problem that's already solved, which is boring, or one whose solution is clear from the beginning, mostly you are stuck. But Zhang is willing to be stuck much longer." 
"I think what he did was brilliant," Deane Yang told me. "If you became a good calculus teacher, a school can become very depended on you. You're cheap and reliable, and there's no reason to fire you. After you've done that a couple of years, you can do it on autopilot; you have a lot of free time to think, as long as you're willing to live modestly...."


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