Terence Tao is UCLA’s first mathematician to receive the prestigious Fields Medal, often described as the “Nobel Prize in Mathematics.”
Tao, 31, was presented the prize today (August 22) at the International Congress of Mathematicians in Madrid. The Fields Medal is awarded by the International Mathematical Union every fourth year.
Tao's capture of the Fields Medal surprised few at UCLA.
“Terry is like Mozart; mathematics just flows out of him, except without Mozart’s personality problems,” said John Garnett, professor and former chair of the mathematics department.
“People all over the world say, ‘UCLA’s so lucky to have Terry Tao,'” said Tony Chan, dean of physical sciences and professor of mathematics. “The way he crosses areas would be like the best heart surgeon also being exceptional in brain surgery.”
A math prodigy from Adelaide, Australia, Tao started learning calculus as a 7-year-old high school student. By 9, he had progressed to university-level calculus; by 11, he was already burnishing his reputation at international math competitions. Tao was 20 when he earned his Ph.D. from Princeton University and joined UCLA’s faculty. By 24, he had become a full professor.
“The best students in the world in number theory all want to study with Terry,” Chan said. Graduate students have come to UCLA from as far as Romania and China.
One area in which Tao specializes is harmonic analysis, an advanced form of calculus that uses equations from physics. Some of his work involves “geometrical constructions that almost no one understands,” Garnett said. Tao is also regarded as the world’s expert on the “Kakeya conjecture,” a perplexing set of five problems in harmonic analysis. And his work with Ben Green of the University of Bristol, England - proving that prime numbers contain infinitely many progressions of all finite lengths - was lauded by Discover magazine as one of the 100 most important scientific discoveries in 2004.
“I don’t have any magical ability,” Tao said. “I look at a problem, and it looks something like one I’ve done before. I think maybe the idea that worked before will work here. . . . After awhile, I figure out what’s going on.”
"Grisha Perelman, where are you?
Three years ago, a Russian mathematician by the name of Grigory Perelman in St. Petersburg announced that he had solved a famous and intractable mathematical problem, known as the Poincaré conjecture, about the nature of space.
After posting a few short papers on the Internet and making a whirlwind lecture tour of the United States, Perelman, known as Grisha, disappeared back into the Russian woods in the spring of 2003, leaving the world's mathematicians to pick up the pieces and decide whether he was right.
Now they say they have finished his work, and the evidence is circulating among scholars in the form of three book-length papers with about 1,000 pages of dense mathematics and prose between them.
As a result, there is a growing feeling, a cautious optimism that they have finally achieved a landmark not just of mathematics, but of human thought.
"It's really a great moment in mathematics," said Bruce Kleiner of Yale University, who has spent the last three years helping to explicate Perelman's work. "It could have happened 100 years from now, or never."
In a speech at a conference in Beijing this summer, Shing-Tung Yau of Harvard said the understanding of three-dimensional space brought about by the Poincaré conjecture could be one of the major pillars of math in the 21st century.
But at the moment of his putative triumph, Perelman is nowhere in sight. He is an odds-on favorite to win a Fields Medal, math's version of the Nobel Prize, when the International Mathematics Union convenes in Madrid next Tuesday. But there is no indication whether he will show up.
Also left hanging, for now, is the $1 million offered by the Clay Mathematics Institute in Cambridge, Massachusetts, for the first published proof of the conjecture, one of seven outstanding questions for which they offered a ransom at the beginning of the millennium.
"It's very unusual in math that somebody announces a result this big and leaves it hanging," said John Morgan of Columbia University in New York, one of the scholars who has also been filling in the details of Perelman's work.
Mathematicians have been waiting for this result for more than 100 years, ever since the French polymath Henri Poincaré posed the problem in 1904. And they acknowledge that it may be another 100 years before its full implications for math and physics are understood. For now, they say, it is just beautiful, like art or a challenging new opera.
Morgan said the excitement came not from the final proof of the conjecture, which everybody felt was true, but the method, "finding deep connections between what were unrelated fields of mathematics."
William Thurston of Cornell University, the author of a deeper conjecture, which includes Poincaré's and which is now apparently proved, said, "Math is really about the human mind, about how people can think effectively, and why curiosity is quite a good guide," saying that curiosity is tied in some way with intuition.
"You don't see what you're seeing until you see it," Thurston said, "but when you do see it, it lets you see many other things."
Depending on who is talking, the Poincaré conjecture can sound either daunting or deceptively simple. It asserts that if any loop in a certain kind of three-dimensional space can be shrunk to a point without ripping or tearing either the loop or the space, the space is equivalent to a sphere.
The conjecture is fundamental to topology, the branch of math that deals with shapes, sometimes described as geometry without the details. To a topologist, a sphere, a cigar and a rabbit's head are all the same because they can be deformed into one another. Likewise, a coffee mug and a doughnut are also the same because each has one hole, but they are not equivalent to a sphere.
In effect, what Poincaré suggested was that anything without holes has to be a sphere. The one qualification was that this "anything" had to be what mathematicians call compact, or closed, meaning that it has a finite extent: No matter how far you strike out in one direction or another, you can get only so far away before you start coming back, the way you can never get more than 12,500 miles, or 20,100 kilometers, from home on Earth.
In the case of two dimensions, like the surface of a sphere or a doughnut, it is easy to see what Poincaré was talking about: Imagine a rubber band stretched around an apple or a doughnut; on the apple, the rubber band can be shrunk without limit, but on the doughnut it is stopped by the hole.
Perelman's first paper, promising "a sketch of an eclectic proof," came as a bolt from the blue when it was posted on the Internet in November 2002.
"Nobody knew he was working on the Poincaré conjecture," said Michael Anderson of the State University of New York in Stony Brook.
Perelman had already established himself as a master of differential geometry, the study of curves and surfaces. Born in St. Petersburg in 1966, he distinguished himself as a high school student by winning a gold medal with a perfect score in the International Mathematical Olympiad in 1982. After getting a doctorate from St. Petersburg State, he joined the Steklov Institute of Mathematics at St. Petersburg.
In a series of postdoctoral fellowships in the United States in the early 1990s, Perelman impressed his colleagues as "a kind of unworldly person," in the words of Robert Greene, a mathematician at the University of California, Los Angeles - friendly, but shy and not interested in material wealth.
"He looked like Rasputin, with long hair and fingernails," Greene said.
Asked about Perelman's pleasures, Anderson said that he talked a lot about hiking in the woods near St. Petersburg looking for mushrooms.
Perelman returned to those woods, and the Steklov Institute, in 1995, spurning offers from Stanford and Princeton, among others. In 1996 he added to his legend by turning down a prize for young mathematicians from the European Mathematics Society.
Until his papers on the Poincaré conjecture started appearing, some friends thought Perelman had left mathematics. Although they were so technical and abbreviated that few mathematicians could read them, they quickly attracted interest among experts. In the spring of 2003, Perelman came back to the United States to give a series of lectures.
But once he was back in St. Petersburg, he did not respond to further invitations. The e-mail gradually ceased.
"He came once, he explained things, and that was it," Anderson said. "Anything else was superfluous."
Recently, Perelman is said to have resigned from Steklov. E-mail messages addressed to him and to the Steklov Institute went unanswered.
In his absence, others have taken the lead in trying to verify and disseminate his work"
Terence Tao has been years ahead of everyone else his entire life. Tao started taking high school classes at age eight; by 11, he was learning calculus and thriving in international mathematics competitions. He was only 21 when he earned his Ph.D. from Princeton University, and joined UCLA's faculty that year. The UCLA College promoted Tao to full professor of mathematics at 24.
"Terry is like Mozart -- except without Mozart's personality problems," said John Garnett, professor and former chair of mathematics in the UCLA College. "Mathematics just flows out of him." "Mathematicians with Terry's abilities appear only once in a generation," said Garnett. "He's probably the best mathematician in the world right now. Terry can unravel an enormously complicated mathematical problem and reduce it to something very simple. We're amazingly lucky to have him at UCLA."
The Fields Medal is considered the Nobel Prize for mathematics, said Tony Chan, Dean of Physical Sciences in the College, and professor and former chair of mathematics. The medal is given every fourth year by the International Mathematical Union, and will be given next summer in Madrid.
No one from UCLA has ever won the Fields Medal, but will that change in 2006?
"Terry is very creative and one of the most talented mathematicians I have seen in the last two decades," Chan said. "He has solved problems that have stumped others. I think the breadth and depth of his work, taken together, should make him a worthy candidate for the Fields Medal. What is even more amazing is that Terry is still so young. If he were a company, he would be Microsoft right before it sent public. If you could invest in him, you would certainly want to, because the payoff will be enormous."
"Terry will be a leading candidate for the 2006 Fields Medal," Garnett said.
While most mathematicians focus on just one branch of mathematics, Tao works in several areas, some completely unrelated to the others, and is a leading figure in four distinct areas, Garnett said.
Tao's branches of mathematics include a theoretical field called harmonic analysis, an advanced form of calculus that uses equations from physics. Some of this work involves, in Garnett's words, "geometrical constructions that almost no one understands."
Tao, 29, also works in a related field, non-linear partial differential equations, and in the entirely distinct fields of algebraic geometry, number theory, and combinatorics.
Tao and colleagues have taken on complex mathematical problems, in one case expanding on work begun by the one of the founders of formal mathematical studies more than 2,200 years ago. Work on prime numbers by Tao and University of Bristol mathematician Ben Green was acknowledged by Discover magazine as one of the 100 most important discoveries in science for 2004.
Green and Tao expanded on theories that originated with the Greek mathematician Euclid. Euclid proved there is an infinite quantity of prime numbers (a number divisible only by itself and one). Green and Tao proved that the prime numbers contain infinitely many progressions of all finite lengths. An example of an equally spaced progression of primes, of length three, is 3, 7, 11; the largest known progression of prime numbers is length 24, with each of the numbers containing more than two dozen digits. Green and Tao's discovery reveals that somewhere in the prime numbers, there is a progression of length 100, and length 1,000, and every other finite length. They also demonstrated that there are an infinite number of such progressions in the primes.
To prove this, Tao and Green spent two years analyzing all four proofs of a theorem named for Hungarian mathematician Endre Szemeredi. Very few mathematicians understand all four proofs, and Szemeredi's theorem does not apply to prime numbers.
"We took Szemeredi's theorem and goosed it so that it handles primes," Tao said. "To do that, we borrowed from each of the four proofs to build an extended version of Szemeredi's theorem. Every time Ben and I got stuck, there was always an idea from one of the four proofs that we could somehow shoehorn into our argument." Tao is also known among mathematics researchers for his work on the "Kakeya conjecture," a perplexing set of five problems in harmonic analysis. One of Tao's proofs extends more than 50 pages, in which he and two colleagues obtained the most precise known estimate of the size of a particular geometric dimension in Euclidean space. The issue involves the most space-efficient way to fully rotate an object in three dimensions, a question that interests theoretical mathematicians.
Tao and colleagues Allen Knutson at UC Berkeley and Chris Woodward at Rutgers solved an old problem (proving a conjecture proposed by former UCLA professor Alfred Horn) for which they developed a method that also solved longstanding problems in algebraic geometry ?describing equations geometrically ?and representation theory.
Speaking of this work, Tao said, "Other mathematicians gave the impression that the puzzle required so much effort that it was not worth making the attempt ?that first you have to understand this 100-page paper and that 100-page paper before even starting. We used a different approach to solve a key missing gap."
Solving some problems comes out of less formal collaborations. Tao found a surprising result to an applied mathematics problem involving image processing with Caltech mathematician Emmanuel Candes in a collaboration forged while they were taking their children to UCLA's Fernald Child Care Center.
"A lot of our work came at the pre-school while we were dropping off our kids," Tao said.
How does Tao describe his success?
"I don't have any magical ability," he said. "I look at a problem, and it looks something like one I've already done; I think maybe the idea that worked before will work here. When nothing's working out; then I think of a small trick that makes it a little better, but still is not quite right. I play with the problem, and after a while, I figure out what's going on.
"Most mathematicians faced with a problem, will try to solve the problem directly. Even if they get it, they might not understand exactly what they did. Before I work out any details, I work on the strategy. Once I have a strategy, a very complicated problem can split up into a lot of mini-problems. I've never really been satisfied with just solving the problem; I want to see what happens if I make some changes.
"If I experiment enough, I get a deeper understanding," said Tao, whose work is supported by the David and Lucille Packard Foundation. "After a while, when something similar comes along, I get an idea of what works and what doesn't work.
"It's not about being smart or even fast," Tao added. "It's like climbing a cliff; if you're very strong and quick and have a lot of rope, it helps, but you need to devise a good route to get up there. Doing calculations quickly and knowing a lot of facts are like a rock climber with strength, quickness and good tools; you still need a plan ?that's the hard part ?and you have to see the bigger picture."
His views about mathematics have changed over the years.
"When I was a kid, I had a romanticized notion of mathematics -- that hard problems were solved in Eureka moments of inspiration," he said. "With me, it's always, 憀et's try this that gets me part of the way. Or, that doesn't work, so now let's try this. Oh, there's a little shortcut here.'
"You work on it long enough and you happen to make progress towards a hard problem by a back door at some point. At the end, it's usually, 'oh, I've solved the problem.'"
Tao concentrates on one math problem at a time, but keeps a couple of dozen others in the back of his mind, "hoping one day I'll figure out a way to solve them. If there's a problem that looks like I should be able to solve it but I can't, that gnaws at me."
Does theoretical mathematics have applications beyond the theory?
"Mathematicians often work on pure problems that may not have applications for 20 years -- and then a physicist or computer scientist or engineer has a real-life problem that requires the solution of a mathematical problem, and finds that someone already solved it 20 years ago," Tao said.
"When Einstein developed his theory of relativity, he needed a theory of curved space. Einstein found that a mathematician devised exactly the theory he needed more than 30 years earlier."
Will Tao become an even better mathematician in another decade or so?
"Experience helps a lot," he said. "I may get a little slower, but I'll have access to a larger database of tricks; I'll know better what will work and what won't. I'll get d閖?vu more often, seeing a problem that reminds me of something."
What does Tao think of his success?
"I'm very happy," he said. "Maybe when I'm in my 60s, I'll look back at what I've done, but now I would rather work on the problems."