From here http://www.college.ucla.edu/news/05/terencetaomath.html

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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, ‘let’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 déjà 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.”