"He can easily be lost in a crowd of UCLA freshmen."īorn in Australia, Tao taught himself arithmetic at age 2. "Terry is as normal as it comes," says Tony Chan of the National Science Foundation, a former chairman of UCLA's mathematics department. Fellow mathematicians find him open and available. He accepts these plaudits with modesty and generally stays out of the public eye, handling most press inquiries by e-mail. Less than a month later, Tao got a $500,000 grant from the MacArthur Foundation. Last year, the International Mathematical Union awarded him the Fields Medal, widely considered the mathematics equivalent of a Nobel Prize. Tao's accomplishments have already earned him nearly every major mathematics prize. "Whenever he touches a subject, it becomes gold very quickly," says Candes. But he saw a connection that no engineer had seen. As a result of Candes and Tao's discovery, engineers are now working on MRI scanners several times faster than today's, and even one-pixel cameras. Not only that, the solution marked the birth of a new field, called compressive sampling. By the next day, Tao had solved the problem himself. Then a couple of minutes later, he allowed that Candes might be on to something. First he told Candes the problem was unsolvable. Why not design a camera that would acquire only a 50th of the data to start with? They record several million pixels (the basic elements of digital pictures), then use computer instructions called a compression algorithm to reduce the amount of data in the picture by 10 or 50 times. Present-day digital cameras go about this in a most inefficient way. For Tao, however, wandering across disciplinary boundaries is commonplace.įor instance, one day in 2004, Emmanuel Candes, an applied mathematician at Caltech, told Tao about a problem he was working on-how to reconstruct images with the least possible information. It was as unexpected as a violinist suddenly winning a major piano competition. Cappell, a professor at NYU's Courant Institute of Mathematical Sciences, calls him "the leading analyst of his generation." Yet the Green-Tao theorem resolved a major question in number theory, a completely separate field. Tao's specialty is analysis, the area of math that includes calculus and differential equations. Green and Tao's discovery was the mathematical sensation of 2004. Somewhere there's a prime number constellation with the same shape as Nash's umbrella. Tao, together with Cambridge University mathematician Ben Green, proved that this canvas contains patterns of every conceivable shape. But it is more illuminating to think of them as stars in the sky, scattered more or less at random over a vast canvas. Prime numbers, those that can be divided only by themselves and 1 (that is, 2, 3, 5, 7, 11, and so on), can be visualized as points on a line. UCLA mathematician Terence Tao, 32, relates to that scene, perhaps because it reminds him of his own wife, Laura, an engineer at NASA's Jet Propulsion Laboratory, but certainly because it illustrates one of his most famous theorems. He does it, and she is utterly charmed, saying, "Do it again." Supplements Tao notes 1.There's a scene in the Oscar-winning film A Beautiful Mind in which Russell Crowe as mathematician John Nash asks his beautiful wife-to-be to gaze into the evening sky and name any shape-an umbrella, for instance-and then says he will find the shape in a constellation. As an application of this lemma, we classify the polynomials over finite fields of large characteristic which are "expanding" in a combinatorial sense. By combining the model-theoretic classification of definable sets in this setting with an ultraproduct argument (to transfer to the characteristic zero setting) combined with algebraic geometry tools (such as the etale fundamental group) and the Lang-Weil inequality, we are able to get much better control on the pseudorandomness and definability properties of the components, and also can exclude the existence of bad components. We present an "algebraic regularity lemma" which covers the case when the graph is definable (with bounded complexity) over a finite field of large characteristic. However, these defects can sometimes be removed if further assumptions are placed on the original graph. While this lemma applies to arbitrary graphs, the dependence on constants can be terrible, and there can also exist "bad" components which are not pseudorandom. Szemeredi's regularity lemma provides a structural description of arbitrary large dense graphs roughly speaking, it decomposes any such graphs into a bounded number of pieces, most of which behave "pseudorandomly". Location MSRI: Simons Auditorium Video Abstract
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