The International Mathematical Union (IMU) doled out seven prizes, including the brand-new, $500,000 Chern Medal Award, in opening ceremonies at its quadrennial International Congress of Mathematicians (ICM) today in Hyderabad, India. Also at the meeting, IMU elected its first woman president, Ingrid Daubechies of Princeton University.
Four mathematicians received the prestigious Fields Medal, long regarded as mathematics' version of the Nobel Prize. Elon Lindenstrauss of Hebrew University of Jerusalem and Ngô Bảo Châu of Université Paris-Sud in Orsay, France, took the prize for analytic work with applications to number theory. Stanislav Smirnov of the University of Geneva, Switzerland, and Cédric Villani of the Henri Poincaré Institute in Paris won for theoretical work in statistical physics.
Lindenstrauss, the ICM citation says, "has made far-reaching advances in ergodic theory," which studies the statistical behavior of dynamical systems. For a seemingly trivial example, imagine a frog making repeated jumps of the same length in the same direction, starting from the corner of a square on an infinite checkerboard. Ergodic theory deals with questions such as, how are the frog's landing spots distributed within the interiors of the squares—and in particular, how close do they come to the squares' corners and edges? Lindenstrauss has made leaps of his own toward understanding a crucial point known as the Littlewood conjecture, which concerns how close such frogs come to landing on edges.
Ngô gave "a brilliant proof" of another long-standing conjecture in number theory known as the "Fundamental Lemma," which lies at the heart of a broad unifying vision of mathematics that Robert Langlands, now at the Institute for Advanced Study in Princeton, New Jersey, initiated in the late 1960s. The Langlands Program, as it's called, ties together virtually all aspects of modern mathematics; among other implications, its realization would make a mere footnote of the famous Fermat's Last Theorem. As its name suggests, the Fundamental Lemma is a technical point, but one that had stymied mathematicians for decades. Ngô's breakthrough makes other advances in the Langlands Program look possible.
Smirnov has brought mathematical rigor to important aspects of statistical physics. Physicists often work with finite "lattice" models—generalizations of two-dimensional checkerboards—as approximations to continuous reality. Generally, they assume that the "scaling limits" they get as their grids become infinitely fine don't carry with them artifacts of the type of lattice they came from. Nobody has proved that assumption for all lattices, but Smirnov has settled the matter in the case of triangular ones. Before this, notes Harry Kesten, an expert in the theory at Cornell University, "nobody knew how to do anything."
Villani's work "makes deep connections between mathematics and physics, in particular regarding the notion of entropy," says Stefan Müller of the University of Bonn in Germany. It brings rigor to another question of statistical physics: How quickly does a highly organized system, such as compressed gas about to be released, reach its disordered equilibrium state? The answer contains a surprise: Entropy (a measure of disorder) increases at different speeds, sometimes quickly, sometimes slowly. Villani has also brought closure to a long-standing question concerning entropy and equilibrium in the "ion gases" known as plasmas. What's more, he has made surprising connections between the theory of gas diffusion and an eminently practical problem in economics known as optimal transport—roughly speaking, how to ship goods from a variety of producers to a variety of consumers most cost effectively.
The Nevanlinna Prize, given for work in mathematical aspects of computer science, went to Daniel Spielman of Yale University for contributions in the areas of linear programming and error-correcting codes, which underlie much of what computers spend their time doing in business applications and telecommunications. In the early 2000s, Spielman and Shang-Hua Teng of Boston University developed a theory that explained why a venerable technique called the "simplex method" works so surprisingly well at solving linear programming problems as they arise in practice, even though mathematicians can easily design artificial problems that confound it. His more recent work in coding theory has led to designs—and patents—that address problems such as packet loss during multicast communications on the Internet.
The Gauss Prize, awarded for work in applied mathematics, went to Yves Meyer, professor emeritus at the École Normale Supérieure de Cachan in France. In the 1980s, Meyer was instrumental in developing the mathematical theory of wavelets, which revolutionized the classical theory of Fourier analysis. Among their myriad applications, wavelets have provided the basis for the new image-compression standard, JPEG 2000, part of the technology that has made it possible, for example, to take what used to go on huge, cumbersome reels of celluloid and put it on thin, shiny disks that slip into DVD players. In addition to Meyers's profound mathematical work, "he has been an inspiration for a whole generation of mathematicians," says Daubechies, herself a key contributor to wavelet theory. "If I had to characterize him with one word, that would be 'enthusiasm.' "
Louis Nirenberg of New York University was honored with the inaugural Chern Medal for his lifetime work in the modern theory of partial differential equations and for his mentoring of students and postdocs. The Chern Award, named after the Chinese mathematician Shiing-Shen Chern, who died in 2004, comes in two parts: $250,000 to the recipient and another $250,000 to one or more organizations in support of research, education, or other mathematical programs, to be nominated by the recipient. (At the time of this writing, Nirenberg's nominations had not been released.) Nirenberg knew Chern personally; the two even collaborated on a joint paper in 1969. In remarks at a 1990 symposium in Chern's honor, Nirenberg, known for his sense of humor, recalled what a "special experience" it was to go to a Chinese restaurant with Chern. "On every such occasion," he said, "one eats infinitely better than if he had not been there—even if there were other Chinese in the party." A quarter million dollars could feed a lot of mathematicians—and probably fill a lot of place mats with work leading to future ICM prizes.
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