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2009 Abel Prize Winner: Geometry Giant Mikhail Leonidovich Gromov

Russian-French mathematician Mikhail Leonidovich Gromov, 65, is the 2009 Abel Prize winner. Gromov's "revolutionary contributions to geometry," according to the Norwegian Academy of Science and Letters, were the basis for this year's honor, which was announced in late March, 2009.

The Abel Committee wrote: "Mikhail Gromov is always in pursuit of new questions and is constantly thinking of new ideas for solutions to old problems. He has produced deep and original work throughout his career and remains remarkably creative. The work of Gromov will continue to be a source of inspiration for many future mathematical discoveries."

Associated with seminal results and concepts within Riemannian geometry, symplectic geometry, string theory, and group theory, Gromov will receive the prize—carrying a cash award of about $950,000—from Norway's King Harald on May 19, 2009, in Oslo.

Gromov, who is currently Jay Gould Professor of Mathematics at New York University's Courant Institute of Mathematical Sciences, obtained his masters degree (1965) and his doctorate (1969) from Leningrad University, where he was Assistant Professor from 1967 to 1974. Since 1982, he has been Permanent Professor at the Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France. He became a French citizen in 1992.

Gromov has played a decisive part in the creation of modern global Riemannian geometry. His solutions of important problems in global geometry relied on concepts such as the convergence of Riemannian manifolds and a compactness principle, which now bear his name. In the 1980s, he introduced the Gromov-Hausdorff distance between two abstract metric spaces, which is measured by embedding the two spaces in a third bigger space. Felix Hausdorff had suggested a way of measuring the distance between the two spaces, and Gromov proved two fundamental results for this construction: a precompactness theorem and a convergence theorem.

Gromov, who is also one of the founders of the field of symplectic geometry, in 1985 extended the concept of holomorphic curves to J-holomorphic curves on symplectic manifolds. This led to the theory of Gromov-Witten invariants, which has become linked to modern quantum field theory. It also led to the creation of symplectic topology, penetrating and transformeding other areas of mathematics.

Gromov's work on groups of polynomial growth introduced ideas that changed the way a discrete infinite group is viewed. He discovered the geometry of discrete groups and solved several outstanding problems. His geometrical approach, in fact, made complicated combinatorial arguments more natural yet powerful.

In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group when viewed as a geometrical object. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length n for increasing values of n. In 1981, Gromov proved that "a finitely generated group G has polynomial growth if and only if it is virtually nilpotent."

The difficult part of the theorem is the converse—and Gromov introduced several new geometric notions in order to convert the geometric information into an algebraic conclusion. His ideas have formed  the basis for strategies for approaching problems in the field of geometric group theory.

Mikhail L. Gromov has received the Kyoto Prize in Basic Sciences (2002), the Balzan Prize (1999), the American Mathematical Society's Leroy P. Steele Prize (1997), the Lobatchewski Medal (1997), and the Wolf Prize (1993). He is a foreign member of the U.S. National Academy of Sciences and the American Academy of Arts and Sciences, and a member of l'Académie française de Sciences.

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News Date: 
Friday, March 27, 2009

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