Institute For Advanced Study Faculty Member Vladimir Voevodsky Wins 2002 Fields Medal

Institute for Advanced Study Faculty member Vladimir Voevodsky received one of two Fields Medals awarded today in Beijing, China, at the International Congress of Mathematicians. The Fields Medal is the world's highest award for achievement in mathematics, and is presented every four years by the International Mathematical Union. Since 1936, the Fields Medal has been presented to 43 individuals, of whom 30 have been either Faculty members or visiting scholars at the Institute for Advanced Study.

Professor Voevodsky will be recognized for developing new cohomology theories for algebraic varieties, thereby providing new insights into number theory and algebraic geometry.

The Fields Medal, first given in 1936 and then not presented again until 1950, is awarded to young mathematicians (aged forty or under) for outstanding mathematical achievement, and recognizes both existing work and the promise of future achievement. Since 1966 the award has usually been shared by four individuals, but in 2002 Professor Voevodsky will share the award with only one other individual, Laurent Lafforgue, Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France, who will be recognized for making a major advance in the Langlands Program, thereby providing new connections between number theory and analysis.

"Vladimir Vovoedsky is a highly talented mathematician who has tackled the most difficult problems in abstract algebraic geometry," stated Phillip A. Griffiths, Director of the Institute for Advanced Study. "His research has influenced the development of algebraic geometry and topology, and led to a solution of several outstanding problems. We are delighted to have a young mathematician of his originality and creativity on the faculty of our School of Mathematics."

Vladimir Voevodsky is known for his work in the homology theory of schemes, algebraic K-theory, and interrelations between algebraic geometry and algebraic topology. He has been concerned with a synthesis of algebraic geometry and homotopy theory, two major branches of modern mathematics.

Voevodsky made one of the most outstanding advances in algebraic geometry in the past few decades by developing new cohomology theories for algebraic varieties. His work is characterized by an ability to handle highly abstract ideas with ease and flexibility and to deploy those ideas in solving quite concrete mathematical problems.

Voevodsky's achievement has its roots in the work of 1966 Fields Medalist Alexandre Grothendieck, a profound and original mathematician who could perceive the deep abstract structures that unite mathematics. Grothendieck realized that there should be objects, which he called "motives," that are at the root of the unity between two branches of mathematics, number theory and geometry. Grothendieck's ideas have had widespread influence in mathematics and provided inspiration for Voevodsky's work.

The notion of cohomology first arose in topology, which can be loosely described as "the science of shapes." Examples of shapes studied are the sphere, the surface of a doughnut, and their higher-dimensional analogues. Topology investigates fundamental properties that do not change when such objects are deformed (but not torn). On a very basic level, cohomology theory provides a way to cut a topological object into easier-to-understand pieces. Cohomology groups encode how the pieces fit together to form the object. There are various ways of making this precise, one of which is called singular cohomology. Generalized cohomology theories extract data about properties of topological objects and encode that information in the language of groups. One of the most important of the generalized cohomology theories, topological K-theory, was developed chiefly by another 1966 Fields Medalist, Michael Atiyah. One remarkable result revealed a strong connection between singular cohomology and topological K-theory.

In algebraic geometry, the main objects of study are algebraic varieties, which are the common solution sets of polynomial equations. Algebraic varieties can be represented by geometric objects like curves or surfaces, but they are far more "rigid" than the malleable objects of topology, so the cohomology theories developed in the topological setting do not apply here. For about forty years, mathematicians worked hard to develop good cohomology theories for algebraic varieties; the best understood of these was the algebraic version of K-theory. A major advance came when Voevodsky, building on a little-understood idea proposed by Andrei Suslin, created a theory of "motivic cohomology." In analogy with the topological setting, there is a strong connection between motivic cohomology and algebraic K-theory. In addition, Voevodsky provided a framework for describing many new cohomology theories for algebraic varieties. His work constitutes a major step toward fulfilling Grothendieck's vision of the unity of mathematics.

One consequence of Voevodsky's work, and one of his most celebrated achievements, is the solution of the Milnor Conjecture, which for three decades was the main outstanding problem in algebraic K-theory. This result has striking consequences in several areas, including Galois cohomology, quadratic forms, and the cohomology of complex algebraic varieties. Voevodsky's work may have a large impact on mathematics in the future by allowing powerful machinery developed in topology to be used for investigating algebraic varieties.

Vladimir Voevodsky was born on June 4, 1966 in Russia. He received his B.S. in mathematics from Moscow State University (1989) and his Ph.D. in mathematics from Harvard University (1992). He held visiting positions at the Institute for Advanced Study, Harvard University, and the Max-Planck-Institut fuer Mathematik before joining the faculty of Northwestern University in 1996. A visiting scholar in the Institute’s School of Mathematics since 1998, he was named a Faculty member in the School in 2002. Voevodsky was a Sloan Fellow in 1996-98, and has twice received grants from the National Science Foundation. He won a Clay Prize Fellowship in 1999 and 2000. He is coauthor (with A. Suslin and E.M. Friedlander) of Cycles, Transfers and Motivic Homology Theories (Princeton University Press, 2000).