Okounkov, Andrei

Andrei Okounkov, (born July 26, 1969, Moscow, Russia, U.S.S.R. [now in Russia]), Russian mathematician awarded a Fields Medal in 2006 “for his contributions bridging probability, representation theory and algebraic geometry.”

Okounkov received a doctorate in mathematics from Moscow State University (1995) and held positions at the Russian Academy of Sciences, the Institute for Advanced Study in Princeton, New Jersey, the University of Chicago, and the University of California, Berkeley. In 2001 he joined the mathematics department at Princeton University but left in 2010 to teach at Columbia University.

Complicated physical systems, such as the energy levels in atomic nuclei, are described by mathematical models using what are called random matrices. These are square arrays of numbers in which each number is chosen at random, perhaps in conformity with some appropriate general requirement on the property of the resulting matrix. Random matrices studied in physics have statistical properties similar to the statistical properties of randomly chosen sequences of numbers, but no explanation was available until Okounkov showed an underlying unity between branches of physics, the probabilistic behaviour of numbers, and algebraic geometry based on the concept of random surfaces.

A random surface is a model of how a crystal erodes or dissolves, and it describes the shape of the crystal as the edges are eaten away. The crystal is thought of as being made of numerous tiny blocks that are gradually removed. Okounkov and his coauthor, American mathematician Richard Kenyon, discovered the remarkable result that the outline of any two-dimensional picture of the crystal is always an algebraic curve and so is defined by polynomial equations (equations of the form p(x) = a₀ + a₁x + a₂x² + ⋯ + a_nxⁿ).

Okounkov has also obtained a significant number of new results in enumerative geometry by a mixture of ingenious combinatorial arguments that draw on his work on randomness and a wide range of ideas from algebra and geometry.