Andrei Okounkov

Native spelling: Андрей Окуньков

fl. 1969, Moscow, Soviet Union

AlgebraGeometryProbability and statisticsFields Medal 2006

Russian-American mathematician who uncovered deep links between probability, representation theory and algebraic geometry, receiving the Fields Medal in 2006.

Andrei Okounkov was born in Moscow in 1969. He first studied economics at Moscow State University before turning to mathematics, and he received his PhD there in 1995 on asymptotic representation theory. After positions at the University of Chicago and the University of California, Berkeley, he became a professor at Princeton University and moved in 2010 to Columbia University, where he is the Samuel Eilenberg Professor of Mathematics.

Okounkov's research rests on the observation that randomness in combinatorial structures is governed by the same algebraic machinery as representation theory. His early work explained the asymptotic shape of large Young diagrams and the spectra of random matrices in terms of symmetric functions. He then helped create the modern enumerative geometry of curves and sheaves: with Rahul Pandharipande he developed the equivariant quantum cohomology of Hilbert schemes, conjectured (later proved in part) the Gromov–Witten/Donaldson–Thomas correspondence, and showed how quantum groups control the enumerative theory of threefolds. This circle of ideas feeds directly into supersymmetric gauge theory and string theory in physics.

He received the Fields Medal at the 2006 International Congress of Mathematicians in Madrid, alongside Grigori Perelman, Terence Tao and Wendelin Werner. He delivered a plenary lecture at the ICM in 2018 and continues to develop geometric and algebraic tools — quantum groups, geometric representation theory, box counting — for both mathematics and theoretical physics.

Workplaces: Columbia University, Princeton University

Russia, United States

Award-winning work

Fields Medal 2006The medal cites his contributions bridging probability, representation theory and algebraic geometry. Okounkov found a deep new interplay between random combinatorial objects — random matrices, random partitions and random surfaces — and classical structures such as the representation theory of symmetric groups and quantum groups. With collaborators he solved problems on the asymptotics of partitions, the Gromov–Witten theory and quantum cohomology of curves (including the celebrated work with Pandharipande on the equivariant quantum cohomology of the Hilbert scheme of points), and the Donaldson–Thomas theory of threefolds. His methods made enumerative problems computable by translating them into questions about integrable systems and symmetric functions.

Contributions, linked to the library

Library topicRepresentation theoryReshaped the asymptotic representation theory of symmetric groups and connected it to integrable systems.Library topicRandom matrix theoryUsed algebraic methods to relate random matrix spectra to partitions and symmetric functions.Library topicAlgebraic geometryBuilt the modern enumerative theory of curves and threefolds, including quantum cohomology of Hilbert schemes and Donaldson–Thomas theory.