# Andrei Yuryevich Okunkov

Andrei Yuryevich Okunkov

Andrei Jurjewitsch Okunkow ( Russian Андрей Юрьевич Окуньков , scientific transliteration Andrej Jur'evič Okun'kov ; English transcription Andrei Okounkov; born June 26, 1969 in Moscow ) is a Russian mathematician who works in the field of representation theory . In particular, he is concerned with the application of this theory in algebraic geometry , mathematical physics , probability theory and the theory of special functions .

## Life

He completed his doctoral thesis in 1995 at Moscow's Lomonosov University , supervised by Alexander Kirillow . In 1994/95 he was at the Dobrushin Laboratory of the Institute for Information Transmission of the Russian Academy of Sciences, was Dickson Instructor at the University of Chicago from 1996 to 1999 and at the Institute for Advanced Study in 1996 and at the MSRI in 1997 . In 1998 he became an assistant professor at the University of California, Berkeley , where he became a professor in 2001, and from 2002 to 2010 he was a professor at Princeton University . He has been a professor at Columbia University since 2010 .

## plant

Since his time in Moscow (with Alexander Jurjewitsch Olschanski in Moscow and the school of Anatoli Werschik in St. Petersburg with their view of partitions in the manner of Young Tableau as stochastic objects) his main area of ​​work has been the representation of groups, especially combinatorial and asymptotic aspects as well as applications in various fields (by Okounkov himself, for example, in the Seiberg-Witten theory, statistical mechanics and real algebraic geometry).

With Rahul Pandharipande he made significant contributions to the theory of the Gromow-Witten invariants in counting algebraic geometry. They gave a description of the Gromow-Witten invariants for curves and proved a conjecture by Eguchi, Hori and Xiong for the case of projective curves (an extension of the conjecture conjectured by Witten and proven by Maxim Konzewitsch about the differential equation systems for the generating function for the invariants from the fall of points to higher-dimensional varieties). This also provided a more transparent proof of the Witten-Konzewitsch conjecture. With Davesh Maulik , Pandharipande and Nikita Alexandrowitsch Nekrassow , he formulated conjectures about the connection between Gromow-Witten invariants and Donaldson-Thomas invariants.

His treatment of a model of statistical mechanics or graph theory, the dimers, also attracted a lot of attention. With Scott Sheffield and Richard Kenyon , he found surprising connections to real algebraic geometry (spectral curves of the dimer model in the form of Harnack curves).

Okounkov bodies or Newton-Okounkov bodies are named after him, convex bodies related to Newton polygons in Euclidean spaces, which are connected with linear families on varieties.

## Honors and memberships

In 2004 he gave a plenary lecture at the 4th European Congress of Mathematicians ( Random surfaces enumerating algebraic curves ) and received the EMS Prize . At the 25th International Congress of Mathematicians in Madrid (2006) he was awarded the Fields Medal . He was also invited speaker there ( random partitions and instanton counting ). In 2012 he was elected a member of the National Academy of Sciences and in 2016 a member of the American Academy of Arts and Sciences . In 2018 he is plenary speaker at the ICM in Rio (On ​​the crossroads of enumerative geometry and geometric representation theory). In 2018 he was on the Fields Medal Award Committee.

## Fonts

• with Richard Kenyon, Scott Sheffield: Dimers and amoebae, Annals of Mathematics, Volume 163, 2006, pp. 1019-1056, Arxiv, 2003
• with Richard Kenyon: Planar dimers and Harnack curves, Duke Math. Journal, Volume 131, 2006, pp. 499-524, Arxiv 2003
• with D. Maulik, N. Nekrasov, R. Pandharipande: Gromov-Witten and Donaldson-Thomas Theory, Part 1,2, Compositio Mathematica, Volume 142, 2006, pp. 1263–1285, 1286–1304, Part 1, Arxiv, 2003 , part 2, Arxiv, 2004
• with Nekrasov: Seiberg-Witten theory and random partitions, in: The unity of mathematics, Progress in Mathematics 244, Birkhäuser 2006, pp. 525-596, Arxiv
• with Nikolai Jurjewitsch Reschetichin , Cumrun Vafa : Quantum Calabi-Yau and classical crystals, in: The unity of mathematics, Progress in Mathematics 244, Birkhäuser 2006, pp. 597–618, Arxiv
• with Pandharipande: Gromov-Witten-Theory, Hurwitz theory and completed cycles, Annals of Mathematics, Volume 163, 2006, pp. 517-560, Arxiv
• with Pandharipande: The equivariant Gromov-Witten theory of , Annals of Mathematics, Volume 163, 2006, pp. 561-605, Arxiv${\ displaystyle P ^ {1}}$
• with Pandharipande: Quantum cohomology of the Hilbert scheme of points in the plane, Inventiones Mathematicae, Volume 179, 2010, pp. 523-557, Arxiv, 2004
• with Pandharipande: Virasoro constraints for target curves, Inventiones Mathematicae, Volume 163, 2006, pp. 47-108, Arxiv
• with Pandharipande: Gromov-Witten theory, Hurwitz numbers and matrix models I, in: Dan Abramovich, Algebraic Geometry, Seattle 2005, Arxiv 2001
• with Alex Eskin : Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Inventiones Mathematicae, Volume 145, 2001, pp. 59-103
• with Spencer Bloch : The character of the infinite wedge representation, Adv. Math., Volume 149, 2000, pp. 1-60, Arxiv
• Why would multiplicities be log-concave?, In: Progress in Math. 213, Birkhauser 2003
• Brunn-Minkowski inequality for multiplicities, Inventiones Mathematicae, Volume 125, 1996, pp. 405-411

## literature

• Giovanni Felder , Honoring Okounkov on the occasion of the Fields Medal, Proc. ICM 2006, volume 1