Vladimir Drinfeld

Short description: Mathematician

Vladimir Drinfeld

Born|  (1954-02-14) February 14, 1954 (age 70)

Kharkov, Ukrainian SSR, Soviet Union

Alma mater| Moscow State University
Known for| Drinfeld center
Drinfeld double
Drinfeld level structure
Drinfeld module
Drinfeld reciprocity
Drinfeld upper half plane
Drinfeld twist
Drinfeld–Sokolov reduction
Drinfeld–Sokolov–Wilson equation
ADHM construction
Manin–Drinfeld theorem
Yetter–Drinfeld category
Chiral algebra
Chiral homology
Quantum groups
Geometric Langlands correspondence
Grothendieck–Teichmüller group
Lie-* algebra
Opers
Quantum affine algebra
Quantized enveloping algebra
Quasi-bialgebra
Quasi-triangular quasi-Hopf algebra
Ruziewicz problem
Tate modules
Awards| Fields Medal (1990)
Wolf Prize (2018)
Shaw Prize (2023)
Scientific career
Fields| Mathematics
Institutions| University of Chicago
Doctoral advisor| Yuri Manin

Vladimir Gershonovich Drinfeld (Ukrainian: Володи́мир Ге́ршонович Дрінфельд; Russian: Влади́мир Ге́ршонович Дри́нфельд; born February 14, 1954), surname also romanized as Drinfel'd, is a renowned mathematician from the former USSR, who emigrated to the United States and is currently working at the University of Chicago.

Drinfeld's work connected algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the geometric Langlands correspondence. Drinfeld introduced the notion of a quantum group (independently discovered by Michio Jimbo at the same time) and made important contributions to mathematical physics, including the ADHM construction of instantons, algebraic formalism of the quantum inverse scattering method, and the Drinfeld–Sokolov reduction in the theory of solitons.

He was awarded the Fields Medal in 1990.[1] In 2016, he was elected to the National Academy of Sciences.[2] In 2018 he received the Wolf Prize in Mathematics.[3] In 2023 he was awarded the Shaw Prize in Mathematical Sciences.[4]

## Contents

* 1 Biography
* 2 Contributions to mathematics
* 3 See also
* 4 Notes
* 5 References
* 6 External links

## Biography

Drinfeld was born into a Jewish[5] mathematical family, in Kharkiv, Ukrainian SSR, Soviet Union in 1954. In 1969, at the age of 15, Drinfeld represented the Soviet Union at the International Mathematics Olympiad in Bucharest, Romania, and won a gold medal with the full score of 40 points. He was, at the time, the youngest participant to achieve a perfect score, a record that has since been surpassed by only four others including Sergei Konyagin and Noam Elkies. Drinfeld entered Moscow State University in the same year and graduated from it in 1974. Drinfeld was awarded the Candidate of Sciences degree in 1978 and the Doctor of Sciences degree from the Steklov Institute of Mathematics in 1988. He was awarded the Fields Medal in 1990. From 1981 till 1999 he worked at the Verkin Institute for Low Temperature Physics and Engineering (Department of Mathematical Physics). Drinfeld moved to the United States in 1999 and has been working at the University of Chicago since January 1999.

## Contributions to mathematics

In 1974, at the age of twenty, Drinfeld announced a proof of the Langlands conjectures for GL2 over a global field of positive characteristic. In the course of proving the conjectures, Drinfeld introduced a new class of objects that he called "elliptic modules" (now known as Drinfeld modules). Later, in 1983, Drinfeld published a short article that expanded the scope of the Langlands conjectures. The Langlands conjectures, when published in 1967, could be seen as a sort of non-abelian class field theory. It postulated the existence of a natural one-to-one correspondence between Galois representations and some automorphic forms. The "naturalness" is guaranteed by the essential coincidence of L-functions. However, this condition is purely arithmetic and cannot be considered for a general one-dimensional function field in a straightforward way. Drinfeld pointed out that instead of automorphic forms one can consider automorphic perverse sheaves or automorphic D-modules. "Automorphicity" of these modules and the Langlands correspondence could be then understood in terms of the action of Hecke operators.

Drinfeld has also worked in mathematical physics. In collaboration with his advisor Yuri Manin, he constructed the moduli space of Yang–Mills instantons, a result that was proved independently by Michael Atiyah and Nigel Hitchin. Drinfeld coined the term "quantum group" in reference to Hopf algebras that are deformations of simple Lie algebras, and connected them to the study of the Yang–Baxter equation, which is a necessary condition for the solvability of statistical mechanical models. He also generalized Hopf algebras to quasi-Hopf algebras and introduced the study of Drinfeld twists, which can be used to factorize the R-matrix corresponding to the solution of the Yang–Baxter equation associated with a quasitriangular Hopf algebra.

Drinfeld has also collaborated with Alexander Beilinson to rebuild the theory of vertex algebras in a coordinate-free form, which have become increasingly important to two-dimensional conformal field theory, string theory, and the geometric Langlands program. Drinfeld and Beilinson published their work in 2004 in a book titled "Chiral Algebras."[6]

## See also

* Drinfeld reciprocity
* Drinfeld upper half plane
* Manin–Drinfeld theorem
* Quantum group
* Chiral algebra
* Quasitriangular Hopf algebra
* Ruziewicz problem

## Notes

1. ↑ O'Connor, J. J.; Robertson, E. F. "Vladimir Gershonovich Drinfeld". Biographies. School of Mathematics and Statistics University of St Andrews, Scotland. http://www-history.mcs.st-and.ac.uk/Biographies/Drinfeld.html.
2. ↑ National Academy of Sciences Members and Foreign Associates Elected, National Academy of Sciences, May 3, 2016, http://www.nasonline.org/news-and-multimedia/news/may-3-2016-NAS-Election.html, retrieved 2016-05-14 .
3. ↑ Jerusalem Post - Wolf Prizes 2018
4. ↑ Shaw Prize 2023
5. ↑ Vladimir Gershonovich Drinfeld
6. ↑ Beilinson, Alexander; Drinfeld, Vladimir (2004) (in English). Chiral Algebras. Providence, R.I.: American Mathematical Society. ISBN 0-8218-3528-9. OCLC 53896661.

## References

* O'Connor, John J.; Robertson, Edmund F., "Vladimir Drinfeld", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Drinfeld.html .
* Victor Ginzburg, Preface to the special volume of Transformation Groups (vol 10, 3–4, December 2005, Birkhäuser) on occasion of Vladimir Drinfeld's 50th birthday, pp 277–278, doi:10.1007/s00031-005-0400-6
* Report by Manin

## External links

* Vladimir Drinfeld at the Mathematics Genealogy Project
* Langlands Seminar homepage

* v
* t
* e

Fields Medalists

*     * 1936  Ahlfors
* Douglas

*     * 1950  Schwartz
* Selberg

*     * 1954  Kodaira
* Serre

*     * 1958  Roth
* Thom

*     * 1962  Hörmander
* Milnor

*     * 1966  Atiyah
* Cohen
* Grothendieck
* Smale

*     * 1970  Baker
* Hironaka
* Novikov
* Thompson

*     * 1974  Bombieri
* Mumford

*     * 1978  Deligne
* Fefferman
* Margulis
* Quillen

*     * 1982  Connes
* Thurston
* Yau

*     * 1986  Donaldson
* Faltings
* Freedman

*     * 1990  Drinfeld
* Jones
* Mori
* Witten

*     * 1994  Bourgain
* Lions
* Yoccoz
* Zelmanov

*     * 1998  Borcherds
* Gowers
* Kontsevich
* McMullen

*     * 2002  Lafforgue
* Voevodsky

*     * 2006  Okounkov
* Perelman (declined)
* Tao
* Werner

*     * 2010  Lindenstrauss
* Ngô
* Smirnov
* Villani

*     * 2014  Avila
* Bhargava
* Hairer
* Mirzakhani

*     * 2018  Birkar
* Figalli
* Scholze
* Venkatesh

* Book
* Category
* Mathematics portal

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