Vladimir Drinfeld

Vladimir (Volodymyr) Drinfeld
Born	(1954-02-14) February 14, 1954 Kharkiv, Ukrainian SSR, Soviet Union
Fields	Mathematics
Institutions	University of Chicago
Alma mater	Moscow State University
Doctoral advisor	Yuri Manin
Doctoral students	Dmytro Arinkin Dmitriy Boyarchenko Masoud Kamgarpour Joaquin Teruji Thomas
Known for	Quantum groups Geometric Langlands correspondence Drinfeld-Sokolov-Wilson equation
Notable awards	Fields Medal (1990)

Vladimir (Volodymyr) Gershonovich Drinfeld (Ukrainian: Володи́мир Гершо́нович Дрі́нфельд, Russian: Влади́мир Гершо́нович Дри́нфельд; born February 14, 1954) is a Ukrainian mathematician 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.

Biography

Drinfeld was born 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, and his since only been surpassed by Sergei Konyagin (1972) and Noam Elkies (1981). 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. 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 done much work in mathematical physics. In collaboration with his advisor Yuri Manin, he constructed the moduli space of Yang–Mills instantons, a result which was proved independently by Michael Atiyah and Nigel Hitchin. Drinfeld coined the term "Quantum group" in reference to Hopf algebras, which 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 conformal field theory, string theory, and the geometric Langlands program. Drinfeld and Beilinson published their work in 2004 in a book titled "Chiral Algebras."

See also

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

References

• .
• 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
• Vladimir Drinfeld's results at the International Mathematical Olympiad
• Langlands Seminar homepage