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Vladimir Gershonovich Drinfel'd (Russian: Владимир Гершонович Дринфельд, Ukrainian: Володимир Гершонович Дрінфельд; born February 14, 1954 in Kharkiv) is a Ukrainian and Soviet mathematician currently working in the USA. The work of Drinfel'd related 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. Drinfel'd introduced the notion of a quantum group (independently discovered by Michio Jimbo at the same time) and made important contributions into mathematical physics, including the ADHM construction of instantons, algebraic formalism of the Quantum Inverse Scattering Method, and the Drinfel'd–Sokolov reduction in the theory of solitons. He was awarded the Fields Medal in 1990.

Biography

In 1969, at the age of 15, Vladimir Drinfel'd 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 entered Moscow State University in the same year, graduating from it in 1974. Drinfel'd was awarded the Candidate of Sciences degree in 1978 and Doctor of Sciences degree from the Steklov Mathematical Institute in 1988. In 1981 Drinfel'd became a researcher at the Institute of Physics of Low Temperatures of the National Academy of Sciences of Ukraine in Kharkiv. Currently, Drinfel'd is the Harry Pratt Judson Distinguished Service Professor at the University of Chicago. In 1992 Drinfel'd was elected a corresponding member of the National Academy of Sciences of Ukraine. Vladimir Drinfel'd was awarded the Fields Medal in 1990.

Mathematical contributions

In 1974, at the age of twenty, Drinfel'd announced a proof of the Langlands conjectures for GL₂ over a global field of positive characteristic. In the course of proving the conjectures, Drinfel'd introduced a new class of objects that he called Elliptic modules and that have since become known also as Drinfel'd modules. Later, in 1983, Drinfel'd 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. Drinfel'd 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.

Drinfel'd later moved to 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. In 1986, he gave a seminal address to the International Congress of Mathematicians at Berkeley, where he 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 Drinfel'd 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.

Drinfel'd has also collaborated with Alexander Beilinson to rebuild the theory of vertex algebras which have become increasingly important to conformal field theory, string theory and the geometric Langlands program. A manuscript circulated for many years, and the work, titled Chiral Algebras, finally appeared in a book form in 2004.