Vladimir Drinfel'd

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Vladimir Drinfel'd
Image:Replace this image male.svg Vladimir Drinfel'd
Born	February 04, 1954
Nationality	Russia
Field	Mathematician
Known for	Drinfel'd module
Notable awards	Fields Medal in 1990

Vladimir Gershonovich Drinfel'd (Russian: Владимир Гершонович Дринфельд) is a mathematician born February 4, 1954 in the Ukrainian SSR.

At the age of 15 he won a gold medal with the perfect score in the International Mathematics Olympiad in 1969, representing the Soviet Union, and entered the Moscow State University the same year, graduating from it in 1974.

Mathematical Contributions

In 1974, Drinfeld announced a proof of the Langlands conjectures for GL₂ over a global field of finite characteristic. In his effort to prove the conjectures, Drinfeld introduced a new class of modular spaces known as Drinfel'd modules. These modules have since become a central topic in number theory. Drinfeld gave a detailed investigation of these objects in his papers Elliptical Modules and Elliptical Modules II.

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 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 which have become increasingly important to conformal field theory, string theory and the geometric Langlands program. This work circulated unpublished for many years, and was recently published in 2004. The monograph is titled as simply Chiral Algebras.

Presently, Drinfel'd, jointly with Mitya Boyarchenko, is trying to develop the theory of character sheaves for unipotent groups. This work was inspired by George Lusztig's work on reductive groups.

For his mathematical achievements, Drinfeld was awarded a Fields Medal in 1990.

Currently, Drinfel'd is the Harry Pratt Judson Distinguished Service Professor at the University of Chicago. Together with Alexander Beilinson, he runs a seminar covering topics such as the Langlands program, algebraic geometry and representation theory.