Vladimir Drinfeld - Contributions To Mathematics

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" and that have since become known also as "shtukas" and 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."