Vladimir Gershonovich Drinfeld (Ukrainian:Володи́мир Ге́ршонович Дрінфельд; born February 14, 1954), surname also romanized asDrinfel'd, is a mathematician from Ukraine, who immigrated to the United States and works at theUniversity of Chicago.
Drinfeld's work connectedalgebraic geometry overfinite fields withnumber theory, especially the theory ofautomorphic forms, through the notions ofelliptic module and the theory of thegeometric Langlands correspondence. Drinfeld introduced the notion of aquantum group (independently discovered byMichio Jimbo at the same time) and made important contributions tomathematical physics, including theADHM construction ofinstantons, algebraic formalism of thequantum inverse scattering method, and the Drinfeld–Sokolov reduction in the theory ofsolitons.
He was awarded theFields Medal in 1990.[1]In 2016, he was elected to theNational Academy of Sciences.[2] In 2018 he received theWolf Prize in Mathematics.[3] In 2023 he was awarded theShaw Prize in Mathematical Sciences.[4]
Drinfeld was born into aJewish[5] mathematical family, inKharkiv,Ukrainian SSR,Soviet Union in 1954. In 1969, at the age of 15, Drinfeld represented the Soviet Union at theInternational Mathematics Olympiad inBucharest, Romania, and won a gold medal with the full score of 40 points. He was, at the time, theyoungest participant to achieve a perfect score, a record that has since been surpassed by only four others includingSergei Konyagin andNoam Elkies. Drinfeld enteredMoscow State University in the same year and graduated from it in 1974. Drinfeld was awarded theCandidate of Sciences degree in 1978 and theDoctor of Sciences degree from theSteklov Institute of Mathematics in 1988. He was awarded theFields Medal in 1990. From 1981 until 1999, he worked at theVerkin Institute for Low Temperature Physics and Engineering (Department of Mathematical Physics). Drinfeld moved to the United States in 1999 and has been working at theUniversity of Chicago since January 1999.
In 1974, at the age of twenty, Drinfeld announced a proof of theLanglands conjectures forGL2 over aglobal 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 asDrinfeld 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 ofnon-abelian class field theory. It postulated the existence of a natural one-to-one correspondence betweenGalois representations and someautomorphic forms. The "naturalness" is guaranteed by the essential coincidence ofL-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 automorphicperverse sheaves or automorphicD-modules. "Automorphicity" of these modules and the Langlands correspondence could be then understood in terms of the action ofHecke operators.
Drinfeld has also worked inmathematical physics. In collaboration with his advisorYuri Manin, he constructed themoduli space ofYang–Millsinstantons, a result that was proved independently byMichael Atiyah andNigel Hitchin. Drinfeld coined the term "quantum group" in reference toHopf algebras that are deformations ofsimple Lie algebras, and connected them to the study of theYang–Baxter equation, which is a necessary condition for the solvability of statistical mechanical models. He also generalized Hopf algebras toquasi-Hopf algebras and introduced the study ofDrinfeld twists, which can be used to factorize theR-matrix corresponding to the solution of the Yang–Baxter equation associated with aquasitriangular Hopf algebra.
Drinfeld has also collaborated withAlexander Beilinson to rebuild the theory ofvertex algebras in a coordinate-free form, which have become increasingly important totwo-dimensional conformal field theory,string theory, and thegeometric Langlands program. Drinfeld and Beilinson published their work in 2004 in a book titled "Chiral Algebras."[6]
