Vladimir Gershonovich Drinfeld (; born February 14, 1954), surname also romanized as Drinfel'd, is a

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

from

Ukraine Ukraine is a country in Eastern Europe. It is the List of European countries by area, second-largest country in Europe after Russia, which Russia–Ukraine border, borders it to the east and northeast. Ukraine also borders Belarus to the nor ...

, who immigrated to the United States and works at the

University of Chicago The University of Chicago (UChicago, Chicago, or UChi) is a Private university, private research university in Chicago, Illinois, United States. Its main campus is in the Hyde Park, Chicago, Hyde Park neighborhood on Chicago's South Side, Chic ...

. Drinfeld's work connected

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

over

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

s with

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, especially the theory of

automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G o ...

s, through the notions of elliptic module and the theory of the

geometric Langlands correspondence In mathematics, the geometric Langlands correspondence relates algebraic geometry and representation theory. It is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing in the original number theoretic ...

. Drinfeld introduced the notion of a

quantum group In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebra ...

(independently discovered by

Michio Jimbo is a Japanese mathematician working in mathematical physics and is a professor of mathematics at Rikkyo University. He is a grandson of the linguist . Career After graduating from the University of Tokyo in 1974, he studied under Mikio Sato a ...

at the same time) and made important contributions to

mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...

, including the

ADHM construction In mathematical physics and gauge theory, the ADHM construction or monad construction is the construction of all instantons using methods of linear algebra by Michael Atiyah, Vladimir Drinfeld, Nigel Hitchin, Yuri I. Manin in their paper "Construc ...

instanton An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. M ...

s, algebraic formalism of the

quantum inverse scattering method In quantum physics, the quantum inverse scattering method (QISM), similar to the closely related algebraic Bethe ansatz, is a method for solving integrable models in 1+1 dimensions, introduced by Leon Takhtajan and L. D. Faddeev in 1979. It can ...

, and the Drinfeld–Sokolov reduction in the theory of

soliton In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such local ...

s. He was awarded the

Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...

in 1990. In 2016, he was elected to the

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...

. In 2018 he received the

Wolf Prize in Mathematics The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts. ...

. In 2023 he was awarded the

Shaw Prize The Shaw Prize is a set of three annual awards presented by the Shaw Prize Foundation in the fields of astronomy, medicine and life sciences, and mathematical sciences. Established in 2002 in Hong Kong, by Hong Kong entertainment mogul and p ...

in Mathematical Sciences.

Biography

Drinfeld was born into a

Jewish Jews (, , ), or the Jewish people, are an ethnoreligious group and nation, originating from the Israelites of History of ancient Israel and Judah, ancient Israel and Judah. They also traditionally adhere to Judaism. Jewish ethnicity, rel ...

Vladimir Gershonovich Drinfeld
/ref> mathematical family, in

Kharkiv Kharkiv, also known as Kharkov, is the second-largest List of cities in Ukraine, city in Ukraine.

Ukrainian SSR The Ukrainian Soviet Socialist Republic, abbreviated as the Ukrainian SSR, UkrSSR, and also known as Soviet Ukraine or just Ukraine, was one of the Republics of the Soviet Union, constituent republics of the Soviet Union from 1922 until 1991. ...

Soviet Union The Union of Soviet Socialist Republics. (USSR), commonly known as the Soviet Union, was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 until Dissolution of the Soviet ...

in 1954. In 1969, at the age of 15, Drinfeld represented the

at the

International Mathematics Olympiad The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre-university students, and is the oldest of the International Science Olympiads. It is widely regarded as the most prestigious mathematical competition in the world ...

Bucharest Bucharest ( , ; ) is the capital and largest city of Romania. The metropolis stands on the River Dâmbovița (river), Dâmbovița in south-eastern Romania. Its population is officially estimated at 1.76 million residents within a greater Buc ...

Romania Romania is a country located at the crossroads of Central Europe, Central, Eastern Europe, Eastern and Southeast Europe. It borders Ukraine to the north and east, Hungary to the west, Serbia to the southwest, Bulgaria to the south, Moldova to ...

, 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, a record that has since been surpassed by only four others including

Sergei Konyagin Sergei Vladimirovich Konyagin (; born 25 April 1957) is a Russian mathematician. He is a professor of mathematics at the Moscow State University. His primary research interest is in applying harmonic analysis to number theoretic settings. Konyagi ...

and

Noam Elkies Noam David Elkies (born August 25, 1966) is a professor of mathematics at Harvard University. At age 26, he became the youngest professor to receive tenure at Harvard. He is also a pianist, chess national master, and chess composer. Early life ...

. Drinfeld entered

Moscow State University Moscow State University (MSU), officially M. V. Lomonosov Moscow State University,. is a public university, public research university in Moscow, Russia. The university includes 15 research institutes, 43 faculties, more than 300 departments, a ...

in the same year and graduated from it in 1974. Drinfeld was awarded the

Candidate of Sciences A Candidate of Sciences is a Doctor of Philosophy, PhD-equivalent academic research degree in all the post-Soviet countries with the exception of Ukraine, and until the 1990s it was also awarded in Central and Eastern European countries. It is ...

degree in 1978 and the

Doctor of Sciences A Doctor of Sciences, abbreviated д-р наук or д. н.; ; ; ; is a higher doctoral degree in the Russian Empire, Soviet Union and many Commonwealth of Independent States countries. One of the prerequisites of receiving a Doctor of Sciences ...

degree from the

Steklov Institute of Mathematics Steklov Institute of Mathematics or Steklov Mathematical Institute () is a premier research institute based in Moscow, specialized in mathematics, and a part of the Russian Academy of Sciences. The institute is named after Vladimir Andreevich Stek ...

in 1988. He was awarded the

in 1990. From 1981 till 1999 he worked at the

Verkin Institute for Low Temperature Physics and Engineering The B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine () is a research institute that conducts basic research in experimental and theoretical physics, mathematics, as well as in the fi ...

(Department of Mathematical Physics). Drinfeld moved to the

United States The United States of America (USA), also known as the United States (U.S.) or America, is a country primarily located in North America. It is a federal republic of 50 U.S. state, states and a federal capital district, Washington, D.C. The 48 ...

in 1999 and has been working at the

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 In mathematics, a global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields: *Algebraic number field: A finite extension of \mathbb *Global functio ...

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 module In mathematics, a Drinfeld module (or elliptic module) is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of comple ...

s). 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 In mathematics, non-abelian class field theory is a catchphrase, meaning the extension of the results of class field theory, the relatively complete and classical set of results on abelian extensions of any number field ''K'', to the general Galois ...

. It postulated the existence of a natural one-to-one correspondence between

Galois representations In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring ...

and some

s. The "naturalness" is guaranteed by the essential coincidence of

L-functions In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may gi ...

. 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 The mathematical term perverse sheaves refers to the objects of certain abelian categories associated to topological spaces, which may be a real or complex manifold, or more general topologically stratified spaces, possibly singular. The concept w ...

or automorphic

D-module In mathematics, a ''D''-module is a module (mathematics), module over a ring (mathematics), ring ''D'' of differential operators. The major interest of such ''D''-modules is as an approach to the theory of linear partial differential equations. S ...

s. "Automorphicity" of these modules and the Langlands correspondence could be then understood in terms of the action of

Hecke operators In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic rep ...

. Drinfeld has also worked in

. In collaboration with his advisor

Yuri Manin Yuri Ivanovich Manin (; 16 February 1937 – 7 January 2023) was a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. Life an ...

, he constructed the

moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...

of Yang–Mills

instantons An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...

, a result that was proved independently by

Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the ...

and

Nigel Hitchin Nigel James Hitchin FRS (born 2 August 1946) is a British mathematician working in the fields of differential geometry, gauge theory, algebraic geometry, and mathematical physics. He is a Professor Emeritus of Mathematics at the University of O ...

. Drinfeld coined the term "

" in reference to

Hopf algebra In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover ...

s that are deformations of

simple Lie algebra In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan. A direct sum of ...

s, and connected them to the study of the

Yang–Baxter equation In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve their ...

, which is a necessary condition for the solvability of statistical mechanical models. He also generalized Hopf algebras to

quasi-Hopf algebra A quasi-Hopf algebra is a generalization of a Hopf algebra, which was defined by the Russian mathematician Vladimir Drinfeld in 1989. A ''quasi-Hopf algebra'' is a quasi-bialgebra \mathcal = (\mathcal, \Delta, \varepsilon, \Phi) for which there ex ...

s and introduced the study of

Drinfeld twist In mathematics, a Hopf algebra, ''H'', is quasitriangularMontgomery & Schneider (2002), p. 72 if there exists an invertible element, ''R'', of H \otimes H such that :*R \ \Delta(x)R^ = (T \circ \Delta)(x) for all x \in H, where \Delta is the cop ...

s, which can be used to factorize the R-matrix corresponding to the solution of the Yang–Baxter equation associated with a

quasitriangular Hopf algebra In mathematics, a Hopf algebra, ''H'', is quasitriangularMontgomery & Schneider (2002), p. 72 if there exists an invertible element, ''R'', of H \otimes H such that :*R \ \Delta(x)R^ = (T \circ \Delta)(x) for all x \in H, where \Delta is the cop ...

. Drinfeld has also collaborated with

Alexander Beilinson Alexander A. Beilinson (born 1957) is the David and Mary Winton Green University professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics. In 1 ...

to rebuild the theory of vertex algebras in a coordinate-free form, which have become increasingly important to

two-dimensional conformal field theory A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations. In contrast to other types of conformal field theories, two-dimensional conformal fi ...

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

, and the

geometric Langlands program In mathematics, the geometric Langlands correspondence relates algebraic geometry and representation theory. It is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing in the original number theoretic ...

. Drinfeld and Beilinson published their work in 2004 in a book titled "Chiral Algebras."

Notes

References

* *

Victor Ginzburg Victor Ginzburg (born 1957) is a Russian American mathematician who works in representation theory and in noncommutative geometry. He is known for his contributions to geometric representation theory, especially, for his works on representations ...

, 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,
Report by Manin

External links

* *
Langlands Seminar homepage
{{DEFAULTSORT:Drinfeld, Vladimir 1954 births 20th-century Ukrainian mathematicians 21st-century Ukrainian mathematicians Moscow State University alumni Fields Medalists Living people Algebraic geometers Number theorists Soviet mathematicians Ukrainian Jews Scientists from Kharkiv International Mathematical Olympiad participants University of Chicago faculty Institute for Advanced Study visiting scholars Members of the United States National Academy of Sciences Corresponding members of the National Academy of Sciences of Ukraine Russian scientists Wolf Prize in Mathematics laureates

Biography

Contributions to mathematics

See also

Notes

References

External links