Vladimir Drinfel'd

Encyclopedia

**Vladimir Gershonovich Drinfel'd**' onMouseout='HidePop("78323")' href="/topics/Kharkiv">Kharkiv

Kharkiv

Kharkiv or Kharkov is the second-largest city in Ukraine.The city was founded in 1654 and was a major centre of Ukrainian culture in the Russian Empire. Kharkiv became the first city in Ukraine where the Ukrainian Soviet Socialist Republic was proclaimed in December 1917 and Soviet government was...

) is a Ukrainian

Ukraine

Ukraine is a country in Eastern Europe. It has an area of 603,628 km², making it the second largest contiguous country on the European continent, after Russia...

and (formerly) Soviet

Soviet Union

The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....

mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

at the University of Chicago

University of Chicago

The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...

.

The work of Drinfeld related algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

over finite field

Finite field

In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

s with number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, especially the theory of automorphic form

Automorphic form

In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...

s, through the notions of elliptic module and the theory of the geometric Langlands correspondence. Drinfeld introduced the notion of a quantum group

Quantum group

In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...

(independently discovered by Michio Jimbo

Michio Jimbo

is a Japanese mathematician, currently a professor at the University of Tokyo. He is a grandson of the linguist Kaku Jimbo.After graduating from the University of Tokyo in 1974, he studied under Mikio Sato at the Research Institute for Mathematical Sciences in Kyoto University...

at the same time) and made important contributions into mathematical physics

Mathematical physics

Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

, including the ADHM construction

ADHM construction

The ADHM construction or monad construction is the construction of all instantons using method of linear algebra by Michael Atiyah, Vladimir G. Drinfel'd, Nigel. J. Hitchin, Yuri I...

of instanton

Instanton

An instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory...

s, algebraic formalism of the Quantum Inverse Scattering Method, and the Drinfeld–Sokolov reduction in the theory of soliton

Soliton

In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...

s. He was awarded the Fields Medal

Fields Medal

The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...

in 1990.

## Biography

In 1969, at the age of 15, Vladimir Drinfeld represented the Soviet UnionSoviet Union

The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....

at the International Mathematics Olympiad in Bucharest

Bucharest

Bucharest is the capital municipality, cultural, industrial, and financial centre of Romania. It is the largest city in Romania, located in the southeast of the country, at , and lies on the banks of the Dâmbovița River....

, Romania

Romania

Romania is a country located at the crossroads of Central and Southeastern Europe, on the Lower Danube, within and outside the Carpathian arch, bordering on the Black Sea...

, and won a gold medal with the full score of 40 points. He entered Moscow State University

Moscow State University

Lomonosov Moscow State University , previously known as Lomonosov University or MSU , is the largest university in Russia. Founded in 1755, it also claims to be one of the oldest university in Russia and to have the tallest educational building in the world. Its current rector is Viktor Sadovnichiy...

in the same year, graduating from it in 1974. Drinfeld was awarded the Candidate of Sciences degree in 1978 and Doctor of Sciences degree from the Steklov Mathematical Institute in 1988. In 1981 Drinfeld became a researcher at the Kharkiv Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine in Kharkiv

Kharkiv

Kharkiv or Kharkov is the second-largest city in Ukraine.The city was founded in 1654 and was a major centre of Ukrainian culture in the Russian Empire. Kharkiv became the first city in Ukraine where the Ukrainian Soviet Socialist Republic was proclaimed in December 1917 and Soviet government was...

. Currently, Drinfeld is the Harry Pratt Judson

Harry Pratt Judson

Harry Pratt Judson was a U.S. educator and historian, born at Jamestown, N. Y., and educated at Williams College , where he was a brother of the Delta Kappa Epsilon fraternity...

Distinguished Service Professor

Professor

A professor is a scholarly teacher; the precise meaning of the term varies by country. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences; a teacher of high rank...

at the University of Chicago

University of Chicago

The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...

. In 1992 Drinfeld was elected a corresponding member of the National Academy of Sciences of the Ukraine. Vladimir Drinfeld was awarded the Fields Medal

Fields Medal

The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...

in 1990.

## Mathematical contributions

In 1974, at the age of twenty, Drinfeld announced a proof of the Langlands conjectures for GL_{2}over a global field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

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 Drinfel'd modules

Drinfel'd module

In mathematics, a Drinfel'd 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 complex multiplication theory...

. 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

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 extension L/K...

. It postulated the existence of a natural one-to-one correspondence between Galois representations and some automorphic form

Automorphic form

In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...

s. 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-module

D-module

In mathematics, a D-module is a module over a 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. "Automorphicity" of these modules and the Langlands correspondence could be then understood in terms of the action of Hecke operators.

Drinfeld later moved to mathematical physics

Mathematical physics

Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

. In collaboration with his advisor Yuri Manin

Yuri I. Manin

Yuri Ivanovitch Manin is a Soviet/Russian/German mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics.-Biography:...

, he constructed the moduli space of Yang–Mills instantons, a result which was proved independently by Michael Atiyah

Michael Atiyah

Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...

and Nigel Hitchin

Nigel Hitchin

Nigel Hitchin is a British mathematician working in the fields of differential geometry, algebraic geometry, and mathematical physics.-Academic career:...

. In 1986, he gave a seminal address to the International Congress of Mathematicians

International Congress of Mathematicians

The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....

at Berkeley, where he coined the term "Quantum group

Quantum group

In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...

" in reference to Hopf algebra

Hopf algebra

In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...

s 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

R-matrix

The term R-matrix has several meanings, depending on the field of study.The term R-matrix is used in connection with the Yang–Baxter equation. This is an equation which was first introduced in the field of statistical mechanics, taking its name from independent work of C. N. Yang and R. J....

corresponding to the solution of the Yang–Baxter equation associated with a quasitriangular Hopf algebra.

Drinfeld has also collaborated with Alexander Beilinson

Alexander Beilinson

Alexander A. Beilinson 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...

to rebuild the theory of vertex algebras which have become increasingly important to conformal field theory

Conformal field theory

A conformal field theory is a quantum field theory that is invariant under conformal transformations...

, string theory

String theory

String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...

and the geometric Langlands program

Langlands program

The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....

. A manuscript circulated for many years, and the work, titled

*Chiral Algebras*, finally appeared in a book form in 2004.