**Vladimir Drinfeld** was born into a Jewish mathematical family. Gershon Ikhelevich Drinfeld (29 February 1908-18 August 2000) was educated at Kiev University and was head of the Mathematics Department at Kharkov University from 1944 to 1962. By 1950 he was deputy director of the Kharkov Institute of Mathematics but it was closed in that year on the orders of Stalin. Gershon Drinfeld also played a major role in the Kharkov Mathematical Society. He worked on differential geometry, particularly on measure and integration.

Vladimir Drinfeld's mathematical career started early ([11] or [12]):-

Drinfeld has written his first published paper when he was a schoolboy. He proved there a nice result in the style of Hardy's classic treatise "Inequalities" and solved a problem to which R A Rankin devoted two notes. This paper still makes interesting reading.

In 1969, at the age of fifteen, he represented the Soviet Union at the International Mathematical Olympiad in Bucharest, Romania, and was awarded a gold medal after obtaining full marks - an incredible achievement. He studied at Moscow State University from 1969 until 1974. He graduated in 1974 and remained at Moscow University to undertake research under Yuri Ivanovich Manin's supervision. Ginzburg writes [5]:-

[

Drinfeld's]vision of mathematics was, to a great extent, influenced by Yu I Manin, his advisor, and by the Algebraic Geometry Seminar(Manin's Seminar)that functioned with regularity at Moscow State University for about two decades.

Drinfeld completed his postgraduate studies in 1977 and he defended his "candidate" thesis in 1978 at Moscow University. The "candidate" thesis is the Russian equivalent of the British or American Ph.D. However, despite being extraordinarily talented, it was difficult for Drinfeld to obtain a position in Moscow. There were basically two reasons for this. Certainly his Jewish origins meant that he suffered from anti-Semitism, but officially the Soviet Union operated a policy that people had their addresses in their passports and were only allowed to work in the town which appeared in this address. Since the address which appeared in Drinfeld's passport was not Moscow, he could not get a job there. He went to Ufa, an industrial centre in the Ural mountains, where he obtained a position teaching mathematics at Bashkir University, one of several universities in the city. In 1981 he moved to Kharkov and lived with his parents. He obtained a position working at the B I Verkin Physical Engineering Institute of Low Temperatures of the National, part of the Ukrainian Academy of Sciences, in Kharkov.

Drinfeld gave an important lecture at the International Congress of Mathematicians in Berkeley in 1986. Entitled *Quantum groups*, the talk reviewed the results obtained by Drinfeld and M Jimbo on Hopf algebras (quantum groups). He discussed the concepts of quantum groups and quantization, and also talked about Poisson groups, Lie bi-algebras and the classical Yang-Baxter equation. In 1988 Drinfeld defended his "doctor" thesis at Steklov Institute, Moscow. The "doctor" thesis is the Russian equivalent of the German habilitation. On 21 August 1990 Drinfeld was awarded a Fields Medal at the International Congress of Mathematicians in Kyoto, Japan:-

... for his work on quantum groups and for his work in number theory.

A Jaffe and B Mazur write in [2] about Drinfeld's work which led to the award of the Fields Medal:-

Drinfeld's interests can only be described as "broad". Not only do they span work in algebraic geometry and number theory, but his most recent ideas have taken a strikingly different direction: he has been doing significant work on mathematical questions motivated by physics, including the relatively new theory of quantum groups.

Drinfeld defies any easy classification ... His breakthroughs have the magic that one would expect of a revolutionary mathematical discovery: they have seemingly inexhaustible consequences. On the other hand, they seem deeply personal pieces of mathematics: "only Drinfeld could have thought of them!" But contradictorily they seem transparently natural; once understood, "everyone should have thought of them!"

Manin ends his address to the International Congress of Mathematicians in Kyoto, Japan (which he could not give in person but was read by Michio Jimbo) with these words:-

I hope that I conveyed to you some sense of broadness, conceptual richness, technical strength and beauty of Drinfeld's work for which we are now honouring him with the Fields Medal. For me, ie was a pleasure and a privilege to observe at a close distance the rapid development of this brilliant mind which taught me so much.

Drinfeld's main achievements are his proof of the Langlands conjecture for *GL*(2) over a functional field; and his work in quantum group theory. Although he only proved a special case of the Langlands conjecture, Drinfeld has introduced important new ideas in his solution and made a real breakthrough. He introduced the idea of an elliptic module in his proof and this notion is leading to a whole new topic within number theory. The interactions between mathematics and mathematical physics studied by Atiyah led to the introduction of instantons - solutions, that is, of a certain nonlinear system of partial differential equations, the self-dual Yang-Mills equations, which were originally introduced by physicists in the context of quantum field theory. Drinfeld and Manin worked on the construction of instantons using ideas from algebraic geometry.

Chari and Thakur write [3]:-

Drinfeld introduced Drinfeld modules and solved a substantial part of the Langlands programme when he was just20years old and completed the GL(2)case when he was24. Drinfeld's work on Langlands conjectures, quantum groups, p-adic uniformizations etc. illustrate his mastery over powerful and involved techniques. On the other hand, his one page proof(jointly with Vladut)giving a sharp asymptotic upper bound for the number of points of a curve defined over a finite field of order p(2^{})^{n}, uses only high-school algebra applied nicely to well-known results. He also gave a one page proof of the fact that any rotation invariant finitely additive measure on the two or three dimensional sphere is proportional to Lebesgue measure by using a clever combination of known results.

In 1992 Drinfeld was elected a member of the Ukrainian Academy of Sciences. He continued to live in Kharkov until 1998 when he emigrated to the United States. In December 1998, he was appointed to the University of Chicago. On Drinfeld's appointment to Chicago, Manin said [10]:-

Drinfeld's work deeply influenced the world of mathematics of the last two decades, Several research monographs, Seminar Notes and hundreds of papers were dedicated to the two new chapters of mathematics created by him - the so-called Drinfeld modules and quantum groups.

Alexander A Beilinson, also a student of Manin's, was appointed to the University of Chicago in 1998, just a short time before Drinfeld. Beilinson and Drinfeld had known each other for many years and had already collaborated on two papers before becoming colleagues in Chicago: *Affine Kac-Moody algebras and polydifferentials* (1994) and *Quantization of Hitchin's fibration and Langlands' program* (1996). Their collaboration in Chicago led to the publication of a jointly authored book *Chiral algebras* published by the American Mathematical Society in 2004. Francisco J Plaza Martin writes in a review:-

This book presents a comprehensive approach to the theory of chiral algebras from the point of view of algebraic geometry. Without a doubt, it will become a standard reference on the subject. ... Chiral algebras arose in mathematical physics in the study of conformal field theory. On the mathematical side, the local theory of chiral algebras overlaps the theory of vertex algebras[R E Borcherds], which are normally studied with representation theory techniques. In these two approaches the "operator product expansion" formalism plays an essential role. As the authors say, their motivation for studying chiral algebras was the understanding of geometric automorphic forms in theD-module setting as well as the description of a spectral decomposition of the category of representations of an affine Kac-Moody algebra.

One of Drinfeld's most recent articles is *Infinite-dimensional vector bundles in algebraic geometry: an introduction*. Drinfeld writes in the introduction to the paper:-

The goal of this work is to show that there is a reasonable algebro-geometric notion of vector bundle with infinite-dimensional locally linearly compact fibers and that these objects appear 'in nature'. Our approach is based on some results and ideas discovered in algebra during the period1958-1972by H Bass, L Gruson, I Kaplansky, M Karoubi, and M Raynaud.

Drinfeld was named Harry Pratt Judson Distinguished Service Professor at the University of Chicago on 1 March 2001. In 2008 he was elected to the American Academy of Arts and Sciences.

**Article by:** *J J O'Connor* and *E F Robertson*

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