Vladimir Gershonovich Drinfeld
BiographyVladimir Drinfeld was born into a Jewish mathematical family. He was the son of the mathematics professor Gershon Ikhelevich Drinfeld (29 February 1908-18 August 2000) and his wife, the classical philologist Frida Iosifovna Lutskaya-Litvak (1921-2011). Since Vladimir's father was a professor of mathematics at Kharkov University, we shall give some details of his life before giving biographical details of his son Vldimir.
Gershon Ikhelevich Drinfeld was born in Starokostiantyniv, Ukraine and showed remarkable mathematical talents from a young age. His teacher, Pavel Maksimovich Semenov, taught through the civil war and encourage Gershon Ikhelevich's passion for mathematics in every possible way. After graduating from seven-year school in 1922, G I Drinfeld was a shoemaker's apprentice and a sawmill worker before he entered Kiev University (then the Kiev Institute of Public Education) in 1927. There he was taught by Mikhail Pylypovych Kravchuk (1892-1942) who invited him to take part in his seminar and introduced him to independent scientific work. His research was supervised by Georgii Vasilovich Pfeiffer and he graduated from the Kiev Institute in 1931. He became head of the Mathematics Department at Kharkov University from 1944 to 1962 but during World War II he worked for the Academy of Sciences of the Ukrainian SSR which was evacuated to Bashkiria. 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 theory and integration.
Vladimir Drinfeld was a child prodigy and Svetlana Jitomirskaya said he :-
... could do maths at age six that really made people's jaws drop.Drinfeld studied at Kharkov Physics and Mathematics School no. 27, a specialised school for talented pupils, and his mathematical career started while he was at this school ( or ):-
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.He submitted this paper, A cyclic inequality (Russian), to Matematicheskie Zametki on 24 November 1969. It was published in February 1971 and, in the same year, an English translation was published in Maths Notes.
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, namely 40 points - an incredible achievement. At that time, he was the youngest competitor to achieve the highest score but three others have achieved this since. He studied at Moscow State University from 1969 until 1974. We note that he was only fifteen years of age when he entered the university. In the interview  he spoke about the difficulties of being a Jew in the USSR at this time:-
... for many years (roughly 1948-1987) anti-Semitism was the (unpublicised) government policy, and not the initiative of individuals. The rector of Moscow State University I G Petrovsky and many others resisted this policy (this required a lot of effort from them). On the other hand, there were influential mathematicians (for example, the then director of the Steklov Institute) who added their personal anti-Semitism to the state one.He graduated in 1974 and remained at Moscow State University to undertake research under Yuri Ivanovich Manin's supervision. Ginzburg writes :-
[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. We note that by 1978, Drinfeld had thirteen papers in print and had proved remarkable results concerning the Langlands conjectures. Despite being extraordinarily talented, however, 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, as he himself described in the above quote, 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 was 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, part of the National Ukrainian Academy of Sciences, in Kharkov. This required much effort from several colleagues :-
I ultimately survived thanks to the fact that in 1981 I was hired at the Kharkov Institute of Low Temperatures. It was not easy: although V A Marchenko and other mathematicians from this institute wanted to take me, and the director of the institute B I Verkin was not against it, but a letter from N N Bogolyubov to the director was also needed in order to protect Verkin from the all-powerful regional committee of the CPSU (and to organise Bogolyubov's letter, it took the efforts of my scientific supervisor Yu I Manin and other people).The appointment, however, had its problems :-
While working at the Institute of Low Temperatures, we had to work on a collective farm in the summer (in 1984 this had to be done for 40 days). This was difficult for me for two reasons. Firstly, I'm not physically strong, and secondly, it was just annoying: I have my own mathematics lesson plan, and the regional committee secretary calls the institute and, well, I think the experimental scientists were as infuriated by all this as I was.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 quantisation, and also talked about Poisson groups, Lie bi-algebras and the classical Yang-Baxter equation. The talk began as follows:-
This is a report on recent works on Hopf algebras (or quantum groups, which is more or less the same) motivated by the quantum inverse scattering method (QISM), a method for constructing and studying integrable quantum systems, which was developed mostly by L D Faddeev and his collaborators. Most of the definitions, constructions, examples, and theorems in this paper are inspired by the QISM. Nevertheless I will begin with these definitions, constructions, etc. and then explain their relation to the QISM. Thus I reverse the history of the subject, hoping to make its logic clearer.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 :-
What is a quantum group? Recall that both in classical and in quantum mechanics there are two basic concepts: state and observable. In classical mechanics states are points of a manifold M and observables are functions on M. In the quantum case states are l-dimensional subspaces of a Hilbert space H and observables are operators in H (we forget the self-adjointness condition). The relation between classical and quantum mechanics is easier to understand in terms of observables. Both in classical and in quantum mechanics observables form an associative algebra which is commutative in the classical case and non-commutative in the quantum case. So quantisation is something like replacing commutative algebras by non-commutative ones.
... for his work on quantum groups and for his work in number theory.A Jaffe and B Mazur write in  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.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 :-
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!"
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, it was a pleasure and a privilege to observe at a close distance the rapid development of this brilliant mind which taught me so much.For more extracts from Manin's address, see THIS LINK.
Drinfeld's main achievements are his proof of the Langlands conjecture for 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. Drinfeld said :-
One of my works (joint with M F Atiyah, Yu I Manin and N J Hitchin) was devoted to the so-called instantons. This was one of the first examples of something useful for theoretical physics being done in algebraic geometry, which impressed physicists. Physicists are smart people; they can count (not always using strict methods) much better than mathematicians, and it is difficult to surprise them with anything. But after our work they realised that they needed algebraic geometry, and after 10 years most physicists had learned it. People working in string theory now know algebraic geometry. In the 1980s I worked on mathematical objects called quantum groups. These objects were invented under the influence of the theory of quantum integrable systems, which was started by physicists and then developed by mathematicians of the Leningrad school of Ludwig Dmitrievich Faddeev. I tried to understand their work, which was not easy. Then I realised that the key role there is played by Hopf algebras, which can be called quantum groups. This approach helped the understanding of many previously obtained results and also obtain a number of new ones. Many people working in this field liked this approach and began to use it. Quantum groups have come into use among mathematical physicists.Chari and Thakur write :-
Drinfeld introduced Drinfeld modules and solved a substantial part of the Langlands programme when he was just 20 years old and completed the case when he was 24. 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 , 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. He spoke about the move to the West in the interview :-
In 1990, I already worked in Kharkov (Physico-Technical Institute of Low Temperatures of the National Academy of Sciences of Ukraine), then I had the opportunity to get a job in the West, but I refused. In 1998, I received several job offers from American universities, and my wife and I decided that we should accept one of them. We lived in Ukraine, whose economy was going downhill (for example, people received advances on their salaries six months late). Our son was nine years old, and we wondered what kind of world he would live in when he grew up. It was clear that this was a world of wild capitalism. We decided to move to a country of more civilised capitalism. It was not easy, since I had elderly parents, I could not leave without them. The University of Chicago was able to employ my mother and provide her with health insurance.On Drinfeld's appointment to Chicago, colleagues expressed their delight :-
Barry Mazur, professor of mathematics at Harvard University, upon learning that Drinfeld had accepted a position at Chicago, said "It's a wonderful appointment." Mazur said he regards Drinfeld and Beilinson as Russia's two most influential mathematicians. "There's no question that Chicago has achieved a great coup there. These are great mathematicians," Mazur said.Alexander A Beilinson, also a student of Manin's, had been 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 :-
A Fields Medal is the equivalent of a Nobel Prize in mathematics, according to Robert Fefferman, Chairman of the Mathematics Department and the Louis Block Professor in Mathematics. The medals are awarded to no fewer than two and no more than four mathematicians under the age of 40 every four years at the International Congress of Mathematicians. Fefferman called Drinfeld "one of the greatest algebraists in the world."
Yuri Manin, director of the Max Planck Institute for Mathematics in Bonn, Germany, offered an equally strong assessment. "Drinfeld's work deeply influenced the world of mathematics of the last two decades," said Manin, who served as Drinfeld's and Beilinson's Ph.D. thesis adviser at Moscow University in the 1980s and was the chairman of the Fields Prize Committee at the Berlin ICM 1998. "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."
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 the D-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 later 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 period 1958-1972 by 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. He was elected to the National Academy of Sciences in 2016.
In 2018 Drinfeld and Beilinson were jointly awarded the Wolf Prize :-
... for their ground-breaking work in algebraic geometry (a field that integrates abstract algebra with geometry), in mathematical physics and in presentation theory, a field which helps to understand complex algebraic structures.The citation continues :-
Drinfeld has contributed greatly to various branches of pure mathematics, mainly algebraic geometry, arithmetic geometry and the theory of representation - as well as mathematical physics. The mathematical objects named after him - the "Drinfeld Modules", the "Drinfeld Chtoucas", the "Drinfeld Upper Half Plane", the "Drinfeld Associator", and so many others that one of his endorsers jokingly said, "one could think that "Drinfeld" was an adjective, not the name of a person".For all the parts of the citation relevant to Drinfeld, see THIS LINK.
In 2023 Drinfeld and Shing-Tung Yau were jointly awarded the Shaw Prize. The citation states :-
The Shaw Prize in Mathematical Sciences 2023 is awarded in equal shares to Vladimir Drinfeld, Harry Pratt Judson Distinguished Service Professor of Mathematics at the University of Chicago, USA and Shing-Tung Yau, Chair Professor at Tsinghua University, PRC, for their contributions related to mathematical physics, to arithmetic geometry, to differential geometry and to Kähler geometry.For further extracts from this citation, see THIS LINK.
In  Victor Ginzburg gives an overview of Drinfeld's contributions up to 2005. He ends with the following examples showing Drinfeld's remarkable insight:-
I would like to finish with a couple of examples that show, I believe, that many of Drinfeld's insights are still awaiting "discovery." One such example is related to symplectic reflection algebras, a notion introduced by P Etingof and myself in 2002. After having worked on the subject for several years, we discovered (in January 2005) that the definition of symplectic reflection algebras was essentially contained in two lines of Drinfeld's paper "Degenerate Affine Hecke Algebras and Yangians," written 15 years earlier! Although the paper itself is very well known, it seems nobody has read those two lines of Drinfeld's very densely written text carefully enough.
The second example is equally amazing. I was preparing for a course on representation theory, which I teach regularly in Chicago. Volodya mentioned to me that he had some old notes with exercises on representation theory, written for his students in Kharkov back in the 1980s. As usual, Volodya's notes were very systematic; they contained both the exercises and the solutions. Somewhere in the middle of the notes, I found a digression on "q-analogues" that contained computations equivalent, essentially, to the important geometric construction of the quantum group discovered by Beilinson, Lusztig, and MacPherson 10 years later!
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Written by J J O'Connor and E F Robertson
Last Update December 2023
Last Update December 2023