Vladimir Drinfeld Extras

Vladimir Drinfeld has been awarded three of the most prestigious mathematical prizes, namely the Field's Medal, the Wolf Prize, and the Shaw Prize. The Mathematical Work of Vladimir Drinfeld which led to the award of the Field's Medal in 1990 was described by Yuri Ivanovich Manin in a lecture delivered to the International Congress of Mathematicians in Kyoto, Japan, by Michio Jimbo (since Manin was unable to obtain permission to attend). We give some extracts below to which we have made minor changes. In 2018 Drinfeld and Beilinson were jointly awarded the Wolf Prize and we give below the parts of the citation most relevant to Drinfeld. In 2023 Drinfeld and Shing-Tung Yau were jointly awarded the Shaw Prize. Again we give the parts of the citation most relevant to Drinfeld. We add a few short extracts regarding these awards.
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The Fields Medal (1990)

The Wolf Prize (2018)

The Shaw Prize (2023)

1. The Fields Medal (1990).
The Fields Medal is awarded every four years on the occasion of the International Congress of Mathematicians to recognise outstanding mathematical achievement for existing work and for the promise of future achievement.

The Fields Medal Committee is chosen by the Executive Committee of the International Mathematical Union and is normally chaired by the IMU President. It is asked to choose at least two, with a strong preference for four, Fields Medalists, and to have regard in its choice to representing a diversity of mathematical fields. A candidate's 40th birthday must not occur before January 1st of the year of the Congress at which the Fields Medals are awarded.

1.1. On the Mathematical Work of Vladimir Drinfeld by Yuri Ivanovich Manin.

Drinfeld wrote 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 an interesting reading. It starts a series of Drinfeld's works which can be considered somewhat isolated in the general context of his mathematical production but which contains worthy results.

The limitations of space and time make it impossible for me to review all Drinfeld's contributions and I shall concentrate upon the two subjects that were Drinfeld's main preoccupation in the last decade. These are Langlands' program and quantum groups. In both domains, Drinfeld's work constituted a decisive breakthrough and prompted a wealth of research.

Langlands' program is a series of conjectures, theorems and insights aimed to an understanding of the Galois groups of local and global fields of dimension one, their finite extensions and completions.

One can convincingly argue that these Galois groups constitute a primary object of number theory, more fundamental than the integers themselves. Anyway, most of the classical themes of number theory, like prime numbers, LL-functions, and modular forms reveal a lot of hidden structure when viewed from the Galois-theoretic angle.

In the classical (global) cases when the structure of these LL-functions is known, they are Mellin's transforms of modular forms which represent, roughly speaking, the de Rham aspect of the cohomology of certain modular spaces, while Galois representations embody their étale cohomology. Thus, Langlands' program is related to Grothendieck's motives, a largely conjectural universal cohomology theory of algebraic varieties.

Drinfeld proved Langlands' conjectures for GL(2)GL(2) over global fields of finite characteristic. His decisive contribution was the discovery of a new class of modular spaces and their detailed investigation. While the classical modular spaces parametrise elliptic curves, abelian varieties or Hodge structures, Drinfeld had an astonishing idea that in order to treat the finite characteristic case one should parametrise objects of a new kind. The first approximation is now called Drinfeld modules (earlier examples of which were introduced by Carlitz). A more general notion indispensable for the complete theory is known under the equally uninspiring names "shtuka" (meaning approximately "a piece of something" in English) and "FF-sheaf" (which Drinfeld uses for the lack of something better; as a mild reproach I must say that in the domain of terminology his imagination produces less brilliant solutions than in the theorem-proving).

Drinfeld's proof of one of his main theorems is long and involved, both technically and logically. Although it is contained in a series of journal papers, the most comprehensive treatment of the whole theory is given in his two still unpublished manuscripts (totalling up to about 800 type-written pages). It is to be hoped that after a final brushing up they will be given to a book publisher. The reason it has not been done to this moment seems to be Drinfeld's preoccupation with another new and fascinating subject - quantum groups.

Formally speaking, quantum groups constitute a vaguely defined subclass of Hopf algebras. Their first examples were discovered by the mathematical physicists of Leningrad school, students and collaborators of L D Faddeev. Drinfeld first summarised the basic definitions and results of this theory, largely conceived or systematised in his own work, in his talk at the Berkeley ICM four years ago. This report and several articles of M Jimbo played a decisive role in the crystallization of this new domain and drew to it attention of many mathematicians.

Until recently, the quantum group theory lacked a classification theorem describing precisely what class of objects we want to consider and what is the structure of this class. Such a theorem was recently formulated and proved in the two important papers by Drinfeld. It can be compared with the first theorems of Lie establishing the relations between Lie algebras and local Lie groups.

Indeed, this theorem is local, even formal, in nature, because Drinfeld considers formal deformations of formal groups (or, in the dual language, of their universal enveloping algebras). He also imposes from the very beginning a Yang-Baxter type condition ("quasi-triangularity"). He proves then that the whole deformation is defined by the lowest order non-trivial data.

In the course of proof, he introduces a new notion of quasi-Hopf algebra, weakening in an appropriate way the coassociativity condition on the comultiplication. He connects quasi-Hopf algebras with the Knizhnik-Zamolodchikov differential equation whose monodromy is related to the Drinfeld-Jimbo RR-operators as was shown by Kohno. Finally, he shows that in a formal situation the quasi-Hopf algebras can be reduced to the usual Hopf algebras by a kind of gauge transformation. The proof involves killing a lot of cohomological obstructions arising in the complex perturbative scheme.

One should mention also the notions of a Poisson-Lie group and a Poisson-Lie action introduced by Drinfeld at an earlier stage of his work on Yang-Baxter equations. They form one of the most basic differential geometric structures connected with Hamiltonian mechanics, and their role in future will certainly grow.

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.
2. The Wolf Prize (2018).
Since 1978, The Wolf Foundation has awarded the acclaimed, international Wolf Prize. The Prize is awarded to outstanding scientists and artists from around the world, (regardless of nationality, race, colour, religion, gender, or political views) for achievements in the interest of humanity and friendly relations among people. The prize laureates are selected by international Jury Committees which comprise world-renowned professionals from all over the world. The prize in each field consists of a certificate and a monetary award of $100,000. The prize presentation takes place at a special ceremony at the Knesset (Israel's Parliament), in Jerusalem.

2.1. Drinfeld and Beilinson awarded the Wolf Prize in 2018.

Drinfeld and Beilinson share the Wolf Prize in 2018 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.

An "algebraic structure" is a set of objects, including the actions that can be performed on those objects, that obey certain axioms. One of the roles of modern algebra is to research, in the most general and abstract way possible, the properties of various algebraic structures (including their objects), many of which are amazingly complicated.

Vladimir Drinfeld, born in Kharkov, Ukraine (1954), represented the USSR at the International Mathematical Olympiad at the age of 15 and won a gold medal. In the same year he also began his studies at the University of Moscow. Since the eighties he has been considered one of the world's leading mathematicians. In 1990 he won the prestigious Fields Medal and in 2008 was elected to the National Academy of Sciences (USA). 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 shtukas", 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".

In the seventies, Drinfeld began his work on the aforementioned "Langlands Program", the ambitious program that aimed at unifying the fields of mathematics. This program was proposed by the American-Canadian mathematician Robert Langlands (winner of the Wolf Prize in 1996) and discovered for the first-time tight and direct links between different branches of mathematics. Numbers theory (the field based on arithmetic, "sums"), algebraic representation theory and another field called "automorphic forms", (which is related to harmonic analysis and assists, for example, in the physical study of waves and frequencies). By means of a new geometrical object he developed, which is now called "Drinfeld shtukas", Drinfeld succeeded in proving some of the connections that had been indicated by the Langlands Program. In the eighties he invented the concept of algebraic "Quantum Group", which led to a profusion of developments and innovations not only in pure mathematics but also in mathematical physics (for example in statistical mechanics).

Drinfeld and Beilinson, together created a geometric model of algebraic theory that plays a key role in both field theory and physical string theory, thereby further strengthening the connections between abstract modern mathematics and physics. In 2004 they jointly published their work in a book that describes important algebraic structures used in quantum field theory, which is the theoretical basis for the particle physics of today. This publication has since become the basic reference book on this complex subject.

2.2. Edward Witten on Drinfeld and Beilinson.

By the late 1980s, Sasha Beilinson and Vladimir Drinfeld had discovered what they called a geometric version of the Langlands program, and it involved ingredients of quantum field theory. Tantalizing. But it was tantalizing because they were using familiar ingredients of quantum field theory in a very unfamiliar way. It looked to me as if somebody had put the pieces at random on a chess board. The pieces were familiar, but the position didn't look like it could happen in a real chess game. It just looked crazy. But anyway, it was clear it had to mean something in terms of physics. I even worked on that for a while at the time.

I think when I worked on it was actually before the work of Beilinson and Drinfeld, driven by other clues. And the Beilinson and Drinfeld work was one of the things that made me stop, because I realized that A, I couldn't understand what they were doing at the time, and B, there were too many things I didn't know that they knew, and that seemed to be part of the story. Anyway, as you can see, my memories from whatever happened in the late 1980s are pretty scrambled.

They wrote a famous paper that was never finished and never published. It's 500 pages long. You can find it online, if you like. They have an incredibly generous acknowledgement of what they supposedly learned from me, which is way exaggerated. Based on a hunch, I told them about a paper of Nigel Hitchin, but I didn't understand anything of what they attributed to me. At any rate, regardless, even if I didn't understand what they did with it, the fact that I was able to point them to the right paper was another sign of the fact that what they were doing had something to do with the physics I knew. But I couldn't make sense of the connection. And this kept nagging at me off and on for a long time.
3. The Shaw Prize (2023).
The Shaw Prize is an international award to honour individuals who have achieved distinguished and significant advances in their fields, who have made outstanding contributions in academic and scientific research or applications, or who in other domains have achieved excellence. Established in 2002 under the auspices of Mr Run Run Shaw, it is managed by The Shaw Prize Foundation in Hong Kong.

3.1. Drinfeld and Shing-Tung Yau awarded the Shaw Prize in 2023.

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.

They share an interest in mathematical physics. Drinfeld launched with Beilinson the geometric Langlands program, which, to quote Witten, has some common features with aspects of quantum field theory, and yet stems from number theory. Yau worked on mathematical problems arising from general relativity and string theory.

Drinfeld invented at an early age the shtukas (coming from Stück in German, meaning "piece") in resonance with the Korteweg-de Vries equation in physics. With it, he solved the arithmetic Langlands program over a function field in rank two, for which he was awarded the Fields Medal in 1990. It was then already noticed that his solution proved at the same time a conjecture of Deligne on the existence of compatible \ell-adic systems in rank two. Remarkably, after the Langlands program over a function field was proven in any rank in 2002 by L Lafforgue, following Drinfeld's method, Drinfeld could extend the existence of compatible \ell-adic systems in any rank from function fields to higher-dimensional varieties. This complete solution to the Deligne conjecture has multiple consequences, even in complex geometry.

In today's pp-adic Hodge theory, and in the dreamed Langlands program over a number field, it is expected that Drinfeld's shtukas should be a key concept as suggested by Scholze's general conjectures exposed in his ICM 2018 plenary address. Moreover, Drinfeld's view on Bhatt-Scholze prismatic cohomology and its systems of coefficients led to a new understanding of the theory and to applications.

Drinfeld's work is a pillar of arithmetic geometry which is at the core of new developments in the field.

3.2. Shaw Prize winners reveal inspiration for groundbreaking work.

Vladimir Drinfeld - who with Yau Shing-tung has shared the Shaw Prize in Mathematical Sciences for their work related to mathematical physics, and arithmetic geometry, differential geometry and Kahler geometry - says he fell in love with mathematics while spending time with his father, a professor of the subject. They enjoyed long walks and holidays filled with discussions about mathematics, which has become his lifelong passion. "The way he taught the subject to me was pleasurable as a child - no stress," Drinfeld says. This passion has inspired him to win numerous prestigious international awards, including the Fields Medal, the Wolf Prize - and now the Shaw Prize.

3.3. Shmuel Weinberger on Drinfeld winning the Shaw Prize.

"Vladimir is a remarkable colleague," said Shmuel Weinberger, the Andrew MacLeish Distinguished Professor of Mathematics and chair of the University of Chicago mathematics department. "Many mathematical breakthroughs were fomented, presented and studied in the Geometric Langlands seminar created by him and Sasha Beilinson; it is a model of strenuous intellectual effort in the service of discovery through deep understanding. He is also a person who cares deeply about others, as well as a profoundly humble one."

Last Updated December 2023