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image European Research News Centre > Pure Science > The mathematician who came in from the cold
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image image image Date published : 11/04/2001
  image The mathematician who came in from the cold
  Winner of the 1998 Fields Medal - the mathematics 'Nobel' - Maxim Kontsevich is a member of the brilliant Russian school of mathematics which was scattered across the West following the collapse of the Soviet Union. We retrace the route of one of 'greater Europe's' masterminds who decided to stay.
Maxim Kontsevich

What Maxim Kontsevich likes about mathematics is what he calls its 'beauty'. And more specifically the beauty of the structures he discovers in it.

Bures-sur-Yvette, south of Paris. This leafy suburb is home to one of the major centres of mathematics and theoretical physics, the Institut des Hautes Etudes Scientifiques (IHES). Although it has only seven permanent researchers, every year this centre of excellence welcomes some 200 scientists from all over the world, selected from among the best in their fields.

Maxim Kontsevich has taught here since 1995.

Springboard of the Olympiads

This quiet mathematician was born in Khimki, near Moscow, in 1964. The route which brought him to France is similar to that taken by many of his fellow countrymen. Born into a cultured family - his father was an expert on Korean language and history, his mother an engineer, and his older brother a researcher in computer imaging - during his last three years at secondary school in Moscow, Kontsevich took special advanced courses in maths and physics, admission to which was by competition only. These subjects have fascinated him since he was an adolescent ('thanks to my brother and some very good books').

His talents were first displayed in the mathematics Olympiads, a high-level competition in which he was ranked second nationally. This success - at the age of 16 - won him a place at Moscow University. His high ranking in the Olympiads allowed him to bypass the rather subjective entrance examinations at a time when university policy had a clearly anti-Semitic slant ('Kontsevich' is pronounced in a way which resembles a Polish Jewish name, a fact which has caused his brother a lot of problems).

After completing his studies, Maxim Kontsevich set to work on his doctoral thesis. He opted for a subject in the field of mathematical physics. At this same time he joined the Institute for Problems of Information Processing, a Moscow laboratory attached to the Academy of Sciences, where he undertook research in mathematical theory.

Witten's conjecture

His career took an international turn in 1990 when he was invited to spend three months at the Max Planck Institute in Bonn (DE). The visit culminated in a conference and seminar at which the guest speakers included Sir Michael Atiyah, 'an eminent British mathematician who spoke of wonderful things, most importantly Witten's conjecture.' This was a major conceptual development based on certain geometrical aspects of string theory. Developed in the late 1980s, this complex approach takes the view that the fundamental particles of physics are not point-like objects but minute, one-dimensional strings existing in a multi-dimensional 'spacetime' (current theory puts the number at 11).

Kontsevich was 'obsessed' by what he had heard and the next day, during a final boat trip on the Rhine for conference participants, he explained to his colleagues how he intended to prove Witten's conjecture. The project sounded so impressive that he was invited there and then to return to the Max Planck Institute as a visitor for a full year.

The young Russian mathematician was to spend a number of periods at the institute and it was in Bonn that he obtained his doctorate in 1992. His German 'visit' in fact lasted until 1994, interrupted by stays of several months in the United States at the invitation of Harvard University, Princeton's famous Institute for Advanced Studies, and the University of California at Berkeley, where he was a professor from 1993 to 1996.

Like many of his compatriots condemned to exile, Kontsevich could have settled permanently in the United States. He had a post at Berkeley, not far from San Francisco where his brother was living. He was in fact on the point of buying a home there when the IHES offered him the post of resident professor. He knew the institute's reputation, having spent a few days there in 1988 during a short working visit to France.

Russian eclecticism

So why did he decide to head back across the Atlantic to Europe? This institute offers total freedom in research, with no hierarchy and virtually no bureaucracy. What is more, the Paris region is the world's leading mathematics centre.' That used to be true of Moscow and Leningrad, but with the collapse of the Soviet Union the famous Russian school found itself scattered across the globe, '…especially in the United States, but also in Europe, where France is the principal host country. At the IHES alone, with Mikhael Gromov and Nikita Nekrassov, the Russians make up half the resident professors...'

Kontsevich is also visiting professor at Rutgers University in the United States, now home to his former teacher Israil Gelfand. Together with his fellow countrymen, he is therefore helping to keep alive, in the diaspora, something of the tradition which brought such renown to the Russian school of mathematics. 'The style is difficult to define, but it can be described as very open, universal and intuitive. You find it more in Europe than in the United States where researchers tend to be very specialised.' This Russian eclecticism is apparent in Kontsevich's own work which covers virtually the whole spectrum of mathematics: 'I have worked on almost 20 different subjects, in many fields.' In 1998, his research won him the Fields Medal, a leading, international prize awarded every four years to four mathematicians under the age of 40.

Often linked to questions originating in string theory and quantum field theory (the theoretical and mathematical field which describes the world of elementary particles), the work of Maxim Kontsevich deals with general mathematical structures that appear in fields which do not seem at first to have a great deal in common. But it is not the possible applications of a particular field of physics or technology which interests him, nor the rigour of mathematical demonstrations. What Maxim Kontsevich likes in mathematics is what he calls its 'beauty'. And especially the beauty of the structures he discovers in it.



Fields, strings and knots

Maxim Kontsevich's research cuts across many fields of pure mathematics, mixing algebra, geometry, analysis, topology, combinatorics, etc. Some of his work is inspired by theoretical physics, in particular string theory and quantum field theory, which applies quantum theory to the interaction between elementary particles. Among other things, this theory helps us understand the interactions between electrons and photons, which are the 'energy packets' of an electromagnetic field. One of physics' main unresolved problems is that we still do not have a coherent theory of this kind which can be applied to gravity, even if preliminary models of quantum gravitation have been proposed and studied. One of Kontsevich's contributions has been to demonstrate the mathematical equivalence of two of the models.

String theories seem the most likely to lead to a unified quantum description of gravitation and the other three fundamental forces. Witten's conjecture, which Kontsevich was able to prove and help bring to wider attention, concerns one of their mathematical aspects.

Kontsevich has also worked on the mathematics of knots, a field which, although is seems to be further from physics, is not without its applications. The big question here is to find the criteria making it possible to state that two complex knots of string are equivalent (meaning that one can be transformed into the other without cutting the string). Kontsevich has found new knot 'invariants' - an invariant being a mathematical object (a number, function or other) which characterises all equivalent knots.


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Fields, strings and knots

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