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A Very Pleasurable Universe
Internationally Acclaimed Mathematician Maxim Kontsevich Chooses Berkeley as His Academic Home
by Robert Sanders
Berkeley can be tumultuous, but to Maxim Kontsevich it is a peaceful haven compared to the turmoil today in Russia.
Kontsevich is one of many elite mathematicians to abandon Russia in the past several years, leaving behind the crime and unpredictability of Moscow.
Courted by both Princeton and Harvard, he is a young international star the mathematics department feels fortunate to have attracted. One eminent scholar praised him as being among the most talented mathematicians to come onto the stage in the last five years.
A mere two weeks after his arrival on campus, his Evans Hall office is a barren expanse of gray linoleum, but he is filled with mathematical fervor. Along with Alexandre Givental and Vera Serganova, two other Russians in the department, he is already sponsoring a weekly seminar on mirror symmetry, his obsession at the moment. And he is immersed in teaching an advanced graduate-level course on deformation theory.
The slight, dark-haired Kontsevich looks younger than his 30 years, and a quiet demeanor fails to hide his playfulness. Play may well be essential to success in mathematics, and he admits to his share.
After leaving Moscow State University in 1985--without a degree--he spent five years at the Institute for Problems of Information Transmission where he devoted half his time to Renaissance and Baroque music (he admits to being "not bad" on the lute and viola da gamba) and the rest to mathematics.
He also enrolled in an intensive French class, where he met his future wife, Katerina Rosanova.
Despite a "very free life" at the institute, he did manage to publish enough to interest the prestigious Max Planck Institute for Mathematics in Bonn, Germany, which invited him to visit for three months in 1990. The culture shock was so immense, he says now, that he had little time for productive mathematics.
Just before his planned return to Moscow, however, he attended a five-day international meeting--the well-known Arbeitstagung--that each year draws the world's leading mathematicians.
The very first talk was by Sir Michael Atiyah on the subject of quantum gravity, a field that emerged from physics as theoreticians tried to model elementary particles as pieces of string knotted in complex ways. Atiyah mentioned an important conjecture by Edward Witten that so intrigued Kontsevich that he forsook the evening party and sketched out a proof, which he presented to the group before the meeting broke up.
Impressed, the institute directors invited him to stay for three years. He finished the proof of the Witten conjecture within a year and worked on various topics in mathematical physics. He also completed a PhD in mathematics from Bonn University.
He was glad to get out of Moscow, which was exciting but stressful during the era of perestroika and the fall of communism. Like him, nearly all the best mathematicians have now fled to the West, primarily the United States, Kontsevich says.
Kontsevich now has more Russian friends in the Bay Area than he has in Russia. Among them is his brother Lenny, a computer vision specialist in San Francisco. Even his parents left Russia to live in Korea. His father, a specialist in Korean history and language, now teaches Russian, while his mother, a mechanical engineer, is retired.
Berkeley lured him for several reasons, not only because of his friends and the pleasant weather. The Mathematical Sciences Research Institute on the hill above campus is much like the very active Max Planck Institute, he said. The faculty here are friendly, with a great diversity of interests, he says. Berkeley's mathematics department ranks among the best in the world and was recently ranked at the top in this country along with Harvard, Princeton, and MIT.
In Kontsevich's eyes, though, nothing will ever match Moscow State University. In 1980, when he entered at the age of 16, it was the largest and best of the two universities for mathematics in the former Soviet Union, a nation known for its contributions to the field. Around 400 students were admitted each year in mathematics alone, while the 100-plus mathematics seminars per week were enough to satisfy any appetite.
"It was a fantastic place--I never met such a concentration of mathematicians," he says. One of his mentors was MacArthur "genius" Award-winner Israel Gelfand, who is now at Rutgers and at 81 going strong.
At the moment Kontsevich is trying to unify two areas of geometry--symplectic and algebraic--in a way that explains the mirror symmetry discovered recently in string theory. This is now one of the most exciting and active areas of research on the boundary between mathematics and physics.
It isn't clear whether this mirror symmetry is related at all to the physics of the real world, but it exhibits what he finds most attractive about mathematics, beauty.
"For me, mathematics is a kind of independent and very pleasurable universe," he says.
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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.
Contact
maxim@ihes.fr
http://www.ams.org/new-in-math/cover/kontsevich.html
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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
