

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

BuressurYvette, 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 highlevel 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 antiSemitic 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 pointlike objects but minute, onedimensional
strings existing in a multidimensional '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/newinmath/cover/kontsevich.html
Box




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.





