**Maxim Kontsevich** was born into a talented family in Khimki (sometimes written Chimki or Himki), about 17 km northwest of the centre of Moscow. His father, Lev Rafailovich Kontsevich, is an expert on the Korean language and Korean history, and he worked as Leading Researcher at the Institute of Oriental Studies of the Russian Academy of Sciences in Moscow. He devised the Kontsevich system for the Cyrillization of the Korean language, the main system in use today for producing Russian versions of Korean texts. Maxim's mother trained as an engineer, and his elder brother Leonid (Lenny) went on to undertake research into computer imaging and works in San Francisco.

Kontsevich attended secondary school in Moscow and became fascinated by mathematics and physics at an early age. This was, he wrote:-

... thanks to my brother and some very good books.

He took special advanced courses in these two subjects during his final three years at secondary school, having won the right to enter these courses which was determined by competition. When he was sixteen years old he was placed second in the national Mathematical Olympiad competition. Because of his success in this competition he was offered a place at Moscow State University without having to sit the entrance examinations. There he was taught by a number of outstanding professors, in particular by Israil Moiseevic Gelfand. In 1983, when he was only nineteen years old, Kontsevich's paper *The growth of the Lie algebra generated by two generic vector fields on the line* (Russian), written jointly with A A Kirillov, was published. In the same year he published *Algebras of intermediate growth *(Russian), written jointly with A A Kirillov and A I Molev. The authors describe the contents as follows:-

We investigate finitely generated associative and Lie algebras for which the dimension of the nth term of the natural filtration grows faster than any polynomial in n but slower than any exponent c^{n}. The associative and Lie algebras generated by two generic vector fields on the real line are considered as examples.

In 1985, he left university to begin research at the Institute for Problems of Information Processing, an Institute attached to the Russian Academy of Sciences. He published further papers *The Virasoro algebra and Teichmüller spaces *(Russian) (1988) and *Jackson networks on countable graphs* (Russian) (1988), the last paper being a joint publication.

Perhaps in retrospect one can say that the most significant event for Kontsevich was an invitation to spend three months at the Max Planck Institute in Bonn in 1990. Towards the end of his visit an international conference was held at the Institute and one of the main speakers at the conference was Michael Atiyah who, in Kontsevich's own words, was :-

... an eminent British mathematician who spoke of wonderful things, most importantly Witten's conjecture.

Kontsevich was inspired by Atiyah's talk, and [11]:-

... 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.

When he returned to Bonn in the next year he registered as a doctoral student at the Rheinische Friedrich-Wilhelms University of Bonn with Don Zagier as his thesis supervisor. He submitted his doctoral thesis *Intersection Theory on the Moduli Space of Curves and the Matrix Airy Function* and was awarded his doctorate in 1992. In his thesis he achieved his aim of proving Witten's conjecture. He published the results in a paper of the same title as his thesis in 1992. Claude Itzykson writes in a review of the paper:-

This article presents the most complete description given by its author of the proof of a conjecture by E Witten. Its subject is the computation of intersection numbers of stable classes, introduced by Mumford, Morita and Miller, on ... a compactification of the moduli space of genus-g algebraic curves with n marked points. Witten conjectured as generating function an asymptotic expansion of matrix integrals investigated by physicists under the name "two-dimensional quantum gravity". A striking consequence is that it satisfies an infinite integrable hierarchy of Korteweg-de Vries equations completed by a so-called "string equation". The author succeeds in exhibiting a second type of matrix Airy integral possessing the same properties. His derivation uses an elegant reduction to a combinatorial problem following ideas of Thurston, Mumford, Harer and Penner.

Clifford Henry Taubes writes [10]:-

... many of the steps in this proof exhibit Kontsevich's unique talent for combinatorial calculations.

This remarkable achievement led to Kontsevich receiving invitations to Harvard University, Princeton's Institute for Advanced Study, and the University of Bonn. He made visits to all three between 1992 and 1995. In 1993, however, he had received an offer of a Professorship at the University of California at Berkeley where he remained until 1996 when he moved to France [11]:-

... 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 Institut des Hautes Études Scientifiques offered him the post of resident professor. He knew the institute's reputation, having spent a few days there in1988during a short working visit to France.

Kontsevich quickly followed his brilliant paper of 1992 with another in the following year entitled *Vassiliev's knot invariants*. J S Birman writes in a review:-

V A Vassiliev[in1990]introduced a family of numerical invariants of knots. ... These invariants were shown by Vassiliev to be determined by a very complicated combinatorial construction. They are very powerful, and subsume the Jones polynomial and all of its generalizations. The paper under review is a research announcement of far-reaching results about the Vassiliev invariants. ... The main theorem has many implications, and is the subject of numerous investigations which are in progress as we write this review. However, what is perhaps even more important than the detailed statement of results is that the author has taken a very fresh and original look at the Vassiliev invariants and tells us how he was led to do it. He also gives an exposition of Vassiliev's theory which, while lacking details, makes it seem both natural and clear in a way which a careful reading of that detailed paper did not do. Curiously, the author has indicated that he does not intend to write a full exposition of his theory, leaving that task to others, notably D Bar-Natan[On the Vassiliev knot invariants, Topology]. That is a pity because when the person who has discovered a phenomenon writes about it he is able to let you in on his way of thinking in a way that another simply cannot do for him. For that reason this little paper should be read and re-read: as an introduction to the subject, for its insights into the intuitive ideas behind it, and(after digesting Bar-Natan's exposition of the proof)to return to it for fresh insights.

At the First European Congress of Mathematics in Paris in 1992 Kontsevich gave the invited address *Feynman diagrams and low dimensional topology* which was published in 1994 in the Proceeding of the conference. He was also awarded a European Mathematical Society Prize at this First European Congress. Also in 1994 he published *Gromov-Witten classes, quantum cohomology, and enumerative geometry* which was written jointly with Yuri Manin. In 1993 he published *Formal (non)commutative symplectic geometry* which was reviewed by Alexander Voronov. His review begins:-

The paper places emphasis on the three fundamental types of algebras - Lie, associative and commutative - as functional models of three hypothetical versions of noncommutative symplectic geometry(in fact, the usual commutative one in the third case). Calculus of differential forms, symplectic forms, Hamiltonian vector fields and Poisson brackets in noncommutative geometry are sketched. As an application of(and a motivation for)these ideas, it is shown how to produce cohomology classes of the moduli spaces of algebraic curves.

Kontsevich was a Plenary Speaker at the International Congress of Mathematicians in 1994 in Zürich. He then proved that any Poisson manifold admits a formal quantization and gave an explicit formula for the flat case. This was one of the four major problems which Kontsevich worked on, leading to him receiving a Fields Medal at the International Congress of Mathematicians in Berlin in 1998. The other three problems have been mentioned above: intersection theory on moduli spaces of curves, low-dimensional topology particularly knots via integrals related to Feynman diagrams, and the enumeration of rational curves.

In 2008 Kontsevich and Witten jointly received the Crafoord Prize in Mathematics from the Royal Swedish Academy of Sciences:-

... for their important contributions to mathematics inspired by modern theoretical physics.

The news release of the Royal Swedish Academy of Sciences described the work for which Kontsevich and Witten were awarded the prize in the following way:-

The laureates in mathematics have used the methodology of physics to develop a revolutionary new mathematics intended for the study of various types of geometrical objects. Their work is not only of great interest in the discipline of mathematics but may also find applications in totally different areas. Its results are of considerable value for physics and research into the fundamental laws of nature. According to string theory, which is an ambitious attempt to formulate a theory for all the natural forces, the smallest particles of which the universe is composed are vibrating strings. This theory predicts the existence of additional dimensions and requires very advanced mathematics. The laureates have resolved several important mathematical problems related to string theory and have in this way paved the way for its further development.

In addition to the honours mentioned above, Kontsevich was awarded the Daniel Iagolnitzer Prize and elected a member of the Academy of Sciences in Paris.

**Article by:** *J J O'Connor* and *E F Robertson*