Maxim Kontsevich (also Максим Концевич, born 1964 August 25) is a Russian-French mathematician, a Professor at IHES, a recipient of the 1998 Fields Medal and the 2008 Crafoord Prize. Some biographical data can be found at the English Wikipedia, the German Wikipedia and his official CV page; some of the articles of MK can be found at the arXiv and most of others at his official publications page (there are also several unlisted preprints at the MPI server from collaboration with Alexander Rosenberg).
Much of his work has never been fully written and has been known via his public lectures and communications to other mathematicians; for example motivic integration? was introduced in his lecture in 1996 at Orsay and never published by him, though the subject influenced and is present implicitly again in his newer (written) works. He has solved the Witten conjecture on the connection between the KdV equation? and quantum gravity? theory (in terms of moduli spaces of Riemann surfaces; cf. also Airy function).
Kontsevich has made contributions to various parts of mathematics and mathematical physics? which inspired much of his research. One should emphasise his work on the mathematical formulation of conformal field theory (esp. 1988–1992 and again from 2002 in connection with the stochastic Loewner equation? with R. Friedrich and Suhov), Gromov-Witten invariants (since collaboration with Manin in Bonn, early 1990s), knot theory? (Vassiliev invariants, quantum groups), -categories, Landau-Ginzburg model?s in algebraic geometry, the AKSZ model in quantum field theory, fundamental works in noncommutative algebraic geometry, introducing homological mirror symmetry, foundations of deformation theory, work on generalizations of determinant? and trace for operator?s (with Vishik), Donaldson-Thomas invariant?s (with Soibelman), tropical geometry?, nonabelian Hodge theory, noncommutative motive?s, various constructions with supersymmetry (esp. in geometry), derived noncommutative algebraic geometry and so on.
Much of Kontsevich’s research is based on insights into the relation between classical physics and quantum physics and quantizing various constructions even in pure mathematics. His most famous work in this area includes the Kontsevich formality theorem? solving the problem of deformation quantization? of Poisson manifolds using homological algebra and operads; this created a number of new directions of research in mathematics. Quantization ideas are also present in works related to the geometry of Weyl algebra, including the progress on the Jacobian conjecture. In addition to finished results, Kontsevich introduced a number of stimulating conjectures which strongly influence modern mathematics.