Maxim Kontsevich

Native spelling: Максим Концевич

fl. 1964, Khimki, Soviet Union

GeometryMathematical physicsAlgebraFields Medal 1998Crafoord Prize in Mathematics 2008Shaw Prize in Mathematical Sciences 2012Breakthrough Prize in Mathematics 2015

Russian-French mathematician whose work on deformation quantization, Gromov–Witten invariants and homological mirror symmetry built bridges between geometry and quantum physics.

Maxim Lvovich Kontsevich was born in Khimki near Moscow and studied mathematics at Moscow State University, where he attended the seminars of Israel Gelfand, before joining the Institute for Information Transmission Problems of the Soviet Academy of Sciences. His doctoral work was guided by the ideas of linking geometry with physics, and in 1991 he proved Witten's conjecture that the intersection numbers of tautological classes on the moduli space of stable curves obey the KdV hierarchy, a result that connected two-dimensional gravity, integrable systems and algebraic geometry.

In the 1990s Kontsevich introduced several of the ideas that now organize the mathematics of mirror symmetry and quantization. His 1994 ICM lecture formulated homological mirror symmetry as an equivalence of triangulated categories, and his 1997 deformation-quantization theorem gave an explicit formula — a sum over graphs with weights computed by integrals on configuration spaces — for quantizing any Poisson manifold. He also produced a wealth of results on graph complexes, Vassiliev-type knot invariants and the cohomology of moduli spaces, including his combinatorial model for the homology of the moduli space of curves.

After periods at the Max Planck Institute in Bonn and a professorship at Berkeley, Kontsevich became a permanent professor at the Institut des Hautes Études Scientifiques near Paris in 1995, taking French citizenship in 1999. He received the Fields Medal in 1998, the Crafoord Prize in 2008, the Shaw Prize in 2012, and the Breakthrough Prize in Mathematics in 2015, and he continues to work at the interface of algebraic geometry, symplectic topology and quantum field theory, including on non-Archimedean geometry and the 'field with one element'.

Workplaces: Institut des Hautes Études Scientifiques (IHÉS), Max Planck Institute for Mathematics, Bonn

Russia, France

Award-winning work

Fields Medal 1998Kontsevich received the medal for contributions to algebraic geometry, topology, and mathematical physics that reshaped the interaction between mathematics and quantum field theory. He proved Witten's conjecture on intersection numbers on the moduli space of curves, providing the first construction of Gromov–Witten-type invariants and linking them to integrable hierarchies. He formulated the homological mirror symmetry conjecture, an equivalence between derived categories of coherent sheaves and Fukaya categories that became the central framework for mirror symmetry, and he proved that every Poisson manifold admits a formal deformation quantization via his explicit star-product formula written as a sum over graphs.

Contributions, linked to the library

Library topicSymplectic geometryHis deformation quantization turns the Poisson bracket into a star product; mirror symmetry links symplectic and complex geometry.Library topicHomological algebraHomological mirror symmetry is an equivalence of derived categories, and his graph complexes are objects of homological algebra.Library topicAlgebraic geometryThe moduli space of curves and Gromov–Witten invariants are central objects of algebraic geometry.