nLab Maxim Kontsevich


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, the 2008 Crafoord Prize, the 2012 Shaw prize and the new prize for fundamental physics 2012.



The Fields Medal 1998 was awarded for solutions of “four problems in geometry” concerning the subjects:

  1. intersection theory on compactified moduli spaces g,n\mathcal{M}_{g,n} of punctured Riemann surfaces and the Witten conjecture (obtained by relating two 2d string/quantum gravity models),

  2. knot invariants,

  3. deformation quantization

    Kontsevich’s formula for the formal deformation quantization of any Poisson manifold given by the 3-point function of open strings in the Poisson sigma-model

  4. quantum cohomology/mirror symmetry;

    Kontsevich’s work for formulating mirror symmetry in string theory by means of A-infinity categories as an equivalence of formalizations of topological string quantum field theory now known as TCFT formulations of the A-model and the B-model:

Cf. T. R. Ramadas, The Work of The Fields Medallists: 1998; 3. Maxim Kontsevich, pdf

Some of the articles of MK can be found at the arXiv and most of others at

Much of Kontsevich’s work has never been fully written and has been known via his public lectures and communications to other mathematicians; most notably, 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 Yuri Manin in Bonn, early 1990s), knot theory (Vassiliev invariants, quantum groups), A A_\infty-categories, Landau-Ginzburg models in algebraic geometry, the AKSZ model in quantum field theory, introduction of formal noncommutative symplectic geometry along with ideas about a generalization of formal Lie calculus and Koszul duality for dg-operads, discovery of graph complex and graph homology and its role in various problems of geometry and topology including moduli spaces and a geometric explanation of the origin of Rozansky-Witten invariants, fundamental works in noncommutative algebraic geometry, introducing homological mirror symmetry, foundations of deformation theory, work on generalizations of determinant and trace for linear operators (with Vishik), Donaldson-Thomas invariants (with Soibelman), tropical geometry, nonabelian Hodge theory, noncommutative motives, 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.

Selected writings

On graph complexes and their quasi-isomorphism to differential forms on (Fulton-MacPherson compactifications of) configuration spaces of points, via assignment of Chern-Simons propagators/Chern-Simons theory-Feynman amplitudes, regarded as differential form on configuration space of points:

  • Maxim Kontsevich, pages 11-12 of Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, 1992, Paris, vol. II, Progress in Mathematics 120, Birkhäuser (1994), 97–121 (pdf)

  • Maxim Kontsevich, around Def. 15 and Lemma 3 in Operads and Motives in Deformation Quantization, Lett. Math. Phys. 48 35-72, 1999 (arXiv:math/9904055)

Specifically on Vassiliev knot invariants:

  • Maxim Kontsevich, Vassiliev’s knot invariants, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 (pdf)

On wall crossing phenomena:

On Wick rotation in quantum field theory in terms of complex metrics:

category: people

Last revised on November 16, 2023 at 05:04:12. See the history of this page for a list of all contributions to it.