KONTSEVICH, MAXIM

E-mail : maxim (AT) ihes.fr

Permanent professor at the Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France.

Research Interests :
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.

Awards and Honors :
He was awarded the Fields Medal in 1998.

Selected Publications :
Kontsevich, Maxim; Zagier, Don Periods. Mathematics unlimited---2001 and beyond, 771--808, Springer, Berlin, 2001. 11-02 (11F67 11G55)

Kontsevich, Maxim; Soibelman, Yan Deformations of algebras over operads and the Deligne conjecture. Conférence Moshé Flato 1999, Vol. I (Dijon), 255--307, Math. Phys. Stud., 21, Kluwer Acad. Publ., Dordrecht, 2000. 18D50 (14G32 16S80 53D55 55P48)

Kontsevich, M. Periods. Mathématique et physique, 28--39, SMF Journ. Annu., 1999, Soc. Math. France, Paris, 1999. 11-02 (11F67 11G55)

Kontsevich, M. L.; Suhov, Yu. M. Statistics of Klein polyhedra and multidimensional continued fractions. Pseudoperiodic topology, 9--27, Amer. Math. Soc. Transl. Ser. 2, 197, Amer. Math. Soc., Providence, RI, 1999. 11K50 (11P21 22D40 37A45)

Kenyon, Richard; Brenier, Yann; Kontsevich, Maxime Mathématique et physique. (French) [Mathematics and physics] SMF Journée Annuelle [SMF Annual Conference], 1999. Société Mathématique de France, Paris, 1999. iv+39 pp. 00B15

Kontsevich, Maxim; Rosenberg, Alexander L. Noncommutative smooth spaces. The Gelfand Mathematical Seminars, 1996--1999, 85--108, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, 2000. 14A22 (16S38)

Pseudoperiodic topology. Edited by Vladimir Arnold, Maxim Kontsevich and Anton Zorich. American Mathematical Society Translations, Series 2, 197. Advances in the Mathematical Sciences, 46. American Mathematical Society, Providence, RI, 1999. xii+178 pp. ISBN: 0-8218-2094-X 57-06

Kontsevich, M. Lyapunov exponents and Hodge theory. The mathematical beauty of physics (Saclay, 1996), 318--332, Adv. Ser. Math. Phys., 24, World Sci. Publishing, River Edge, NJ, 1997. 58F11 (82C28)

Kontsevich, Maxim Formality conjecture. Deformation theory and symplectic geometry (Ascona, 1996), 139--156, Math. Phys. Stud., 20, Kluwer Acad. Publ., Dordrecht, 1997. 58F06

Kontsevich, Maxim Homological algebra of mirror symmetry. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 120--139, Birkhäuser, Basel, 1995. 32J25 (14D07 14J32 18E30 32G05)

Kontsevich, Maxim Rigorous results in topological $\sigma$-model. XIth International Congress of Mathematical Physics (Paris, 1994), 47--59, Internat. Press, Cambridge, MA, 1995. 14N10 (14H10 14L30 58D10)

Kontsevich, Maxim Enumeration of rational curves via torus actions. The moduli space of curves (Texel Island, 1994), 335--368, Progr. Math., 129, Birkhäuser Boston, Boston, MA, 1995. 14N10 (14D22 14L30)

Kontsevich, Maxim; Vishik, Simeon Geometry of determinants of elliptic operators. Functional analysis on the eve of the 21st century, Vol. 1 (New Brunswick, NJ, 1993), 173--197, Progr. Math., 131, Birkhäuser Boston, Boston, MA, 1995. 58G26 (58G15)

Kontsevich, Maxim Feynman diagrams and low-dimensional topology. First European Congress of Mathematics, Vol. II (Paris, 1992), 97--121, Progr. Math., 120, Birkhäuser, Basel, 1994. 57R57 (14H15 32G15 57M25)

Kontsevich, Maxim Vassiliev's knot invariants. I. M. Gel\cprime fand Seminar, 137--150, Adv. Soviet Math., 16, Part 2, Amer. Math. Soc., Providence, RI, 1993. 57M25 (81T40)

Kontsevich, Maxim Formal (non)commutative symplectic geometry. The Gel\cprime fand Mathematical Seminars, 1990--1992, 173--187, Birkhäuser Boston, Boston, MA, 1993. 58H15 (17B65 58F05)

Kontsevich, M.; Manin, Yu. Gromov-Witten classes, quantum cohomology, and enumerative geometry [MR 95i:14049]. Mirror symmetry, II, 607--653, AMS/IP Stud. Adv. Math., 1, Amer. Math. Soc., Providence, RI, 1997. 14N10 (53C15 58A10 58F05)


Kontsevich, Maxim Deformation quantization of algebraic varieties Lett. Math. Phys. 56 3 (2001) 271-294 53D55, 14Axx, 16S37

Kontsevich, Maxim Operads and motives in deformation quantization Lett. Math. Phys. 48 1 (1999) 35-72 53D55, 14F42, 16S80, 18D50, 18G55, 55P48, 81S10

Kontsevich, Maxim Rozansky-Witten invariants via formal geometry Compositio Math. 115 1 (1999) 115-127 57R57, 17B66, 32J18, 53C26, 53D35, 57R32, 57R56

Kontsevich, M.;Manin, Yu. Relations between the correlators of the topological sigma-model coupled to gravity Comm. Math. Phys. 196 2 (1998) 385-398 14H10, 14D15, 14D20, 14N10, 58D10

Kontsevich, Maxim Product formulas for modular forms on ${rm O}(2,n)$ (after R. Borcherds) Asterisque 245 (1997) 41-56 11F55, 11E45, 11F22, 11F37, 14J28, 17B67, 57R55

Barannikov, Sergey;Kontsevich, Maxim Frobenius manifolds and formality of Lie algebras of polyvector fields Internat. Math. Res. Notices 4 (1998) 201-215 14D07, 14J32, 17B66, 32G13, 58D29

Alexandrov, M.;Schwarz, A.;Zaboronsky, O.;Kontsevich, M. The geometry of the master equation and topological quantum field theory Internat. J. Modern Phys. A 12 7 (1997) 1405-1429 81T70, 58D29, 58F05

Kontsevich, M.;Manin, Yu. Gromov-Witten classes, quantum cohomology, and enumerative geometry Comm. Math. Phys. 164 3 (1994) 525-562 14N10, 53C15, 58D10, 58F05

Kontsevich, Maxim Intersection theory on the moduli space of curves and the matrix Airy function Comm. Math. Phys. 147 1 (1992) 1-23 32G15, 14H15, 58F07, 81T40

Kontsevich, M. L. Intersection theory on the moduli space of curves Funct. Anal. Appl. 25 2 (1991) 123-129 32G15, 14H15, 81T40

Kelcprime bert, M. Ya.;Kontsevich, M. L.;Rybko, A. N. Jackson networks on countable graphs Theory Probab. Appl. 33 2 (1988) 358-361 60K25, 90B15

Kontsevich, M. L. The Virasoro algebra and Teichm?ller spaces Funktsional. Anal. i Prilozhen. 21 2 (1987) 78-79 17B65, 32G15, 81E13, 81G20

Kirillov, A. A.;Kontsevich, M. L. The growth of the Lie algebra generated by two generic vector fields on the line Vestnik Moskov. Univ. Ser. I Mat. Mekh. 4 (1983) 15-20 17B65

Kirillov, A. A.;Kontsevich, M. L.;Molev, A. I. Algebras of intermediate growth Akad. Nauk SSSR Inst. Prikl. Mat. Preprint 39 (1983) 1-19 17B65, 16A99

Kontsevich, M.;Manin, Yu. Quantum cohomology of a product Invent. Math. 124 1-3 (1996) 313-339 14N10,14H10

Kontsevich, Maxim Mirror symmetry in dimension $3$ Asterisque 237 (1996) 275-293 14J32, 14D07, 14N10, 32G20, 32J17, 81T30

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