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 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)