Maxim Kontsevich

Deformation quantization of Poisson manifolds. 1.

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the ‘Formality conjecture’), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.

Intersection theory on the moduli space of curves and the matrix Airy function

We show that two natural approaches to quantum gravity coincide. This identity is nontrivial and relies on the equivalence of each approach to KdV equations. We also investigate related mathematical problems.

Homological Algebra of Mirror Symmetry

Mirror symmetry (MS) was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeros). The name comes from the symmetry among Hodge numbers. For dual Calabi-Yau manifolds V, W of dimension n (not necessarily equal to 3) one has

Enumeration of Rational Curves Via Torus Actions

\dim {H^p}(V,{\Omega ^q}) = \dim {H^{n - p}}(W,{\Omega ^q}).

Connected components of the moduli spaces of Abelian differentials with prescribed singularities

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Formal (Non)-Commutative Symplectic Geometry

The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov-Witten classes, and a discussion of their properties for Fano varieties. Cohomological Field Theories are defined, and it is proved that tree level theories are determined by their correlation functions. Application to counting rational curves on del Pezzo surfaces and projective spaces are given.

Feynman Diagrams and Low-Dimensional Topology

This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry.

Homological mirror symmetry and torus fibrations

The algebraic world of associative algebras has many deep connections with the geometric world of two-dimensional surfaces. Recently, D. Tamarkin discovered that the operad of chains of the little discs operad is formal, i.e. it is homotopy equivalent to its cohomology. From this fact and from Delignes conjecture on Hochschild complexes follows almost immediately my formality result in deformation quantization. I review the situation as it looks now. Also I conjecture that the motivic Galois group acts on deformation quantizations, and speculate on possible relations of higher-dimensional algebras and of motives to quantum field theories.