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115
Intersection theory, integrable hierarchies and topological field theory
, 1992
"... In these lecture notes we review the various relations between intersection theory on the moduli space of Riemann surfaces, integrable hierarchies of KdV type, matrix models, and topological field theory. We focus in particular on the question why matrix integrals of the type considered by Kontsevic ..."
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Cited by 95 (5 self)
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In these lecture notes we review the various relations between intersection theory on the moduli space of Riemann surfaces, integrable hierarchies of KdV type, matrix models, and topological field theory. We focus in particular on the question why matrix integrals of the type considered by Kontsevich naturally appear as τfunctions of integrable hierarchies related to topological minimal models.
Modular Operads
 COMPOSITIO MATH
, 1994
"... We develop a "higher genus" analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the Feynman transform, which generalizes Kontsevich's graph complexes and also the bar construction for operads. ..."
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Cited by 68 (5 self)
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We develop a "higher genus" analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the Feynman transform, which generalizes Kontsevich's graph complexes and also the bar construction for operads. We calculate the Euler characteristic of the Feynman transform, using the theory of symmetric functions: our formula is modelled on Wick's theorem. We give applications to the theory of moduli spaces of pointed algebraic curves.
Random matrices and random permutations
 Internat. Math. Res. Notices
, 2000
"... We prove the conjecture of Baik, Deift, and Johansson which says that with respect to the Plancherel measure on the set of partitions λ of n, the rows λ1,λ2,λ3,... of λ behave, suitably scaled, like the 1st, 2nd, 3rd, and so on eigenvalues of a Gaussian random Hermitian matrix as n → ∞. Our proof is ..."
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Cited by 62 (7 self)
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We prove the conjecture of Baik, Deift, and Johansson which says that with respect to the Plancherel measure on the set of partitions λ of n, the rows λ1,λ2,λ3,... of λ behave, suitably scaled, like the 1st, 2nd, 3rd, and so on eigenvalues of a Gaussian random Hermitian matrix as n → ∞. Our proof is based on an interplay between maps on surfaces and ramified coverings of the sphere. We also establish a connection of this problem with intersection theory on the moduli spaces of curves. 1
Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over Q, Asian
 Journal of Mathematics
, 1998
"... Abstract. It is well known that there is a bijective correspondence between metric ribbon graphs and compact Riemann surfaces with meromorphic Strebel differentials. In this article, we prove that Grothendieck’s correspondence between dessins d’enfants and Belyi morphisms is a special case of this c ..."
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Cited by 37 (9 self)
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Abstract. It is well known that there is a bijective correspondence between metric ribbon graphs and compact Riemann surfaces with meromorphic Strebel differentials. In this article, we prove that Grothendieck’s correspondence between dessins d’enfants and Belyi morphisms is a special case of this correspondence through an explicit construction of Strebel differentials. For a metric ribbon graph with edge length 1, an algebraic curve over Q and a Strebel differential on it is constructed. It is also shown that the critical trajectories of the measured foliation that is determined by the Strebel differential recover the original metric ribbon graph. Conversely, for every Belyi morphism, a unique Strebel differential is constructed such that the critical leaves of the measured foliation it determines form a metric ribbon graph of edge length 1,
Transitive factorisations into transpositions and holomorphic mappings on the sphere
 Proc. A.M.S
, 1997
"... Abstract. We determine the number of ordered factorisations of an arbitrary permutation on n symbols into transpositions such that the factorisations have minimal length and such that the factors generate the full symmetric group on n symbols. Such factorisations of the identity permutation have bee ..."
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Cited by 33 (6 self)
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Abstract. We determine the number of ordered factorisations of an arbitrary permutation on n symbols into transpositions such that the factorisations have minimal length and such that the factors generate the full symmetric group on n symbols. Such factorisations of the identity permutation have been considered by Crescimanno and Taylor in connection with a class of topologically distinct holomorphic maps on the sphere. As with Macdonald’s construction for symmetric functions that multiply as the classes of the class algebra, essential use is made of Lagrange inversion. 1.
Asymptotics of the partition function for random matrices via RiemannHilbert techniques, and applications to graphical enumeration
 Internat. Math. Research Notices
, 2003
"... Abstract. We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed RiemannHilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N ..."
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Cited by 33 (6 self)
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Abstract. We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed RiemannHilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N expansion for the partition function, for a general class of probability measures on matrices, originally conjectured by Bessis, Itzykson, and Zuber. We prove that the coefficients in the asymptotic expansion are analytic functions of parameters in the original probability measure, and that they are generating functions for the enumeration of labelled maps according to genus and valence. Central to the analysis is a large N expansion for the mean density of eigenvalues, uniformly valid on the entire real axis.
N = 2 topological gauge theory, the Euler characteristic of moduli spaces, and the Casson invariant”, Commun.Math.Phys
 Aspects of NT ≥ 2 topological gauge theories and Dbranes”, Nucl.Phys. B492
, 1993
"... We discuss gauge theory with a topological N = 2 symmetry. This theory captures the de Rham complex and Riemannian geometry of some underlying moduli space M and the partition function equals the Euler number χ(M) of M. We explicitly deal with moduli spaces of instantons and of flat connections in t ..."
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Cited by 25 (8 self)
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We discuss gauge theory with a topological N = 2 symmetry. This theory captures the de Rham complex and Riemannian geometry of some underlying moduli space M and the partition function equals the Euler number χ(M) of M. We explicitly deal with moduli spaces of instantons and of flat connections in two and three dimensions. To motivate our constructions we explain the relation between the MathaiQuillen formalism and supersymmetric quantum mechanics and introduce a new kind of supersymmetric quantum mechanics based on the GaussCodazzi equations. We interpret the gauge theory actions from the AtiyahJeffrey point of view and relate them to supersymmetric quantum mechanics on spaces of connections. As a consequence 1