Let M n g be the moduli space of n-pointed Riemann surfaces of genus g. Denote by M n g the Deligne-Mumford compactification of M n g. In the present paper, we calculate the orbifold and the ordinary Euler characteristic of M n g for any g and n such that n> 2 − 2g. 1

. We derive semiclassical asymptotics for the orthogonal polynomials on the line with the weight exp(\GammaN V (z)), where V (z) = tz 2 2 + gz 4 4 ; g ? 0; t ! 0, is a double-well quartic polynomial. Simultaneously we derive semiclassical asymptotics for the recursive coefficients of the orthogonal polynomials. The proof of the asymptotics is based on the analysis of the appropriate matrix Riemann-Hilbert problem. As an application of the semiclassical asymptotics, we prove the universality of the local distribution of eigenvalues in the matrix model with the double-well quartic interaction in the presence of two cuts. Contents 1. Main Result. 2. Universality of the Local Distribution of Eigenvalues in the Matrix Model. 3. The Lax Pair for the Freud Equation. 4. The Stokes Phenomenon. 5. The Riemann-Hilbert Problem. 6. Formal Asymptotic Expansion for R n . 7. The Bohr-Sommerfeld Quantization Condition. 8. Semiclassical Approximation Near Turning Point. 9. Connection Formula Be...

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.

Using mirror symmetry, we show that Chern-Simons theory on certain manifolds such as lens spaces reduces to a novel class of Hermitian matrix models, where the measure is that of unitary matrix models. We show that this agrees with the more conventional canonical quantization of Chern-Simons theory. Moreover, large N dualities in this context lead to computation of all genus A-model topological amplitudes on toric Calabi-Yau manifolds in terms of matrix integrals. In the context of type IIA superstring compactifications on these Calabi-Yau manifolds with wrapped D6 branes (which are dual to M-theory on G2 manifolds) this leads to engineering and solving F-terms for N = 1 supersymmetric gauge theories with superpotentials involving certain multi-trace operators

this paper, we will use the following operators e

Abstract. We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate it to a nonlinear hierarchy of ordinary differential equations. 1.

The express purpose of these Lecture Notes is to go through some aspects of the simplicial quantum gravity model known as the Dynamical Triangulations approach. Emphasis has been on lying the foundations of the theory and on illustrating its subtle and often unexplored connections with many distinct mathematical fields ranging from global riemannian geometry, moduli theory, number theory, and topology. Our exposition will concentrate on these points so that graduate students may find in these notes a useful exposition of some of the rigorous results one can establish in this field and hopefully a source of inspiration for new exciting problems. We also illustrate the deep and beautiful interplay between the analytical aspects of dynamical triangulations and the results of MonteCarlo simulations. The techniques described here are rather novel and allow us to address successfully many high points of great current interest in the subject of simplicial quantum gravity while requiring very lit-1 tle in the way of fancy field theoretical arguments. As a consequence, these

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Abstract. The asymptotic expansion of a Hermitian matrix integral known as the Penner model is rigorously calculated. 1. Introduction. The purpose of this paper is to establish an asymptotic analysis of a Hermitian matrix integral known as the Penner model, and to calculate its asymptotic expansion. It was proved by Penner [7] that this asymptotic series gives the orbifold Euler characteristic of the moduli spaces of pointed algebraic curves. The formula

Abstract. In these lectures three different methods of computing the asymptotic expansion of a Hermitian matrix integral is presented. The first one is a combinatorial method using Feynman diagrams. This leads us to the generating function of the reciprocal of the order of the automorphism group of a tiling of a Riemann surface. The second method is based on the classical analysis of orthogonal polynomials. A rigorous asymptotic method is established, and a special case of the matrix integral is computed in terms of the Riemann ζ-function. The third method is derived from a formula for the τ-function solution to the KP equations. This method leads us to a new class of solutions of the KP equations that are transcendental, in the sense that theycannot be obtained bythe celebrated Krichever construction and its generalizations based on algebraic geometryof vector bundles on Riemann surfaces. In each case a mathematicallyrigorous wayof dealing with asymptotic series in an infinite number of variables is established. Contents