This paper is based on the author's talk at the 2012 Workshop on Geometric Methods in Physics held in Bialowieza, Poland. The aim of the talk is to introduce the audience to the Eynard-Orantin topological recursion. The formalism is originated in random matrix theory. It has been predicted, and in some cases it has been proven, that the theory provides an effective mechanism to calculate certain quantum invariants and a solution to enumerative geometry problems, such as open Gromov-Witten invariants of toric Calabi-Yau threefolds, single and double Hurwitz numbers, the number of lattice points on the moduli space of smooth algebraic curves, and quantum knot invariants. In this paper we use the Laplace transform of generalized Catalan numbers of an arbitrary genus as an example, and present the Eynard-Orantin recursion. We examine various aspects of the theory, such as its relations to mirror symmetry, Gromov-Witten invariants, integrable hierarchies such as the KP equations, and the Schroedinger equations.

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 $\zeta$-function. The third method is derived from a formula for the $\tau$-function solution to the KP equations. This method leads us to a new class of solutions of the KP equations that are \emph{transcendental}, in the sense that they cannot be obtained by the celebrated Krichever construction and its generalizations based on algebraic geometry of vector bundles on Riemann surfaces. In each case a mathematically rigorous way of dealing with asymptotic series in an infinite number of variables is established.

This article is based on a talk delivered at the RIMS--OCAMI Joint International Conference on Geometry Related to Integrable Systems in September, 2007. Its aim is to review a recent progress in the Hitchin integrable systems and character varieties of the fundamental groups of Riemann surfaces. A survey on geometric aspects of these character varieties is also provided as we develop the exposition from a simple case to more elaborate cases.

We present in this paper a differential version of Mirzakhani's recursion relation for the Weil-Petersson volumes of the moduli spaces of bordered Riemann surfaces. We discover that the differential relation, which is equivalent to the original integral formula of Mirzakhani, is a Virasoro constraint condition on a generating function for these volumes. We also show that the generating function for psi and kappa_1 intersections on the moduli space of stable algebraic curves is a 1-parameter solution to the KdV hierarchy. It recovers the Witten-Kontsevich generating function when the parameter is set to be 0.

We present a new set of axioms for 2D TQFT formulated on the category of cell graphs with edge-contraction operations as morphisms. We construct a functor from this category to the endofunctor category consisting of Frobenius algebras. Edge-contraction operations correspond to natural transformations of endofunctors, which are compatible with the Frobenius algebra structure. Given a Frobenius algebra A, every cell graph determines an element of the symmetric tensor algebra defined over the dual space A⁎. We show that the edge-contraction axioms make this assignment depending only on the topological type of the cell graph, but not on the graph itself. Thus the functor generates the TQFT corresponding to A.

A classification of commutative integral domains consisting of ordinary differential operators with matrix coefficients is established in terms of morphisms between algebraic curves.

We show that the Poincare polynomial associated with the orbifold cell decomposition of the moduli space of smooth algebraic curves with distinct marked points satisfies a topological recursion formula of the Eynard-Orantin type. The recursion uniquely determines the Poincare polynomials from the initial data. Our key discovery is that the Poincare polynomial is the Laplace transform of the number of Grothendieck's dessins d'enfants.

We introduce the notion of volume of the representation variety of a finitely presented discrete group in a compact Lie group using the push-forward measure associated to a map defined by a presentation of the discrete group. We show that the volume thus defined is invariant under the Andrews-Curtis moves of the generators and relators of the discrete group, and moreover, that it is actually independent of the choice of presentation if the difference of the number of generators and the number of relators remains the same. We then calculate the volume of the representation variety of a surface group in an arbitrary compact Lie group using the classical technique of Frobenius and Schur on finite groups. Our formulas recover the results of Witten and Liu on the symplectic volume and the Reidemeister torsion of the moduli space of flat G-connections on a surface up to a constant factor when the Lie group G is semisimple.

In \cite{MP} we have shown that if a compact Riemann surface admits a Strebel differential with rational periods, then the Riemann surface is the complex model of an algebraic curve defined over the field of algebraic numbers. We will show in this article that even if all geometric data are defined over $\overline{\mathbb{Q}}$, the Strebel differential can still have a transcendental period. We construct a Strebel differential $q$ on an arbitrary complete nonsingular algebraic curve $C$ defined over $\overline{\mathbb{Q}}$ such that (i) all poles of $q$ are $\overline{\mathbb{Q}}$-rational points of $C$; (ii) the residue of $\sqrt{q}$ at each pole is a positive integer; and (iii) $q$ has a transcendental period.

A geometric quantization using a partial differential equation version of topological recursion is established for the compactified cotangent bundle of a smooth projective curve of an arbitrary genus. In this quantization, the Hitchin spectral curve of a rank 2 meromorphic Higgs bundle on the base curve corresponds to a quantum curve, which is a Rees D-module on the base. The topological recursion then gives an all-order asymptotic expansion of its solution, thus determining a state vector corresponding to the spectral curve as a meromorphic Lagrangian. We propose a generalization of the topological recursion for a singular spectral curve. We show that the PDE version of the topological recursion automatically selects the normal ordering of the canonical coordinates, and determines the unique quantization of the spectral curve. The quantum curve thus constructed has the semi-classical limit that agrees with the original spectral curve. Typical examples of our construction includes classical differential equations, such as Airy, Hermite, and Gauß hypergeometric equations. The topological recursion gives an asymptotic expansion of solutions to these equations at their singular points, relating Higgs bundles and various quantum invariants.