**Abstract: **

Lecture 1. (Monday June 27, B. Eynard)** Introduction to two-matrix and
multi-matrix models.**

We will review definitions and motivations for the 2-matrix model,
and explain the relationship with biorthogonal polynomials, showing
how to compute eigenvalue correlations in terms of kernels.
We will also comment on generalizations of this model: the chain of matrices.
The main goal is to be able to compute large N correlation functions,
i.e. we need the kernels in the large N limit, and in various scaling
regimes. Thus, one needs large N asymptotics for large N biorthogonal
polynomials, and we will introduce the recursion and differential
equations that are needed for that purpose.

Lecture 2. (Tuesday June 28, M. Bertola) **Differential equations, finite N
spectral curve and duality.**

The recursion relations and the differential and deformation equations
satisfied by paired sequences of polynomials that are biorthogonal
with respect to exponential polynomial measures will be derived,
as well as the generalized Christoffel-Darboux relations.
The differential-deformation-recursion relations are expressed on finite
"windows" of biorthogonal polynomials, and their integral transforms, using
the method of "folding". It will be shown that these differential
systems are in duality, in the sense that they all share
the same spectral curve, and that they represent dual pairs
of differential operators with polynomial coefficients, whose
generalized monodromy (i.e. Stokes data at infinity) is preserved
under both the recursions and the deformations induced by
changes of the polynomial potentials defining the biorthogonality
measures.

Lecture 3. (Wednesday June 29, J. Harnad)
**Integral representations of the fundamental systems. **

For paired sequences of polynomials that are biorthogonal with respect to
measures determined by polynomial potentials, integral representations of
the fundamental systems satisfying the recursion and differential equations
will be derived. These will be shown to be bilinearly paired with integral
representations of the fundamental
systems associated to the dual sequence of Fourier-Laplace transforms of the
polynomials. They satisify differential systems
which determine the spectral curve and deformation equations
that determine the partition function. The integral representations lead
to a Riemann-Hilbert characterization determining the biorthogonal
sequences, which may be used as the starting point in large N asymptotic
analysis. These integral representations will be related to
another one, recently proposed by Kujlaars and Mclaughlin, based on
multi-orthogonal polynomials, in which the jump matrices are nonconstant
functions.

Lecture 4. (Wednesday June 29, M. Bertola)
**Dual Riemann-Hilbert approach to biorthogonal polynomials. **

The integral representations of the fundamental systems of
solutions to the differential-deformation-recursion relations
introduced in the previous lecture will be analyzed via the
steepest descent method to deduce the large argument asymptotics
of the biorthogonal polynomial sequences and their integral
transforms. Together with the local jump discontinuitites across the
integration contours, this provides a Riemann-Hilbert
characterization of these fundamental systems and their duals,
giving a starting point for the Riemann-Hilbert
approach to the study of large N asymptotics of biorthogonal polynomials.

Lecture 5. (Thursday June 30, B. Eynard) **
Formal large N expansion for the 2-matrix model
and algebraic geometry.**

The formal 2-matrix model can be defined as a combinatoricial generating
function for enumerating discrete surfaces.
By definition, this model has a well defined 1/N expansion.
We will show in this lecture how to solve recursively the Schwinger-Dyson
equation, and find a formula for all terms in the 1/N expansion.
To leading order, the Schwinger-Dyson equation is an algebraic equation,
and all the other orders can be written in terms of geometric properties
of the leading order algebraic curve.
To do this, we will introduce some basics of analysis on algebraic curves.

Lecture 6. (Friday July 1, B. Eynard)
** Large N asymptotics of biorthogonal polynomials. **

We will present a conjecture for the asymptotics of biorthogonal
polynomials. This conjecture is in terms of algebraic geometry of the
large N limit of the spectral curve.