1 Riemann-Hilbert Problems

1.1 What Is a Riemann-Hilbert Problem?

1.2 Examples

Nonlinear Schrödinger Equation
Korteweg-de Vries Equation
Boussinesq Equation
Burger's Equation
Toda Equations
Other Problems

2 Jacobi Operators

2.1 Jacobi Matrices

2.2 The Spectrum of Jacobi Matrices

2.3 The Toda Flow

2.4 Unbounded Jacobi Operators

2.5 Appendix: Support of a Measure

3 Orthogonal Polynomials

3.1 Construction of Orthogonal Polynomials

3.2 A Riemann-Hilbert Problem

3.3 Some Symmetry Considerations

3.4 Zeros of Orthogonal Polynomials

4 Continued Fractions

4.1 Continued Fraction Expansion of a Number

4.2 Measure Theory and Ergodic Theory

4.3 Application to Jacobi Operators

4.4 Remarks on the Continued Fraction Expansion of a Number

5 Random Matrix Theory

5.1 Introduction

5.2 Unitary Ensembles

5.3 Spectral Variables for Hermitian Matrices

5.4 Distribution of Eigenvalues

5.5 Distribution of Spacings of Eigenvalues

5.6 Further Remarks on the Nearest-Neighbor Spacing Distribution and Universality

6 Equilibrium Measures

6.1 Scaling

6.2 Existence of the Equilibrium Measure \mu^{V}

6.3 Convergence of \lambda_{x*}

6.4 Convergence of (1/n)R1(x1)dx1

6.5 Convergence of \eta_{x*}

6.6 Variational Problem for the Equilibrium Measure

6.7 Equilibrium Measure for V(x)=tx2m

6.8 Appendix: The Transfinite Diameter and Fekete Sets

7 Asymptotics for Orthogonal Polynomials

7.1 Riemann-Hilbert Problem: The Precise Sense

7.2 Riemann-Hilbert Problem for Orthogonal Polynomials

7.3 Deformation of a Riemann-Hilbert Problem

7.4 Asymptotics of Orthogonal Polynomials

7.5 Some Analytic Considerations of Riemann-Hilbert Problems

7.6 Construction of the Parametrix

7.7 Asymptotics of Orthogonal Polynomials on the Real Axis

8 Universality

8.1 Universality

8.2 Asymptotics of Ps




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