Preface   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   Bibliography To go back to the main page, click here.