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NCTS Short Course on Riemann-Hilbert Method in Integrable Systems
 
10:10-12:00, 14:10-16:00 on 8/2, 8/5; 10:10-12:00 on 8/6
College of Science, NSYSU

Speaker:
Peter David Miller (University of Michigan)


Organizers:
Chun-Kong Law (National Sun Yat-sen University)


一、 課程背景與目的:
Riemann-Hilbert method is a complex analytic method for the solution of Riemann-Hilbert problem. Through the ingenious extensions of Deift, Xin, Its and Fokas, it has become a powerful method for the asymptotic analysis of integrable systems, Painleve equations and random matrices. As an expert in the method, Prof. Peter Miller has solved several intricate asymptotics problems in orthogonal polynomials, Painleve equations and integrable systems. The introductory course is suitable for graduate students and researchers with background in differential equations.   
 
二、 課程之大綱:
 
Tentative Schedule:  Summer, 2019.

August 2nd (Fri)

 10:10-12:00

 14:10-16:00

August 5th (Mon)

 10:10-12:00

 14:10-16:00

August 6th (Tue)

 10:10-12:00

 
LECTURE 1: RIEMANN-HILBERT PROBLEMS AND LAX PAIRS
Abstract: The inverse-scattering transform can be used to study the initial-value problem for certain nonlinear wave equations, and the most important part of this analysis frequently leads to a Riemann-Hilbert problem of complex function theory. This lecture will explain how Riemann-Hilbert problems arise in this setting, and will then reveal why Riemann-Hilbert problems are fundamentally related to integrability by means of Lax pairs arising from the dressing construction.
 
LECTURE 2: SOME THEORY OF RIEMANN-HILBERT PROBLEMS
Abstract: A Riemann-Hilbert problem is fundamentally a problem of complex analysis, a kind of boundary-value problem for the Cauchy-Riemann equations. However, as with many problems of elliptic partial differential equations, a Riemann-Hilbert problem can be recast as a singular integral equation. This lecture will highlight some of the key ideas of the connection between Riemann-Hilbert problems and integral equations, with emphasis on the small-norm setting and how to achieve it by deformation techniques.
 
LECTURE 3: ASYMPTOTIC ANALYSIS OF RIEMANN-HILBERT PROBLEMS, PART I
Abstract: The Deift-Zhou steepest descent method is a powerful set of techniques applicable to Riemann-Hilbert problems that are generalizations of the classical steepest descent method for the asymptotic expansion of certain contour integrals. This lecture will focus on the Fokas-Its’-Kitaev Riemann-Hilbert problem characterizing orthogonal polynomials with exponentially varying weights and the asymptotic limit of large degree as an example of the steepest descent method. The goal of this lecture is to deform the Riemann-Hilbert problem to the point where it appears at a formal level to be asymptotically simple.
 
LECTURE 4: ASYMPTOTIC ANALYSIS OF RIEMANN-HILBERT PROBLEMS, PART II
Abstract: This lecture picks up where Lecture 3 left off. The deformed Riemann-Hilbert problem suggests an approximate solution, known as a parametrix. The parametrix will be constructed explicitly with the help of elementary and special functions. Then by comparing the parametrix to the exact solution we will arrive at a Riemann-Hilbert problem of small-norm type (cf., Lecture 2). Estimates on the solution of the latter problem yield explicit leading-order asymptotic formulae for the orthogonal polynomials and related quantities of interest in applications such as random matrix theory.
 
LECTURE 5: FINALE AND DISCUSSION
 
 





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