FSF3707 Riemann-Hilbert Methods in Asymptotic Analysis 7.5 credits

• Linear ordinary differential equations and method of stationary phase classical linear steepest descent. (Case study: the Airy equation)
• Riemann-Hilbert problems: Monodromy of ordinary differential equations and the isomonodromy approach
• Riemann-Hilbert problems: general theory
• Riemann-Hilbert problems: The Painleve II equation

1. The Hastings-McLeod Solution
2. Connection formulas
3. The Vanishing Lemma and pole free solutions

• Discrete Painlevé equations and orthogonal polynomials
• Steepest descent for the Riemann-Hilbert problem for orthogonal polynomials
• Double scaling  limits
• Small  dispersion of KdV  (if time  permits)

The overall purpose of the course is to discuss the Riemann­ Hilbert  approach in the asymptotic analysis of special functions/orthogonal  polynomials and differential equations.

After the course, the student is expected to explain and work with the following concepts:

• Monodromy for differential equations
• Riemann-Hilbert problems
• Isomonodromic deformations
• Painlevé equations
• Lax pairs
• Discrete Painlevé equations and orthogonal polynomials
• Deift/Zhou steepest descent  for Riemann-Hilbert  problems

1. g-functions
2. Global  parametrix
3. Local parametrices

• Double scaling limits

After the course, the student should have sufficient skills to independently and efficiently read research papers on the topic.

A Master degree including at least 30 university credits (hp) in in Mathematics (including SF1628 Complex analysis or equivalent).

Some knowledge in linear  analysis  and  operator theory.

• .S. Fokas, A. Its, A. A.Kapaev; V.Y. Novokshenov, Painlev transcendents. The Riemann-Hilbert approach. Mathematical Surveys and Monographs, 128. American Mathematical Society, Providence, RI, 2006. xii+553 pp.

• P.A. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in .Mathematics, 3. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. viii+273pp.

We will also use (part of) papers and lectures notes from the arXiv.

If the course is discontinued, students may request to be examined during the following two academic years.

• INL1 - Assignment, 7.5 credits, grading scale: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

Homeworks, and presentation/oral exam.

Homeworks, and presentation/oral exam.

Maurice Duits

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.