On the origins of Riemann-Hilbert problems in mathematics

Thomas Bothner*

*Corresponding author for this work

Research output: Contribution to journalReview article (Academic Journal)peer-review

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Abstract

This article is firstly a historic review of the theory of Riemann-Hilbert problems with particular emphasis placed on their original appearance in the context of Hilbert’s 21st problem and Plemelj’s work associated with it. The secondary purpose of this note is to invite a new generation of mathematicians to the fascinating world of Riemann-Hilbert techniques and their modern appearances in nonlinear mathematical physics. We set out to achieve this goal with six examples, including a new proof of the integro-differential Painlev ́e-II formula of Amir, Corwin, Quastel [6] that enters in the description of the KPZ crossover distri- bution. Parts of this text are based on the author’s Szeg ̋o prize lecture at the 15th International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA) in Hagenberg, Austria.
Original languageEnglish
Pages (from-to)R1-R73
Number of pages71
JournalNonlinearity
Volume34
Issue number4
DOIs
Publication statusPublished - 22 Feb 2021

Keywords

  • Riemann-Hilbert problem
  • Hilbert’s 21st problem
  • Fuchsian systems
  • monodromy group
  • singular integral equations
  • defocusing nonlinear Schrödinger equation
  • Painlevé-II equation
  • spin- 1 2XY model
  • orthogonal polynomials
  • nonlinear steepest descent method
  • random matrices
  • random permutations
  • KPZ equation

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