On the origins of Riemann-Hilbert problems in mathematics

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 Painlevé-II formula of Amir, Corwin, Quastel \cite{ACQ} that enters in the description of the KPZ crossover distribution. Parts of this text are based on the author's plenary lecture at the $15$th International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA) in Hagenberg, Austria.

arXiv comment56 pages, 9 figures, to appear in Nonlinearity. Version 2 corrects typos and updates literature

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Cited by in corpus (9)

• Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations
• Degenerate Riemann-Hilbert-Birkhoff problems, semisimplicity, and convergence of WDVV-potentials
• Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation
• Integrability in the weak noise theory
• Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, Circular $\beta$-ensemble and double confluent Heun equation
• Jánossy densities and Darboux transformations for the Stark and cylindrical KdV equations
• Kubo-Martin-Schwinger relation for an interacting mobile impurity
• Maxima of log-correlated fields: some recent developments
• Integrable equations associated with the finite-temperature deformation of the discrete Bessel point process