On the origins of RiemannHilbert problems in mathematics
Abstract
This article is firstly a historic review of the theory of RiemannHilbert 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 RiemannHilbert 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 integrodifferential PainlevéII 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 DOI:
 10.48550/arXiv.2003.14374
 arXiv:
 arXiv:2003.14374
 Bibcode:
 2020arXiv200314374B
 Keywords:

 Mathematical Physics;
 Mathematics  History and Overview;
 Mathematics  Probability;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 Primary 30E25;
 Secondary 45M05;
 60B20
 EPrint:
 56 pages, 9 figures, to appear in Nonlinearity. Version 2 corrects typos and updates literature