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The Riemann-Hilbert method. Toeplitz determinants as a case study

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The Riemann-Hilbert method is one of the primary analytic tools of modern theory
of integrable systems. The origin of the method goes back to Hilbert's 21st prob-
lem and classical Wiener-Hopf method. In its current form, the Riemann-Hilbert
approach exploits ideas which goes beyond the usual Wiener-Hopf scheme, and
they have their roots in the inverse scattering method of soliton theory and in the
theory of isomonodromy deformations. The main \bene ciary” of this, latest ver-
sion of the Riemann-Hilbert method, is the global asymptotic analysis of nonlinear
systems. Indeed, many long-standing asymptotic problems in the diverse areas of
pure and applied math have been solved with the help of the Riemann-Hilbert
One of the recent applications of the Riemann-Hilbert method is in the theory
of Toeplitz determinants. Starting with Onsager's celebrated solution of the two-
dimensional Ising model in the 1940's, Toeplitz determinants have been playing
an increasingly important role in the analytic apparatus of modern mathematical
physics; speci cally, in the theory of exactly solvable statistical mechanics and
quantum eld models.
In these two lectures, the essence of the Riemann-Hilbert method will be pre-
sented taking the theory of Topelitz determinants as a case study. The focus will
be on the use of the method to obtain the Painleve type description of the tran-
sition asymptotics of Toeplitz determinants. The RIemann-Hilbert view on the
Painleve functions will be also explained.

This talk is part of the Isaac Newton Institute Seminar Series series.

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