skip to content
 

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

Presented by: 
Alexander Its
Date: 
Friday 25th October 2019 - 14:00 to 15:30
Venue: 
INI Seminar Room 2
Abstract: 
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 \beneciary" 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
technique.
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; specically, 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.
The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons