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SUMMARY:The Riemann-Hilbert method. Toeplitz determinants as a case study
- Alexander Its (Indiana University-Purdue University Indianapolis)
DTSTART:20191024T130000Z
DTEND:20191024T143000Z
UID:TALK132949@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:The Riemann-Hilbert method is one of the primary analytic tool
s 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 whi
ch goes beyond the usual Wiener-Hopf scheme\, and
they have their roots
in the inverse scattering method of soliton theory and in the
theory o
f isomonodromy deformations. The main \\bene\;ciary" of this\, latest
ver-
sion of the Riemann-Hilbert method\, is the global asymptotic anal
ysis of nonlinear
systems. Indeed\, many long-standing asymptotic probl
ems in the diverse areas of
pure and applied math have been solved with
the help of the Riemann-Hilbert
technique.
One of the recent applic
ations of the Riemann-Hilbert method is in the theory
of Toeplitz deter
minants. Starting with Onsager'\;s celebrated solution of the two-
d
imensional Ising model in the 1940'\;s\, Toeplitz determinants have bee
n playing
an increasingly important role in the analytic apparatus of m
odern mathematical
physics\; speci\;cally\, in the theory of exactl
y solvable statistical mechanics and
quantum \;eld models.
In th
ese two lectures\, the essence of the Riemann-Hilbert method will be pre-<
br>sented taking the theory of Topelitz determinants as a case study. The
focus will
be on the use of the method to obtain the Painlev\;e typ
e description of the tran-
sition asymptotics of Toeplitz determinants.
The RIemann-Hilbert view on the
Painlev\;e functions will be also
explained.
LOCATION:Seminar Room 2\, Newton Institute
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