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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:The Riemann-Hilbert method. Toeplitz determinants
as a case study - Alexander Its (Indiana Universit
y-Purdue University Indianapolis)
DTSTART;TZID=Europe/London:20191024T140000
DTEND;TZID=Europe/London:20191024T153000
UID:TALK132898AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/132898
DESCRIPTION:The Riemann-Hilbert method is one of the primary a
nalytic tools of modern theory
of integrable sy
stems. The origin of the method goes back to Hilbe
rt'\;s 21st prob-
lem and classical Wiener-H
opf method. In its current form\, the Riemann-Hilb
ert
approach exploits ideas which goes beyond t
he usual Wiener-Hopf scheme\, and
they have the
ir roots in the inverse scattering method of solit
on theory and in the
theory of isomonodromy def
ormations. The main \\bene\;ciary" of this\, l
atest ver-
sion of the Riemann-Hilbert method\,
is the global asymptotic analysis of nonlinear
systems. Indeed\, many long-standing asymptotic p
roblems in the diverse areas of
pure and applie
d math have been solved with the help of the Riema
nn-Hilbert
technique.
One of the recent appl
ications of the Riemann-Hilbert method is in the t
heory
of Toeplitz determinants. Starting with O
nsager'\;s celebrated solution of the two-
d
imensional Ising model in the 1940'\;s\, Toepli
tz determinants have been playing
an increasing
ly important role in the analytic apparatus of mod
ern mathematical
physics\; speci\;cally\, i
n the theory of exactly solvable statistical mecha
nics 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
Painlev\;e type description of the tran-
si
tion asymptotics of Toeplitz determinants. The RIe
mann-Hilbert view on the
Painlev\;e functio
ns will be also explained.
LOCATION:Seminar Room 2\, Newton Institute
CONTACT:INI IT
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