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 END:VEVENT END:VCALENDAR