The Riemann-Hilbert method. Toeplitz determinants as a case study
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| Description: |
Its, A
Thursday 24th October 2019 - 14:00 to 15:30 |
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| Created: | 2019-10-25 10:42 |
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| Collection: | Complex analysis: techniques, applications and computations |
| Publisher: | Isaac Newton Institute |
| Copyright: | Its, A |
| Language: | eng (English) |
| Distribution: |
World
(downloadable)
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| Explicit content: | No |
| Aspect Ratio: | 16:9 |
| Screencast: | No |
| Bumper: | UCS Default |
| Trailer: | UCS Default |
| 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. |
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