126
reads

Alexander Its

2011-05-13
11:00:00 - 12:00:00

308 , Mathematics Research Center Building (ori. New Math. Bldg.)

The classical Painlev'e equations are getting increasingly involved in many areas of mathematics. Indeed, it is now clear that the Painlev'e transcendents play the same role in nonlinear problems that the 'linear' special functions, such as Airy functions, Bessel functions etc., play in linear science. During the last twenty to twenty five years, great progress in the theory of Painlev'e equations themselves has been achieved. This progress has been based on the so-called Riemann-Hilbert, or Isomonodromy, Method.

The Riemann-Hilbert method reduces a particular problem at hand to the {it Riemann-Hilbert problem} of analytic factorization of a given matrix-valued function defined on an oriented contour in the complex plane. The main benefit of this reduction arises in asymptotic analysis.

The principal goal of this three-lecture series is to demonstrate the power of the Riemann-Hilbert approach to the asymptotic analysis of the Painlev'e equations considering the second Painlev'e equation as {it a case study} and using the third Painlev'e equation as another illustrative example. This would allow us to present the Riemann-Hilbert scheme in a rather elementary, although sufficiently general, manner. Simultaneously, the Painlev'e equations will be introduced as an intrinsic part of the general Fuchsian monodromy theory. We are also planning to outline some of the other applications of the Riemann-Hilbert technique which range from integrable PDEs of KdV type (the area where the Riemann-Hilbert method was in fact originated) to exactly solvable quantum field and statistical mechanics models and (most recently) to the theory of orthogonal polynomials, matrix models, and Toeplitz and Hankel determinants.