[geometry-ml:01491] Alexander Its氏, Elizabeth Its氏

[Martin Guest]

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日時：2012年7月6日(金）15:00--17:30
講演者：  Alexander Its, Distinguished Professor
         Elizabeth Its, Senior Lecturer
         (Indiana University - Purdue University Indianapolis, USA)
於：首都大学東京 南大沢キャンパス, Building 8, 6th Floor, Room 618

組織委員：
Martin Guest (首都大学東京)


日程：

15:00--16:00
SPEAKER: Elizabeth Its

TITLE: A Riemann-Hilbert approach to the boundary problems
for linear PDEs. The elastodynamic  equation in the quarter plane.
A case study.

ABSTRACT: The Riemann-Hilbert method was originated
in the theory of integrable nonlinear PDEs. In  the 90s,
the method was extended to a number of new
areas,  and since then it has played an important role
in solving  a number of long-standing  problems
of analysis and mathematical physics.
In the talk, we will present some recent developments
in  the Riemann-Hilbert approach obtained back
in the PDE theory. This time, the Riemann-Hilbert techniques is
applied to linear problems but  in the domains which
do not allow a direct separation of variables. We will focus
on the solution of the boundary value problem for the
elastodynamic  equation in the quarter plane. We shall show
that the problem is reduced to a matrix Riemann-Hilbert
problem with a shift posed on a torus. This is
a joint work with  Alexander Its and Julius Kaplunov.


16:30--17:30
SPEAKER: Alexander Its

TITLE: Special Functions and Integrable Systems

ABSTRACT: The recent developments in the theory of integrable systems
have revealed its intrinsic relation to the theory of special functions.
Perhaps the most generally known aspects of this relation are
the group-theoretical, especially the quantum-group theoretical,
and the algebra-geometrical ones. In the talk we will discuss
the analytic side of the Special Functions-Integrable Systems
connection. This aspect of the relation between the two theories
is less known to the general mathematical community,
although it goes back to the  classical works of Fuchs, Garnier and
Schlesinger on the isomonodrony deformations of the systems of linear differential equations with rational coefficients. Indeed, the monodromy theory of linear systems provides a unified framework for  the  linear (hypergeometric type) and nonlinear (Painleve type) special functions  and, simultaneously, builds a base for the new powerful technique of the asymptotic analysis  - the Riemann-Hilbert method.

In this survey talk, which is based on the works of many authors
spanned over more than two decades, the isomonodromy point of view
on special function will be outlined. We will also review the history of
the Riemann-Hilbert method as well as its most recent applications in the theory of orthogonal polynomials and random matrices.