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Boundary relations

Description and Aim Numerous physical and
technical processes can be described by differential equations. The
mathematical treatment of the corresponding initialvalue and boundaryvalue
problems uses methods from operator theory and functional analysis. Typically
the properties of the solutions of initialvalue and boundaryvalue problems
depend on the spectral properties of the associated differential operators.
Hence operator theoretic methods in spectral theory are intimately connected
with the analysis of differential equations and their applications. These
operator theoretic methods give rise to the extension theory in the title. On
the other hand the theory of boundary triplets offers a convenient technique
for dealing with boundaryvalue problems involving ordinary differential
operators, partial differential operators, and equations with operatorvalued
coefficients. Boundary triplets were first introduced some thirty years ago and
nowadays they are extensively used also in mathematical physics, for instance,
for Schrödinger operators on quantum graphs and for scattering problems. Their
popularity has been widened due to the fundamental contributions by Derkach and Malamud on the area of extension theory of
abstract symmetric operators who associated with a boundary triplet an analytic
object, the socalled Weyl function. The original notion
of boundary triplet is sometimes too restrictive because the corresponding Weyl function belongs to a class of functions, which is in
general too small when dealing with infinitedimensional problems. This
objection led to introduction of the new notions of boundary relations and
associated Weyl families with a proper generality for
dealing with various problems appearing in the extension theory of general
symmetric operators. The theory of boundary triplets and boundary relations is
presently being further developed with applications to, for instance, elliptic
operators and to nonstandard boundary conditions. Parallel to the development
of boundary relations, analogous notions have been independently developed in the
area of system theory. The aim of the workshop is to bring together
specific mathematicians from operator theory, functional analysis, and system
theory involved with these recent developments. The aim is to pay attention to
the general field of extension theory with a special emphasis on boundary
relations and boundary triplets and their applications to boundary value
problems for ordinary and partial differential equations, as well as to
consider the intimate connections with system theory. So the workshop brings
together people working in these areas, with an aim to create new views and
ideas for some further progress in the outlined area of the workshop. [Back] 