Over the past few years a certain shift of focus within the theory of
algebras of generalized functions (in the sense of J. F. Colombeau) has
taken place. Originating in infinite dimensional analysis and initially
applied mainly to problems in nonlinear partial differential equations
involving singularities, the theory has undergone a change both in in-
ternal structure and scope of applicability, due to a growing number of
applications to questions of a more geometric nature. The present book
is intended to provide an in-depth presentation of these develop- ments
comprising its structural aspects within the theory of generalized
functions as well as a (selective but, as we hope, representative) set
of applications. This main purpose of the book is accompanied by a
number of sub- ordinate goals which we were aiming at when arranging the
material included here. First, despite the fact that by now several
excellent mono- graphs on Colombeau algebras are available, we have
decided to give a self-contained introduction to the field in Chapter 1.
Our motivation for this decision derives from two main features of our
approach. On the one hand, in contrast to other treatments of the
subject we base our intro- duction to the field on the so-called special
variant of the algebras, which makes many of the fundamental ideas of
the field particularly transpar- ent and at the same time facilitates
and motivates the introduction of the more involved concepts treated
later in the chapter.
