250113 VO Colombeau Algebras (Sel. topics mod. Analysis) (2005W)

This lecture offers an introduction into the at present highly active field of
nonlinear theories of generalized functions. As a follow-up to this lecture,
composition of a diploma resp. doctoral thesis in this branch of mathematics
should be possible, with one of the members of the group DIANA supervising
(DIANA: DIfferential Algebras and Nonlinear Analysis; M. Grosser, G. Hörmann,
M. Kunzinger, R. Steinbauer, Wien; [M. Oberguggenberger, Innsbruck; James
Vickers, Southampton, Claudia Garetto, Innsbruck at present]).

The lecture starts with a brief review of the theory of distributions. The
first part then gives an introduction into the basic questions of
multiplication of distributions by studying numerous examples; the need for a
"nonlinear'' theory is highlighted. In parts two and three, respectively, the
"special" and the "full" variants of Colombeau algebras are presented, each
of which contains the distributions as a linear subspace. The concluding
section offers an outlook at diffeomorphism-invariant Colombeau algebras and
at those defined on differentiable manifolds.

Prerequisites: Knowledge of the material covered by the introductory lectures
of the first part of the diploma curriculum (analysis; linear algebra) and of
the topology of metric spaces; proficiency in the fields of functional
analysis and the theory of measure and integration might be useful now and
then, yet is not strictly required.

Examinations: oral, according to individual arrangement.