Colombeau Algebras: Structure Theory and Regularity

Algebras of nonlinear generalized functions were introduced in the 1980s by J.-F. Colombeau in order to treat mathematical questions involving nonlinear operations, differentiation and singularities. These algebras constitute an extension of both classical analysis and L. Schwartz` distribution theory. Colombeau algebras are employed successfully for the study of singular partial differential equations, non-smooth differential geometry and singular Fourier integral operators. They have found numerous applications e.g. in mathematical geophysics and general relativity. Up to today, many aspects of this theory have been developed only for simplified variants of these algebras, so-called special Colombeau algebras. These suffer from certain restrictions, for example it is impossible to have a geometric (coordinate invariant) embedding of distributions into special algebras. For this reason, a geometric theory of nonlinear generalized functions was developed since the late 1990s in the context of full Colombeau algebras, which permit a coordiante-invariant treatment of singular problems in a geometric context. The development of this geometric theory of full Colombeau algebras in the most general case, which is for generalized sections of vector bundles, was made possible only recently through the introduction of a functional analytic approach which anchors Colombeau theory in the theoretical foundations of distribution theory. The aim of the present project is to further develop this functional analytic approach and hence connect Colombeau theory to functional analytic methods which played only a minor role in it so far. One major goal in this is the development of a structure theory for Colombeau algebras that can explain the multitude of existing variants, their properties and their relations to each other, and only in the context of which one can introduce and study important operations on nonlinear generalized functions. As a result, methods will be developed for naturally adapting Colombeau algebras to specific applications and placing them in closer relation to classical distribution theory. The other main aim is to develop microlocal analysis and regularity theory for Colombeau algebras based on this functional analytic approach. This way, more natural geometric approaches to this field become possible, which gives new tools and methods for the analysis of nonlinear generalized functions.

This research project was situated in the field of nonlinear generalized functions in the sense of Colombeau. This is a mathematical theory developed to handle problems which at the same time involve nonlinearities (i.e., the output of a physical system can be very disproportional to its input), differentiation (i.e., keeping track of the momentarily change of a system), and singularities (i.e., phenomena whose description includes terms like "infinitely small" or "infinitely large"). The main aim of this project was to extend the theory of nonlinear generalized functions such that it can be applied to geometric questions. Usually, geometric phenomena can be described in arbitrarily many but equivalent ways (coordinate systems), so any mathematical theory applied to them should be able to give the same results regardless of the particular choice of description. This requirement, however, poses serious difficulties in the context of nonlinear generalized functions. These difficulties were successfully tackled in the course of this project, which involved a fundamental reworking of the whole theory. This not only led to a much better structural understanding but gave rise to a very flexible general structure of nonlinear generalized function spaces which can be adapted easily to specific situations and needs. On these grounds, geometric questions in the context of nonlinear generalized functions, and in particular of general relativity, have been successfully studied and better understood.