Global Analysis in Algebras of Generalized Functions

Algebras of generalized functions have recently found an increasing number of applications in a geometric context. Both in the structural theory (non-smooth differential geometry) and in applications in mathematical physics, in particular in non-smooth general relativity, important progress has been made. The precursor of the present project, Geometric Theory of Generalized Functions, has made a substantial contribution to this fast developing field. The aim of this research project is to build on these foundations to advance global analysis in algebras of generalized functions in close interaction with relevant applications. Our investigations will be carried out in the framework of Colombeau`s theory of algebras of generalized functions, a nonlinear extension of the linear theory of distributions in the sense of Laurent Schwartz, which provides a framework for analyzing problems that simultaneously involve nonlinearities, differentiation, and singularities. The basic ideas underlying the construction are regularization of singular objects via convolution and asymptotic estimates in terms of a regularization parameter for quantifying the strength of singularities. There are two main versions of the construction, namely special and full algebras, the latter distinguished by the existence of canonical embeddings of spaces of distributions. For both variants of the theory we intend to lay the foundations of a global analysis of generalized functions. Main directions of investigation will be pseudo-Riemannian geometry (relevant for applications in general relativity), algebraic approaches to non-smooth differential geometry, and the study of algebras of generalized functions on manifolds with additional structure. As in the precursor project, special attention will be given to applications, in particular in general relativity, the field in which many of the concepts relevant to the project have had their origin.

Algebras of generalized functions have recently found an increasing number of applications in a geometric context. Both in the structural theory (non-smooth differential geometry) and in applications in mathematical physics, in particular in non-smooth general relativity, important progress has been made. The present project has taken up this line of research and has led to a number of new developments, both foundational, and in applications. Our investigations were carried out in the framework of Colombeaus theory of algebras of generalized functions, a nonlinear extension of the linear theory of distributions in the sense of Laurent Schwartz, which provides a framework for analyzing problems that simultaneously involve nonlinearities, differentiation, and singularities. The basic ideas underlying the construction are regularization of singular objects via convolution and asymptotic estimates in terms of a regularization parameter for quantifying the strength of singularities. There are two main versions of the construction, namely special and full algebras, the latter distinguished by the existence of canonical embeddings of spaces of distributions. For both variants of the theory the project has laid the foundations of a global analysis of generalized functions. The main directions of investigation were pseudo-Riemannian geometry (relevant for applications in general relativity), algebraic approaches to non-smooth differential geometry, and the study of algebras of generalized functions on manifolds with additional structure. Special attention was given to applications, in particular in general relativity, the field in which many of the concepts relevant to the project have had their origin.