Geometry of Algebras of Generalized Functions

The nonlinear theory of generalized functions in the sense of J. F. Colombeau experienced rapid growth since its introduction in the early eighties. Its use lies in the fact that it allows simultaneous treatment of differentiation, nonlinear operations, and singular objects while consistently extending both classical analysis as well as the theory of distributions of L. Schwartz. In the context of numerous applications in the description of singular phenomena in mathematical physics a geometric formulation of this theory was developed in the late nineties. This allowed to treat questions involving nonlinear operations and singular objects with a global (i.e., coordinate-invariant) approach. Our main emphasis lies on the so-called full variant of the theory, which - in contrast to the special variant - renders a geometric embedding of Schwartz distributions possible. Tensorial objects hold an important role in both global analysis as well as applications in physics. On manifolds, however, these can be differentiated only in the presence of an additional structure, namely a connection. This leads to the notion of a covariant derivative which is fundamental for the entire field of (semi-)Riemannian geometry. The present project aims at extending the geometric theory of generalized functions, which has been co-developed in a previous project, to tensorial objects in such a way that the notion of covariant derivative respectively connection can be introduced. Apart from treating fundamental questions on the abstract theory of generalized functions, this primarily allows the formulation of a nonlinear distributional (semi-)Riemannian geometry in the setting of an invariant theory of nonlinear generalized functions. In particular, one can treat applications in general relativity in the case of weakly regular space-time metrics.

In this project the theory of nonlinear generalized functions in the sense of Colombeau has been fundamentally restructured and extended in various respects. Most importantly, a vector-valued formulation of the theory on differentiable manifolds, allowing for the use of covariant derivative operators, has been developed.The theory of linear generalized functions or distributions introduced by L.Schwartz in the 1950s does not allow for nonlinear operations like multiplication to be defined consistently and thus suffers from severe shortcomings when one wants to apply it to PDEs with singular coefficients or data, or to nonlinear field theories like general relativity. To overcome these problems, in the 1980s J. F. Colombeau developed a nonlinear extension of Schwartz' theory that can simultaneously handle (a) nonlinear operations, (b) differentiation and (c) singularities.The full theory of Colombeau algebras was, at first, confined to flat Euclidean space. By introducing a simplified version it could be implemented also on manifolds and hence be applied to geometric problems in settings as required, for example, in general relativity. This simplification, however, entailed the loss of certain desirable properties, as e.g. the possibility of embedding distributions canonically.In contrast, a geometric formulation of the full theory took much longer to attain and was much more involved at the technical level. In the first respective approaches, it was restricted to the case of scalar generalized functions and was not capable of handling vector-valued functions in full generality in a geometric context. This important final step was successfully achieved in this project via a reformulation of the foundations of the theory, employing methods of vector-valued distributions and topological tensor products as key ingredients. This way, a single conceptual framework for the construction of a wide variety of Colombeau-type algebras (including the bulk of the ones considered so far) originated, shedding light on many structural questions. One of the principal features of the new theory developed in this project consists in allowing for a covariant derivative for nonlinear generalized functions. This is an indispensable prerequisite for applying Colombeau theory to questions of singular (semi-)Riemannian geometry.This development has been complemented by research on Riemannian and Lorentzian metrics of low regularity. In this context the singularity theorems of Hawking and Penrose have been extended to the setting of C1,1-metrics. Moreover, impulsive gravitational waves have been studied and geodesic completeness has been established for large classes of these exact solutions of Einsteins equations.