# Geometry of Algebras of Generalized Functions

### project team

- Grosser, Michael, (Project Lead)
- Nigsch, Eduard, (Scientific Project Staff)
- Kunzinger, Michael (Affiliated Project Staff)
- Steinbauer, Roland (Affiliated Project Staff)
- Vickers, James A. G. (Affiliated Project Staff)

### project time

4/07/11 → 15/08/16

## Abstract

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.

Generalized functions, introduced by L.Schwartz in the 1950s, arise naturally in the abstract formulation of a wide range of discontinuous, singular phenomena in physical models. Prime examples for this are shock waves in elasticity or concentrated sources in field theories like electrodynamics or general relativity. However, Schwartz' theory 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 of distributions 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 [WEG: a] 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.

Finally, for applications of Colombeau algebras in the theory of Fourier Integral Operators, an adapted version of symplectic geometry has been developed in the context of Colombeau algebras.