Geometric Theory of Generalized Functions

Algebras of generalized functions in the sense of J. F. Colombeau were introduduced in the early 1980ies as a tool for analyzing mathematical problems involving (a) differentiation, (b) nonlinear operations and (c) singular objects. They therefore simultaneously extend classical analysis (able to handle (a) and (b)) and distribution theory (designed to treat (a) and (c)). Initially, the main field of applications of Colombeau algebras naturally lay in the field of nonlinear partial differential equations involving singularities. Starting from the mid 1990ies, moreover, also applications in a more geometric context, in particular in Lie group analysis of differential equations and Einstein`s theory of general relativity began to emerge. Coordinate invariance of the construction therefore attained central importance in the further development of the theory. Since the original version of Colombeau`s construction did not enjoy this property, a number of authors contributed to a fundamental restructuring of the construction in the late 1990ies which resulted in the so-called "geometric theory of generalized functions". This setting, finally, allows to consider problems involving (a), (b), (c) in a global context. The aim of the present project is twofold: first, to further develop the geometric theory of generalized functions itself (singular ordinary differential equations, generalized connections, embedding properties, ); and second, to apply it to problems of general relativity (spherical impulsive gravitational waves, dust solutions of Einstein`s equations, ) and symmetry group analysis of differential equations (globalization of the existing theory, study of group invariant generalized functions), the fields where much of the incentive for its development originated.

Algebras of generalized functions in the sense of J. F. Colombeau were introduduced in the early 1980ies as a tool for analyzing mathematical problems involving (a) differentiation, (b) nonlinear operations and (c) singular objects. They therefore simultaneously extend classical analysis (able to handle (a) and (b)) and distribution theory (designed to treat (a) and (c)). Initially, the main field of applications of Colombeau algebras naturally lay in the field of nonlinear partial differential equations involving singularities. Starting from the mid 1990ies, moreover, also applications in a more geometric context, in particular in Lie group analysis of differential equations and Einstein`s theory of general relativity began to emerge. Coordinate invariance of the construction therefore attained central importance in the further development of the theory. Since the original version of Colombeau`s construction did not enjoy this property, a number of authors contributed to a fundamental restructuring of the construction in the late 1990ies which resulted in the so-called "geometric theory of generalized functions". This setting, finally, allows to consider problems involving (a), (b), (c) in a global context. The aim of the present project is twofold: first, to further develop the geometric theory of generalized functions itself (singular ordinary differential equations, generalized connections, embedding properties, ...); and second, to apply it to problems of general relativity (spherical impulsive gravitational waves, dust solutions of Einstein`s equations, ...) and symmetry group analysis of differential equations (globalization of the existing theory, study of group invariant generalized functions), the fields where much of the incentive for its development originated.