Regularity Theory in Algebras of Generalized Functions

The theory of generalized functions has a long and successful history and is today an indispensable tool in many branches of mathematics, most notably in the theory of partial differential equations (PDE), harmonic analysis, and mathematical physics. Its linear branch, the theory of distributions, initiated by S. Sobolev and L. Schwartz, supplies both the language and a repository of techniques and methods for analyzing and solving linear PDEs, most notably the theory of pseudodifferential and Fourier integral operators, and the resulting microlocal analysis. Starting from the early 1980s, a nonlinear theory of generalized functions has been developed by J.F. Colombeau and his coworkers. This theory provides a framework for addressing nonlinear problems in the presence of singularities, by embedding the space of Schwartz distributions into suitable algebras of generalized functions (Colombeau algebras) that possess optimal permanence property with respect to classical operations. In particular, differentiation and the product of smooth functions are preserved under the embedding. The basic idea of Colombeaus construction is the regularization of distributions through convolution with a mollifier and the expression of analytic properties by asymptotic estimates in terms of a regularization parameter. Colombeau algebras quickly found manifold applications to problems in partial differential equations involving singularities and/or nonlinearities. In addition, the theory has been successfully applied to several other fields, including low-regularity differential geometry, general relativity, or nonstandard analysis. The aim of this project is to advance regularity theory in algebras of generalized functions by intro- ducing new mathematical tools into the field, while at the same time integrating various branches of the existing theory into one common line of research. The project will address regularity issues in the nonlinear theory of generalized functions from three interconnected vantage points: algebraic, spectral theoretical, and using Fréchet techniques. The ultimate aim of this line of research is to obtain a satisfactory theory of the geometrical propagation of singularities, as modelled by Fourier integral operator methods, for nonsmooth hyperbolic problems. The core team of the proposed project will consist of Shantanu Dave, Michael Kunzinger, Eduard Nigsch, and Hans Vernaeve, all of whom are experienced researchers who over last years have made substantial contributions to the field.

The goal of this project was the development of new approaches to the regularity theory in algebras of generalized functions, a theory that had been founded by J.F. Colombeau in the 1980ies and that has since then found manifold applications in analysis, geometry and the theory of differential equations. These new methods were intended to be algebraic and geometric in nature and were expected to have strong ties to differential geometry. These goals have been reached, leading to the development of the following new branches of research: 1.) Regularity theory of Rockland operators: this is a geometric approach to regularity theory that is based on works of Alain Connes and others. The original problem is studied by S. Dave and S. Haller on filtered manifolds, using, among others, methods from spectral theory. This is then applied to the very general class of Rockland differential operators. 2.) New definition and re-development of Colombeau algebras: in this branch of the project, an entirely new and original approach to the nonlinear theory of generalized functions was developed, which in many respects parallels classical analysis. The main difference consists in the fact that here the underlying numbers contain infinitely small and infinitely large quantities (Colombeau-Robinson ring of generalized numbers). This allows one to exactify well-known heuristic calculations from physics, leading to a very intuitive re-invention and development of Colombeau's theory. In particular, a seamless transfer of results from classical analysis becomes feasible in many cases. Moreover, it was possible to put this new approach in the framework of A. Grothendieck's topos theory. Further works applied these constructions to ordinary and partial differential equations. This the new theory has already turned out to be extremely fruitful, both theoretically and in applications, with a number of follow-ups being developed currently. 3.) Functional analytic approach to Colombeau-algebras: This part of the project, mainly developed by Eduard Nigsch, was dedicated to introducing new functional analytic methods into the theory of algebras of generalized functions. In this approach, generalized functions are viewed as maps from smoothing operators into algebras of classical smooth functions. This openes the possibility to unify all known versions of Colombeau's theory in a consistent framwork. Moreover, centrally important operations of Riemannian and Lorentzian geometry, like convariant derivatives could be defined in this theory. This was then applied to relevant problems of General Relativity in follow-up works, in particular to the study of strin-type singular spacetimes. The methods developed in the course of the project have also found applications in problems of ordinary and partial differential equations, and distribution theory.