The present research project concerns the application of nilpotent infinitesimal methods to differential geometry
and generalized functions. The project is subdivided into three parts. In the first one we want to develop intrinsic
differential geometry both for ordinary manifolds and spaces of smooth mappings using the theory of Fermat reals.
The theory of Fermat reals has been developed by the applicant as a cartesian closed categorical framework,
including diffeological spaces, where nilpotent infinitesimal methods are available. In the second part we want to
generalize F. Colombeau`s construction of algebras of generalized functions, so as to include also non polynomial
smooth operations. In the last part of the project we plan to connect the first two sections, finding the first
relationships between nilpotent infinitesimals and these generalized functions. Indeed, we plan to prove
infinitesimal Taylor formulas, the Fermat-Reyes theorem for generalized functions, and to define generalized
functions with values in diffeological spaces, in particular in spaces of mappings.
The project will be performed at the mathematics department of the University of Vienna, in collaboration with
Prof. M. Kunzinger and the DIANA research group, thereby profiting from the outstanding expertise of the host
institute in closely related fields, like non-linear theory of generalized functions and the convenient settings of
global analysis (developed by Prof. A. Kriegl and Prof. P.W. Michor who are also part of the host institution).