Analysis and Geometry based on generalized numbers

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. The theory of Fermat reals proposes itself as an alternative (in classical logic) to Synthetic Differential Geometry and, more generally, as a good cartesian closed categorical framework, where nilpotent infinitesimals are available, and applicable to diffeological spaces. It is a theory of infinitesimals that does not need any background in formal logic, and hence seems more easily acceptable by differential geometers and physicists. The proposed project is structured to develop two interacting threads depending on the base ring of numbers: Colombeau generalized numbers and Fermat reals. Its aim is to develop an extension of Colombeau`s theory of generalized functions which permits to see them as a category of smooth set-theoretical functions on a ring of generalized numbers, with applications also to infinite dimensional spaces like spaces of smooth mappings and to Differential Geometry of spaces with singular points. This theory has strong analogies with ordinary smooth functions and very good categorical properties, so that several classical theorems can be generalized. Infinite dimensional spaces will be studied in a quasi-topos embedding the category of diffeological spaces, in particular that of smooth manifolds. We expect close connections of the functorial extension of every diffeological space with new generalized points and the related preservation properties with the semantics of intuitionistic logic. The first application of this new theory is in the intrinsic study, using infinitesimal methods, of classical topics of Differential Geometry of smooth manifolds, spaces of smooth mappings and spaces with singular subsets, like unions of manifolds of different dimensions.

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