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.

The general aim of this project was to develop several branches of non- Archimedean Geometry and Analysis, i.e. topics where the role of infinitesimals numbers is central. This general aim has been realized, even if some particular topic revealed more difficulties than expected.The greatest part of the project concerns the development of a new theory of generalized functions, called generalized smooth functions, and to a further advancing of the theory of Fermat reals. All these theories are framed in a non-Archimedean context, i.e. they use different forms of infinitesimal and infinite numbers other than the usual real numbers. Even if these new type of functions can also take infinite of infinitesimal values, several classical results of differential and integral calculus are still true. Frequently, the statements of these results is simply a reformulation of classical analogues but simply replacing the classical field of real numbers with the new non- Archimedean ring of scalars. New important general results about solutions of both singular nonlinear ordinary and partial differential equations have been achieved.The main result of this project has been to arrive at mathematical structures where to formalize several informal calculations carried out with infinitesimals both in Mathematics and in Physics, but using a modern and rigorous mathematical language.