Non-Archimedean Geometry and Analysis

The general aim of the proposed project is to develop several branches of non-Archimedean Geometry and Analysis, i.e. topics where the role of a ring containing infinitesimals is central. The greatest part of the project concerns the intrinsic study, using the infinitesimal language of the ring of Fermat reals, of classical topics of Differential Geometry of smooth manifolds, infinite dimensional spaces of smooth mappings and spaces with singular subsets. Since at singular points some geometrical quantities can be infinite, e.g. curvature, we will need to develop a theory of infinities as reciprocals of nilpotent infinitesimals. This also permits to introduce generalized functions as quasi-standard smooth functions, i.e. as ordinary smooth functions with some infinitesimal or infinite parameter. We also aim at finding the relationship between these generalized functions and Benci`s theory of ultrafunctions and at transferring to and improving for ultrafunctions some of the results of Colombeau theory. In the third part of the project, we want to show that nilpotent infinitesimals in the ring of Fermat reals can be used as a new framework for the formalization of several methods used in perturbation theory. In the last part of the project, we aim at generalizing the construction of the ring of Fermat reals to a stochastic context using the notion of stochastic little-oh symbol between stochastic processes and to develop limit theorems and first properties of stochastic processes in non-Archimedean probability. The main aim of this part is to arrive at a mathematical structure where to formalize several informal calculations carried out with stochastic differentials both in Physics and in Economics and to a new approach to Probability Theory which is more general and, at the same time, simpler than the classical one.

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 intrinsic study, using the infinitesimal language of the ring of Fermat reals, of classical topics of Differential Geometry of smooth manifolds, infinite dimensional spaces of smooth mappings and spaces with singular subsets. This has been faced by defining and studying a general way to relate the standard world (without infinitesimals) with our new universe which contains infinitesimal numbers. This general way is a suitable functor between these two universes (categories), and we strongly generalize the old definition of Fermat functor into a better version having improved preservation properties.We also advanced the theory of ultrafunctions, generalized the notion of Colombeau algebra and deepened notions of combinatorial number theory. 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.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.