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.

No additional funding sources recorded.