Paths to a Foundation of Non Archimedean Geometry" In this thesis, historical bases of Non Archimedean Geometry (NAG) are revised, with special emphasis to those from Hilbert and Veronese. A definition of Non Standard Geometry (NSG) is suggested and general Hilbert -Veronesian (H-V) and Birkhoffian models of Non Standard non Archimedean Geometry (NSAG) are builted and studied. Analytic and synthetic realizations of the models are presented or suggested and studied. Some elementary applications to solve standard problems and situations are worked out. /Abstract - Abstract of the Ph.D. Thesis submitted by Paulino Lima Fortes to the University of Évora and entitled "Caminhos para uma Fundamentação da Geometria Não Arquimediana (Paths to a Foundation of Non Archimedean Geometry)" In this thesis, historical bases of Non Archimedean Geometry (NAG) are revised, with special emphasis to those from Hilbert and Veronese. A definition of Non Standard Geometry (NSG) is suggested and general Hilbert -Veronesian (H-V) and Birkhoffian models of Non Standard non Archimedean Geometry (NSAG) are builted and studied. Analytic and synthetic realizations of the models are presented or suggested and stud ied. Some elementary applications to solve standard problems and situations are worked out.

CEMAT - Center for Computational and Stochastic Mathematics