Prof. Dr. Adel Khalfallah (King Fahd University of Petroleum & Minerals KSA): Linking Complex Analytic to Nonstandard Algebraic Geometry
Abstract: The value of nonstandard mathematics is in serving as a "guiding star" and often offering a conceptually simple and elegant interpretation and generalization of classical theory and sometimes leads to new concrete standard results. Not so much is known about nonstandard complex analysis, unlike nonstandard real analysis, topology and metric spaces theory. Only some very speciﬁc applications of model theory are used to be known as for instance the Lefschetz principle, the theorem of Tarski-Seidenberg or some simple proofs of Hilbert’s Nullstellensatz. Recently, in collaboration with S. Kosarew, we started a program to develop a theory of analytic geometry using nonstandard methods. One of our fundamental constructions is that of a category of certain ringed spaces, called bounded schemes, which contains the category of algebraic C-schemes and which admits an essentially surjective functor, called the standard part functor, to the category of complex spaces. The advantage of this new more algebraic category is that it allows us to apply many constructions of standard algebraic geometry which are not evident in the analytic context. We obtain analytic results just by taking the standard part functor.
Keywords: Nonstandard analysis; Bounded schemes; internal polynomials; convex subrings; Colombeau’s algebras
(Organisator: Prof. Dr. Alexander Schmitt)