Applied nonstandard analysis

The present research project focuses on applications of Nonstandard Analysis to Mathematical Analysis, Differential Geometry, and Combinatorial Number Theory. The project is divided in three parts. In the first part we plan to develop a topological approach to Nonstandard Analysis. This approach has two advantages with respect to the classical one: it allows to obtain nonstandard constructions without having to appeal to formal logic (so to make nonstandard methods easily available to a wider range of scientists) and it is particularly suitable for applications to PDEs and Differential Geometry. The second, and main, part of the project concerns the theory of ultrafunctions. Ultrafunctions are a new family of generalized functions, constructed by means of nonstandard analysis, that have been introduced by Professor V. Benci from the University of Pisa and applied by Professor V. Benci and the main applicant to study topics in Calculus of Variations, Partial Differential Equations, Functional Analysis and Mathematical Physics. In this research project we plan to apply the theory of ultrafunctions to Differential Geometry and to study evolution problems. In particular, we plan to develop three topics. We want to find the relationship between ultrafunctions, Colombeau generalized functions and Robinson`s asymptotic functions, so to transfer to and improve for ultrafunctions some of the results of Colombeau and Robinson`s theories. Then we want to develop a theory of ultrafunctions valued in manifolds; this would allow to apply ultrafunctions to various problems in Differential Geometry and Relativity Theory. Finally, we want to develop the notion of ultrafunction solutions of evolution problems. This will be performed by studying some particular evolution equations by means of ultrafunctions, in particular by studying in details Burger`s inviscid equation. In the third part of the project we plan to develop new nonstandard techniques in Combinatorial Number Theory. In particular, we want to study the partition regularity of nonlinear diophantine equations, the partition regularity of infinite matrices and the combinatorial properties of high density sets. The first and the second topic will be studied by means of a new nonstandard technique, based on iterated hyperextensions, that has been developed by the main applicant in some recent works. This technique allows to study these problems by means of rather simple manipulations of symbols. This should allow to apply new algebraic and computational approaches to these problems. To study the combinatorial properties of high density sets we plan to highlight the relationship between the known nonstandard approaches (usually based on Loeb measures) and F-finite embeddabilities, that are a family of (pre)orders, defined on subsets and ultrafilters on N, with good combinatorial properties.

Final summary for Lise-Meitner project M 1876-N35 Applied nonstandard analysis L. Luperi Baglini In this project we obtained three main results, all related with the development of new non-Archimedean methods in mathematics. We believe that this is an important topic: in fact, in many cases it is very dicult to construct classical models or to use classical techniques in the presence of problems where innite or innitesimal quantities naturarly arise, whilst non-Archimedean models are usually easier to construct than the corresponding classical ones, and they are quite near to the intuition. The rst main result of this project regards that branch of combinatorial number theory involved with the partition regularity of nonlinear Diophantine equations. This is a topic where the knowledge is scarce and fragmented; thanks to a collaboration with Professor M. Di Nasso (University of Pisa) we have been able to unify and extend most of the known results under two rather simple and general statements. The relevance of this fact is that it allows to unify many dierent results that had been previously obtained by means of various dierent techniques, whilst at the same time introducing some new ideas based on NSA that could be employed to start a systematic study of other questions in this eld. In particular, the peculiar manipulation of ultralters that is done by means of their generators could lead to some interesting applications by means of computational methods. The second result is the introduction of a new theory of generalized solutions of PDE called generalized ultrafunction solutions. This has been done thanks to a collaboration with Professor V. Benci (University of Pisa). These new generalized solutions have some very useful formal properties that allow to reproduce many classical results in a much more general setting. In particular, we have been able to prove an existence and uniqueness theorem for generalized ultrafunctions solutions under very mild hypotheses on the PDE to be studied (such solutions exist even in the presence of a classical blow up). To prove the usefulness of this approach, we performed a detailed study of Burgers` equation, which is a PDE that is used to model many real-world situations such as, e.g., trac ows. The third result is the study of many basic properties and applications of Colombeau generalized functions and their extensions by means of generalized smooth functions. In particular, thanks to a collaboration with PhD A. Lecke (University of Vienna) and PhD P. Giordano (University of Vienna) we have been able to extend many basic variational principles to this generalized context, which led to applications to the study of geodesics in low regularity settings. This could easily lead to applications to physics in the very next future. 1