Singularities, geometry and differential equations

Abstract

Singularity theory has wide applications to other areas in mathematics and in particular to differential geometry and to the qualitative study of ordinary and partial differential equations. These branches of mathematics also feed back into and enrich singularity theory. The following project, which has well defined objectives, aims at consolidating research activities in the state of São Paulo in the areas of dynamical systems and singularity theory. The interaction will promote the development of the following points 1) geometric properties of differential equations and bifurcations, 2) generic properties of submanifolds in euclidean spaces. The main focus of the research proposal on "geometric properties of differential equations and bifurcations" is to study the singularities of certain classes of differential equations in IRn. These classes include reversible systems, discontinuous systems and some mathematical models from Biomathematics. The purpous of the proposal on "generic properties of submanifolds in euclidean spaces" is to study the geometric properties of smooth and singular submanifolds in euclidean spaces which are defined by applications whose singularities are finitely determined. The principal problems to be considered in the first instance are those concerning embedded surfaces in R4 and singular surfaces in R3. The main approach that will be taken for dealing with the proposed projects is singularity theory. The development of the projects requires new results in this theory and the expertise of the researchers involved is very relevant for achieving the set objectives. The project is designed to allow maximum collaboration between the participants. It should be stressed that all the members involved in the project have wide experience in research and collaborative work between the members already exists. (AU)