Singularities, geometry and differential equations

Abstract

Singularity theory has wide applications to various areas of mathematics and in particular to differential geometry and qualitative theory of differential equations. These branches of mathematics feedback, in turn, into and enrich singularity theory. The aim of this project is the development of methods of classification of real and complex singularities, with special attention to invariants and equisingularity conditions for families of sets and mappings. The following project, with well defined objectives, aims at consolidating research activities in the state of Sao Paulo in geometric aspects of dynamical systems and singularity theory. The interaction will promote the development of in the following areas: 1) topology and classification of singularities; 2)multiplicities, integral closure and equisingularity; 3) singularities in differential geometry and implicit differential equations; 4) applications of singularity theory to bifurcation problems. The purpose of the project topology and classification of singularities and the study of their invariants’ is to study invariants and the topology of singularities, with special attention to non isolated singularities of sets and mappings. The expected results are on topological triviality of families of analytic singularities and their Milnor fibrations, as well as in the metric theory of singularities. In the research project ‘multiplicity, integral closure and equisingularity’, we shall study numerical invariants for equisingularity of families of holomorphic mappings. The algebraic aspects of singularity theory are the main interest of this project, which intends to extend the concept of multiplicities to ideals and modulus of non-finite co length. The research project on ‘singularities in differential geometry and implicit equations’ are motivated by the investigation of new results on the geometry of high co dimensional embeddings of manifolds in euclidean and hyperbolic spaces. The study of implicit equations, emphasizing the differential equations of differential geometry, is another direction of this research project. The purpose of the project ‘an application of singularity theory to bifurcation problems’ is to apply methods of singularity theory to the classification and recognition of multiparameter bifurcation problems, and to investigate the geometry of bifurcation sets. The proposal is designed to allow maximum collaboration between the participants. We observe that all the members involved in the project have valuable experience in research related to the proposal, and collaboration between the members of the group has already given fruitful results. We also aim to consolidate the interchange among the singularity group in the state of Sao Paulo and researchers in the state of Parana, Ceara and Paraiba, as well as in other countries, in special with groups in Portugal, United Kingdom, Spain, Germany, France, Japan, United States and Poland. (AU)