Membros Principais: Míriam Manoel, Maria Aparecida Soares Ruas, Patrícia Hernandes Baptistelli, Regilene de Lazari Oliveira, Carles Bivià-Ausina, Maria Elenice Rodrigues Hernandes
Descrição: This project aims to advance the frontiers of knowledge in singularity theory through the development of fundamental topics in the field. Particular emphasis is given to themes such as the recognition of the topology and geometry of real and complex singularities, the characterization of families of singular sets that satisfy some equisingularity condition, and the Lipschitz geometry of singularities. The invariants will be investigated in diverse forms—geometric, algebraic, and topological. Special classes of singularities, such as matrix singularities and determinantal and toric varieties, will also be studied. Recently obtained pioneering results motivate new research directions in this area.
The central theme of the proposal is the study of real and complex singularities of sets, mappings, differential equations, and vector fields. The main driving question concerns the structure of various classes of singular objects and their generic perturbations, through the development of classification methods and algorithms for recognizing singularities. The investigation of invariants of singularities and the topology of the fibrations associated with them is a fundamental part of the project, which also aims at applications of the theory, particularly in geometry and dynamical systems.
The Brazilian team is specialized in this research area and has made substantial and pioneering contributions in the study of singular invariants, in applications of singularity theory to the geometry and topology of submanifolds, and in the qualitative theory of ODEs and bifurcations. The main goal of the project is to foster interaction between the singularity research activities of the São Carlos group and other centers in Brazil and abroad, with the aim of advancing the following themes: classification, equisingularity, and invariants; geometry and topology of manifolds and singular varieties; singularities in differential geometry; and singularities of vector fields.
The researchers involved have extensive experience in the relevant research areas, and prior collaboration among them has already led to fundamental progress in singularity theory and its applications. The interaction between researchers from the various centers is evidenced by the team's publications.
