Singularities of differentiable mappings: theory and applications
Introduction to the study of differential equations: a dynamic approach
Grant number: | 03/03107-9 |
Support type: | Research Projects - Thematic Grants |
Duration: | December 01, 2003 - November 30, 2007 |
Field of knowledge: | Physical Sciences and Mathematics - Mathematics - Geometry and Topology |
Principal researcher: | Carlos Teobaldo Gutierrez Vidalon |
Grantee: | Carlos Teobaldo Gutierrez Vidalon |
Home Institution: | Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil |
Pesquisadores principais: | Maria Aparecida Soares Ruas |
Associated grant(s): | 06/52081-0 - Jacques Elie Furter | Brunel University - Inglaterra,
AV.EXT 06/52268-3 - Numerical control of equisingularity for map germs from cn to c3, n >-3., AR.EXT 06/51744-6 - Partial hyperbolicity for synplectic diffeomorphisms., AR.EXT |
Associated scholarship(s): | 07/08056-4 - Stable invariants, Milnor numbers and equisingularity in families of map germs.,
BP.PD 06/60600-8 - Cr - closing lemma problems for vector fields on surfaces., BP.PD |
Abstract
Singularity theory has wide applications to various areas in Mathematics and in particular to differential Geometry and to the qualitative study of ordinary and partial differential equations. These branches of Mathematics feed back, in turn, into and enrich singularity theory. The following project, with well defined objectives, aims at consolidating research activities in the state of São Paulo in geometric aspects of dynamical systems and singularity theory. The interaction will promote development in the following areas: 1) qualitative theory of differential equations and applications; 2) generic properties of submanifolds in Euclidean spaces; 3) classification of singularities, the study of their topology as their invariants. In the research proposal on "qualitative theory of differential equations and applications" we shall study problems related to the Global Asymptotic Stability Conjecture and the injectivity of maps from Rn to Rn which are local diffeomorphisms. Flows in dimensional manifolds and transformations of the interval will also be studied M well M the singularities of certain classes of differential equations in Rn using singularity theory. The aim of the proposal on "generic properties of submanifolds in Euclidean and hyperbolic 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 problems to be considered are those concerning embedded surfaces in R4 and singular surfaces in R3. Geometric properties of submanifolds in hyperbolic spaces will also be studied. The project "classification of singularities, the study of their topology as their invariants" is, in some parts, motivated by the problems highlighted above and is a fundamental part of our proposal. Some of the achievements of our group in this area are obtained using geometric methods and Newton polyhedron. The development of this project requires new results in singularity theory and the expertise of the researchers involved is very relevant for achieving the set objectives. 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. (AU)
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