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Singularities, geometry and differential equations

Grant number: 08/54222-6
Support type:Research Projects - Thematic Grants
Duration: December 01, 2008 - November 30, 2012
Field of knowledge:Physical Sciences and Mathematics - Mathematics
Principal Investigator:Maria Aparecida Soares Ruas
Grantee:Maria Aparecida Soares Ruas
Home Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
Co-Principal Investigators:Marcelo José Saia
Associated grant(s):11/21893-8 - The center and isochronicity problems for two-dimensional systems, AV.EXT
11/20621-4 - Invariants and topology of maps on singular varieties, AV.EXT
11/03185-6 - Characteristic classes of singular varieties, AV.EXT
+ associated grants 11/06175-1 - Topology and singularities of singular varieties, AV.EXT
10/03525-9 - Invariants of singularities of Differentiable maps and multiplicities of rings and modules, AV.BR
09/00099-1 - 2nd Meeting IST-IME, AR.EXT
08/58465-0 - Wojciech domitrz | warsaw university of technology - polonia, AV.EXT - associated grants
Associated scholarship(s):12/22000-0 - Averaging Method for Filippov Systems, BP.MS
12/01139-0 - Topology of non-isolated singularities, BP.PD
11/21898-0 - The Center-Focus problem in cubic nilpotent and degenerate polynomial systems, BP.DR
+ associated scholarships 11/19406-1 - Planar dynamical systems with symmetries in space and time, BP.IC
11/10653-6 - Newton non-degenerate polynomial ideals, BP.DR
11/09139-6 - Algebraic methods for the systematic study of (Gamma,sigma)-equivariant applications, BP.PD
11/02851-2 - The Euler obstruction of a function, BP.MS
10/20301-7 - Curves in Minkowski space, BP.DR
10/17790-6 - Group theory applied to crystal structures, BP.IC
10/01895-3 - Topological classification of singularities, BP.PD
10/11668-4 - Coefficient ideals and modules/Milnor number in caracteristic zero for maps, BP.PD
09/12266-0 - From Milnor number to the lê's Milnor number, BP.MS
10/01501-5 - Topological invariants of stable maps, BP.PD
09/12285-4 - Centers and limit cycles, BP.MS
09/03201-1 - Indices of vector fields on singular varieties, BP.MS
09/02989-4 - On limit cycles that bifurcates from a center in a planar polynomial system, BP.MS
09/05364-5 - Polar varieties of surfaces, BP.IC
08/10563-4 - Global singular fibrations of polynomial mappings, BP.DR - associated scholarships

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)

Scientific publications (10)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
SINHA, R. OSET; RUAS, M. A. S.; ATIQUE, R. WIK. ON THE SIMPLICITY OF MULTIGERMS. MATHEMATICA SCANDINAVICA, v. 119, n. 2, p. 197-222, 2016. Web of Science Citations: 1.
OSET SINHA, R.; RUAS, M. A. S.; WIK ATIQUE, R. Classifying codimension two multigerms. MATHEMATISCHE ZEITSCHRIFT, v. 278, n. 1-2, p. 547-573, OCT 2014. Web of Science Citations: 5.
DIAS, FABIO SCALCO; SINHA, RAUL OSET; SOARES RUAS, MARIA APARECIDA. A FORMULA RELATING INFLECTIONS, BITANGENCIES AND THE MILNOR NUMBER OF A PLANE CURVE. Proceedings of the American Mathematical Society, v. 142, n. 7, p. 2353-2368, JUL 2014. Web of Science Citations: 1.
MARTINS, LUCIANA F.; NABARRO, ANA CLAUDIA. PROJECTIONS OF HYPERSURFACES IN R-4 WITH BOUNDARY TO PLANES. Glasgow Mathematical Journal, v. 56, n. 1, p. 149-167, JAN 2014. Web of Science Citations: 4.
JANECZKO, S.; JELONEK, Z.; RUAS, M. A. S. SYMMETRY DEFECT OF ALGEBRAIC VARIETIES. ASIAN JOURNAL OF MATHEMATICS, v. 18, n. 3, p. 525-544, 2014. Web of Science Citations: 0.
SOARES RUAS, MARIA APARECIDA; PEREIRA, MIRIAM DA SILVA. CODIMENSION TWO DETERMINANTAL VARIETIES WITH ISOLATED SINGULARITIES. MATHEMATICA SCANDINAVICA, v. 115, n. 2, p. 161-172, 2014. Web of Science Citations: 8.
FERNANDES, ALEXANDRE; RUAS, MARIA. RIGIDITY OF BI-LIPSCHITZ EQUIVALENCE OF WEIGHTED HOMOGENEOUS FUNCTION-GERMS IN THE PLANE. Proceedings of the American Mathematical Society, v. 141, n. 4, p. 1125-1133, APR 2013. Web of Science Citations: 0.
CALLEJAS-BEDREGAL, R.; SAIA, M. J.; TOMAZELLA, J. N. EULER OBSTRUCTION AND POLAR MULTIPLICITIES OF IMAGES OF FINITE MORPHISMS ON ICIS. Proceedings of the American Mathematical Society, v. 140, n. 3, p. 855-863, MAR 2012. Web of Science Citations: 1.
DIAS, L. R. G.; RUAS, M. A. S.; TIBAR, M. Regularity at infinity of real mappings and a Morse-Sard theorem. Journal of Topology, v. 5, n. 2, p. 323-340, 2012. Web of Science Citations: 13.
AHMED, IMRAN; SOARES RUAS, MARIA APARECIDA. INVARIANTS OF RELATIVE RIGHT AND CONTACT EQUIVALENCES. HOUSTON JOURNAL OF MATHEMATICS, v. 37, n. 3, p. 773-786, 2011. Web of Science Citations: 1.

Please report errors in scientific publications list by writing to: cdi@fapesp.br.