Singularities of differentiable mappings: theory and applications

• Grant number: 14/00304-2
• Support Opportunities: Research Projects - Thematic Grants
• Start date: June 01, 2014
• End date: May 31, 2019
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Maria Aparecida Soares Ruas
• Grantee: Maria Aparecida Soares Ruas
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Pesquisadores principais: Farid Tari ; Marcelo Jose Saia
• Associated researchers: Ana Claudia Nabarro ; Bruna Orefice Okamoto ; Eliris Cristina Rizziolli ; Grazielle Feliciani Barbosa ; João Carlos Ferreira Costa ; João Nivaldo Tomazella ; Luciana de Fátima Martins ; Michelle Ferreira Zanchetta Morgado ; Míriam Garcia Manoel ; Nivaldo de Góes Grulha Júnior ; Raimundo Nonato Araújo dos Santos ; Raúl Adrián Oset Sinha ; Regilene Delazari dos Santos Oliveira ; Roberta Godoi Wik Atique ; Serguei Agafonov ; Victor Hugo Jorge Pérez
• Associated research grant(s): 18/09297-0 - Topology and geometry of real and complex singularities, AV.EXT; 18/07040-1 - Topology and invariants of maps between varieties, AV.EXT; 17/01410-9 - Global dynamical aspects of Abel equations of the third kind, AV.EXT; + associated grants 17/06176-4 - Mathematical Congress of the Americas 2017, AR.EXT; 16/19436-1 - The Zariski's approach for resolution of singularities, AV.EXT; 16/08929-7 - Methods of commutative algebra and algebraic geometry in singularity theory., AV.BR; 15/06697-9 - Characteristic classes and intersection homology, AV.EXT - associated grants
• Associated scholarship(s): 19/00194-6 - Geometry of singular surfaces in $\mathbb{R}^4$, BP.PD; 19/02068-8 - Equivariant characteristic classes of singular hypersurfaces, BP.DR; 18/17712-7 - Geometry of singular surfaces, BP.DR; + associated scholarships 18/18791-8 - Curves and introduction to singularity theory, BP.IC; 18/19525-0 - Introduction to algebraic topology, BP.IC; 17/23555-9 - Singularities of Hamiltonian Systems with constraints, BP.PD; 18/07176-0 - Invariant algebraic curves and integrability in differentiable polynomial systems, BP.IC; 17/26168-6 - An Introduction to algebraic geometry, BP.IC; 17/20243-6 - An introduction to dynamical systems and bifurcations applied to the Hodgkin-Huxley neuronal model, BP.IC; 17/15369-0 - Polar singularity of a flat curve germ , BP.PD; 17/19553-0 - Introduction to complex analytical geometry, BP.IC; 16/23906-3 - Vanishing homology and multiple-point spaces of singular maps, BP.PD; 17/08540-5 - The Poincaré-Hopf Theorem, BP.IC; 16/14580-7 - Applications of singularity theory: differential geometry and algebraic geometry, BP.PD; 16/21226-5 - Differential geometry of the focal set, BP.MS; 16/14330-0 - Lipschitz geometry of stratified spaces, BP.PD; 16/11732-0 - Singularities of smooth function germs, BP.IC; 15/25191-9 - The Euler obstruction and generalizations, BP.DR; 16/02701-4 - Flat and round singularity theory, BE.PQ; 16/10580-2 - Basic Mathematics for Aeronautical Engineering, BP.IC; 15/12667-5 - Lipschitz Geometry of tame singular sets and mappings., BP.PD; 15/16746-7 - Equisingularity and the Theory of Integral Closure of modules., BE.PQ; 15/07612-7 - Uniform isochronous centers in planar polynomial differential systems of degree 5, BP.PD; 15/08026-4 - Bilipschitz geometry and surface singularities resolution, BP.PD; 15/08149-9 - Variations in multiplicities and limited polynomials, BP.PD; 15/05380-1 - Introduction to the singularity theory of differentiable maps, BP.IC; 15/02102-0 - An introduction to singularity theory, BP.IC; 15/04807-1 - Functions in complex variables, BP.IC - 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 to develop methods of classification of real and complex singularities with special attention to invariants and equisingularity conditions for families of sets and mappings, and to apply singularity theory to problems in differential geometry and differential equations. The 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 the development in the following areas: 1) geometry and classification of singularities; 2) topology of singular sets and mappings; 3) algebraic methods in singularity theory; 4) singularities in differential geometry; 5) applications to bifurcation problems and to the qualitative theory of differential equations. (AU)

Scientific publications

(56)

(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)

MENCINGER, MATEJ; FERCEC, BRIGITA; FERNANDES, WILKER; OLIVEIRA, REGILENE. Linearizability problem of persistent centers. Electronic Journal of Qualitative Theory of Differential Equations, n. 37, p. 1-27, 2018. (14/00304-2)

OLIVEIRA, REGILENE D. S.; REZENDE, ALEX C.; SCHLOMIUK, DANA; VULPE, NICOLAE. GEOMETRIC AND ALGEBRAIC CLASSIFICATION OF QUADRATIC DIFFERENTIAL SYSTEMS WITH INVARIANT HYPERBOLAS. Electronic Journal of Differential Equations, NOV 28 2017. (14/00304-2)

ARTES, JOAN C.; OLIVEIRA, REGILENE D. S.; REZENDE, ALEX C.. Structurally Unstable Quadratic Vector Fields of Codimension Two: Families Possessing Either a Cusp Point or Two Finite Saddle-Nodes. Journal of Dynamics and Differential Equations, JUL 2020. (17/20854-5, 18/21320-7, 14/00304-2)

LLIBRE, JAUME; OLIVEIRA, REGILENE D.; RODRIGUES, CAMILA AP B.. Limit cycles for two classes of control piecewise linear differential systems. SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES, JAN 2020. (17/20854-5, 14/00304-2)

BRANDER, DAVID; TARI, FARID. Families of spherical surfaces and harmonic maps. Geometriae Dedicata, v. 201, n. 1, p. 203-225, AUG 2019. (16/02701-4, 14/00304-2)

KASEDOU, MASAKI; NABARRO, ANA CLAUDIA; SOARES RUAS, MARIA APARECIDA. Singular 4-Webs of Asymptotic Lines of Spacelike Surfaces in de Sitter 5-space. BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, v. 51, n. 1, p. 293-315, MAR 2020. (16/19139-7, 14/00304-2)

REMIZOV, A. O.; TARI, F.. Singularities of the geodesic flow on surfaces with pseudo-Riemannian metrics. Geometriae Dedicata, v. 185, n. 1, p. 131-153, DEC 2016. (14/04944-6, 14/00304-2)

OLIVEIRA, REGILENE; VALLS, CLAUDIA. Polynomial integrability of Hamiltonian systems with homogeneous potentials of degree -k. Physics Letters A, v. 380, n. 46, p. 3876-3880, DEC 1 2016. (14/00304-2)

ROMANOVSKI, VALERY G.; FERNANDES, WILKER; OLIVEIRA, REGILENE. Bi-center problem for some classes of Z(2)-equivariant systems. Journal of Computational and Applied Mathematics, v. 320, p. 61-75, AUG 15 2017. (14/00304-2)

LLIBRE, JAUME; OLIVEIRA, REGILENE; VALLS, CLAUDIA. Phase portraits for some symmetric Riccati cubic polynomial differential equations. Topology and its Applications, v. 234, p. 220-237, FEB 1 2018. (14/00304-2)

LLIBRE, JAUME; OLIVEIRA, REGILENE. Quadratic systems with an invariant conic having Darboux invariants. COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, v. 20, n. 4, JUN 2018. (14/00304-2)

BRASSELET, JEAN-PAUL; CHACHAPOYAS, NANCY; RUAS, MARIA A. S.. Generic sections of essentially isolated determinantal singularities. INTERNATIONAL JOURNAL OF MATHEMATICS, v. 28, n. 11, OCT 2017. (15/06697-9, 10/09736-1, 14/00304-2, 11/20082-6)

DIAS, FABIO SCALCO; TARI, FARID. ON THE GEOMETRY OF THE CROSS-CAP IN MINKOWSKI 3-SPACE AND BINARY DIFFERENTIAL EQUATIONS. TOHOKU MATHEMATICAL JOURNAL, v. 68, n. 2, p. 293-328, JUN 2016. (11/21240-4, 11/01946-0, 14/00304-2)

OLIVEIRA, REGILENE; VALLS, CLAUDIA. Chaotic Behavior of a Generalized Sprott E Differential System. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, v. 26, n. 5, MAY 2016. (14/00304-2)

ARTES, JOAN C.; OLIVEIRA, REGILENE D. S.; REZENDE, ALEX C.. Topological Classification of Quadratic Polynomial Differential Systems with a Finite Semi-Elemental Triple Saddle. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, v. 26, n. 11, OCT 2016. (14/00304-2)

BARBOSA, G. F.; GRULHA, JR., N. G.; SAIA, M. J.. Minimal Whitney stratification and Euler obstruction of discriminants. Geometriae Dedicata, v. 186, n. 1, p. 173-180, FEB 2017. (13/11258-9, 14/00304-2, 15/16746-7)

MARTINEZ-ALFARO, J.; MEZA-SARMIENTO, I. S.; OLIVEIRA, R. D. S.. Singular levels and topological invariants of Morse Bott integrable systems on surfaces. Journal of Differential Equations, v. 260, n. 1, p. 688-707, JAN 5 2016. (14/00304-2)

SALARINOGHABI, MOSTAFA; TARI, FARID. FLAT AND ROUND SINGULARITY THEORY OF PLANE CURVES. QUARTERLY JOURNAL OF MATHEMATICS, v. 68, n. 4, p. 1289-1312, DEC 2017. (14/00304-2)

BRUCE, J. W.; TARI, F.. Frame and direction mappings for surfaces in R-3. PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, v. 149, n. 3, p. 795-830, JUN 2019. (14/00304-2)

AHMED, IMRAN; SOARES RUAS, MARIA APARECIDA. Determinacy of determinantal varieties. MANUSCRIPTA MATHEMATICA, v. 159, n. 1-2, p. 269-278, MAY 2019. (14/00304-2)

LLIBRE, JAUME; OLIVEIRA, REGILENE; VALLS, CLAUDIA. On the Darboux integrability of a three-dimensional forced-damped differential system. JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS, v. 24, n. 4, p. 473-494, 2017. (14/00304-2)

SOARES RUAS, MARIA APARECIDA; TRIVED, SAURABH. BI-LIPSCHITZ GEOMETRY OF CONTACT ORBITS IN THE BOUNDARY OF THE NICE DIMENSIONS. ASIAN JOURNAL OF MATHEMATICS, v. 23, n. 6, p. 953-968, DEC 2019. (14/00304-2, 15/12667-5)

FARNIK, MICHAL; JELONEK, ZBIGNIEW; SOARES RUAS, MARIA APARECIDA. Finite A-determinacy of generic homogeneous map germs in C-3. JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, v. 73, n. 1, p. 211-220, JAN 2021. (14/00304-2)

ARTES, JOAN C.; OLIVEIRA, REGILENE D. S.; REZENDE, ALEX C.. Structurally Unstable Quadratic Vector Fields of Codimension Two: Families Possessing Either a Cusp Point or Two Finite Saddle-Nodes. Journal of Dynamics and Differential Equations, v. 33, n. 4, p. 1779-1821, DEC 2021. (18/21320-7, 14/00304-2, 17/20854-5)

OLIVEIRA, REGILENE; REZENDE, ALEX C.; SCHLOMIUK, DANA; VULPE, NICOLAE. Characterization and bifurcation diagram of the family of quadratic differential systems with an invariant ellipse in terms of invariant polynomials. REVISTA MATEMATICA COMPLUTENSE, v. 35, n. 2, p. 53-pg., 2021-05-22. (14/00304-2, 17/20854-5)

GAFFNEY, TERENCE; RUAS, MARIA APARECIDA SOARES. EQUISINGULARITY AND EIDS. Proceedings of the American Mathematical Society, v. 149, n. 4, p. 16-pg., 2021-04-01. (14/00304-2)

DEOLINDO-SILVA, JORGE LUIZ; TARI, FARID. ON THE DIFFERENTIAL GEOMETRY OF HOLOMORPHIC PLANE CURVES. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v. 373, n. 10, p. 17-pg., 2020-10-01. (14/00304-2)

MENCINGER, MATEJ; FERCEC, BRIGITA; FERNANDES, WILKER; OLIVEIRA, REGILENE. Linearizability problem of persistent centers. Electronic Journal of Qualitative Theory of Differential Equations, v. N/A, n. 37, p. 27-pg., 2018-01-01. (14/00304-2)

DOMITRZ, W.; RIOS, P. DE M.; RUAS, M. A. S.. SINGULARITIES OF AFFINE EQUIDISTANTS: PROJECTIONS AND CONTACTS. JOURNAL OF SINGULARITIES, v. 10, p. 15-pg., 2014-01-01. (10/15179-8, 13/04630-9, 08/54222-6, 14/00304-2)

LLIBRE, JAUME; OLIVEIRA, REGILENE; RODRIGUES, CAMILA AP B.. On the periodic solutions of the Michelson continuous and discontinuous piecewise linear differential system. COMPUTATIONAL & APPLIED MATHEMATICS, v. 37, n. 2, p. 12-pg., 2018-05-01. (14/00304-2, 12/22000-0)

LLIBRE, JAUME; OLIVEIRA, REGILENE; RODRIGUES, CAMILA AP B.. On the periodic solutions of the Michelson continuous and discontinuous piecewise linear differential system. COMPUTATIONAL & APPLIED MATHEMATICS, v. 37, n. 2, p. 1550-1561, MAY 2018. (14/00304-2, 12/22000-0)

OLIVEIRA, REGILENE; REZENDE, ALEX C.; SCHLOMIUK, DANA; VULPE, NICOLAE. Characterization and bifurcation diagram of the family of quadratic differential systems with an invariant ellipse in terms of invariant polynomials. REVISTA MATEMATICA COMPLUTENSE, MAY 2021. (14/00304-2, 17/20854-5)

MARTINEZ-ALFARO, JOSE; MEZA-SARMIENTO, INGRID S.; OLIVEIRA, REGILENE D. S.. SINGULAR LEVELS AND TOPOLOGICAL INVARIANTS OF MORSE-BOTT FOLIATIONS ON NON-ORIENTABLE SURFACES. TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS, v. 51, n. 1, p. 183-213, MAR 2018. (14/00304-2)

HASEGAWA, MASARU; TARI, FARID. On Umbilic Points on Newly Born Surfaces. BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, v. 48, n. 4, p. 679-696, DEC 2017. (16/02701-4, 14/00304-2, 13/02543-1)

RODRIGUES HERNANDES, M. E.; RUAS, M. A. S.. Parametrized monomial surfaces in 4-space. QUARTERLY JOURNAL OF MATHEMATICS, v. 70, n. 2, p. 473-485, JUN 2019. (14/00304-2)

FERNANDES, WILKER; OLIVEIRA, REGILENE; ROMANOVSKI, VALERY G.. Isochronicity of a Z(2)-equivariant quintic system. Journal of Mathematical Analysis and Applications, v. 467, n. 2, p. 874-892, NOV 15 2018. (14/00304-2)

KERNER, DMITRY; PEDERSEN, HELGE MOLLER; RUAS, MARIA A. S.. Lipschitz normal embeddings in the space of matrices. MATHEMATISCHE ZEITSCHRIFT, v. 290, n. 1-2, p. 485-507, OCT 2018. (15/08026-4, 14/00304-2)

GAFFNEY, TERENCE; GRULHA, JR., NIVALDO G.; RUAS, MARIA A. S.. The local Euler obstruction and topology of the stabilization of associated determinantal varieties. MATHEMATISCHE ZEITSCHRIFT, v. 291, n. 3-4, p. 905-930, APR 2019. (14/00304-2, 17/09620-2, 15/16746-7)

BRASSELET, JEAN-PAUL; MIWA LIBARDI, ALICE KIMIE; RIZZIOLLI, ELIRIS CRISTINA; SAIA, MARCELO JOSE. Cobordism of maps of locally orientable Witt spaces. PUBLICATIONES MATHEMATICAE-DEBRECEN, v. 94, n. 3-4, p. 299-317, 2019. (14/00304-2, 16/24707-4, 15/06697-9)

BIVIA-AUSINA, CARLES; SOARES RUAS, MARIA APARECIDA. Mixed Bruce-Roberts numbers. PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, v. 63, n. 2, p. 456-474, MAY 2020. (14/00304-2)

BENEDINI RIUL, P.; RUAS, M. A. S.; OSET SINHA, R.. THE GEOMETRY OF CORANK 1 SURFACES IN R-4. QUARTERLY JOURNAL OF MATHEMATICS, v. 70, n. 3, p. 767-795, SEP 2019. (14/00304-2)

EYRAL, CHRISTOPHE; SOARES RUAS, MARIA APARECIDA. On the Zariski multiplicity conjecture for weighted homogeneous and Newton nondegenerate line singularities. INTERNATIONAL JOURNAL OF MATHEMATICS, v. 30, n. 11, OCT 2019. (14/00304-2)

DIAS, FABIO SCALCO; TARI, FARID. ON VERTICES AND INFLECTIONS OF PLANE CURVES. JOURNAL OF SINGULARITIES, v. 17, p. 70-80, JAN-DEC 2018. (14/00304-2)

EYRAL, CHRISTOPHE; RUAS, MARIA APARECIDA SOARES. TOPOLOGICAL TRIVIALITY OF LINEAR DEFORMATIONS WITH CONSTANT LE NUMBERS. KODAI MATHEMATICAL JOURNAL, v. 39, n. 1, p. 189-201, MAR 2016. (14/00304-2)

MEREU, ANA C.; OLIVEIRA, REGILENE; RODRIGUES, CAMILA A. B.. Limit cycles for a class of discontinuous piecewise generalized Kukles differential systems. NONLINEAR DYNAMICS, v. 93, n. 4, p. 2201-2212, SEP 2018. (14/00304-2, 12/18780-0)

OLIVEIRA, REGILENE; VALLS, CLAUDIA. Global dynamical aspects of a generalized Chen-Wang differential system. NONLINEAR DYNAMICS, v. 84, n. 3, p. 1497-1516, MAY 2016. (14/00304-2)

REEVE, GRAHAM MARK; TARI, FARID. MINKOWSKI SYMMETRY SETS OF PLANE CURVES. PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, v. 60, n. 2, p. 461-480, MAY 2017. (14/00304-2, 12/05326-9)

NISHIMURA, T.; OSET SINHA, R.; RUAS, M. A. S.; WIK ATIQUE, R.. Liftable vector fields over corank one multigerms. MATHEMATISCHE ANNALEN, v. 366, n. 1-2, p. 573-611, OCT 2016. (13/02381-1, 14/00304-2)

LLIBRE, JAUME; OLIVEIRA, REGILENE D. S.; VALLS, CLAUDIA. Final evolutions for simplified multistrain/two-stream model for tuberculosis and dengue fever. CHAOS SOLITONS & FRACTALS, v. 118, p. 181-186, JAN 2019. (17/20854-5, 14/00304-2)

RUAS, M. A. S.; SILVA, O. N.. Whitney equisingularity of families of surfaces in C-3. MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, v. 166, n. 2, p. 353-369, FEB 2019. (14/00304-2)

ITIKAWA, JACKSON; LLIBRE, JAUME; MEREU, ANA CRISTINA; OLIVEIRA, REGILENE. LIMIT CYCLES IN UNIFORM ISOCHRONOUS CENTERS OF DISCONTINUOUS DIFFERENTIAL SYSTEMS WITH FOUR ZONES. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, v. 22, n. 9, p. 3259-3272, NOV 2017. (12/18780-0, 15/07612-7, 14/00304-2)

OSET SINHA, RAUL; TARI, FARID. ON THE FLAT GEOMETRY OF THE CUSPIDAL EDGE. OSAKA JOURNAL OF MATHEMATICS, v. 55, n. 3, p. 393-421, JUL 2018. (14/00304-2)

THUY, NGUYEN THI BICH; RUAS, MARIA APARECIDA SOARES. ON SINGULAR VARIETIES ASSOCIATED TO A POLYNOMIAL MAPPING FROM C-n TO Cn-1. ASIAN JOURNAL OF MATHEMATICS, v. 22, n. 6, p. 1157-1171, DEC 2018. (14/00304-2, 13/18706-7)

SINHA, R. OSET; RUAS, M. A. S.; ATIQUE, R. WIK. The extra-nice dimensions. MATHEMATISCHE ANNALEN, NOV 2021. (14/00304-2, 15/04409-6)

OLIVEIRA, REGILENE D. S.; REZENDE, ALEX C.; SCHLOMIUK, DANA; VULPE, NICOLAE. GEOMETRIC AND ALGEBRAIC CLASSIFICATION OF QUADRATIC DIFFERENTIAL SYSTEMS WITH INVARIANT HYPERBOLAS. Electronic Journal of Differential Equations, v. N/A, p. 122-pg., 2017-11-28. (14/00304-2)

LLIBRE, JAUME; OLIVEIRA, REGILENE D.; RODRIGUES, CAMILA AP B.. Limit cycles for two classes of control piecewise linear differential systems. SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES, v. 14, n. 1, p. 17-pg., 2020-01-17. (14/00304-2, 17/20854-5)