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New frontiers in Singularity Theory

Grant number: 19/21181-0
Support Opportunities:Research Projects - Thematic Grants
Duration: March 01, 2020 - February 28, 2026
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal Investigator:Regilene Delazari dos Santos Oliveira
Grantee:Regilene Delazari dos Santos Oliveira
Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
Pesquisadores principais:
( Atuais )
Marcelo Jose Saia ; Maria Aparecida Soares Ruas ; Míriam Garcia Manoel ; Raimundo Nonato Araújo dos Santos
Pesquisadores principais:
( Anteriores )
Nivaldo de Góes Grulha Júnior ; Regilene Delazari dos Santos Oliveira
Associated researchers: Aldicio José Miranda ; Alex Carlucci Rezende ; Andres Daniel Duarte ; Benjamin Bode ; Carles Bivià-Ausina ; Dahisy Valadão de Souza Lima ; Eliris Cristina Rizziolli ; Fernando Martins Antoneli Junior ; Grazielle Feliciani Barbosa ; Guillermo Penafort Sanchis ; Hellen Monção de Carvalho Santana ; Ingrid Sofia Meza Sarmiento ; Isabel Salgado Labouriau ; Jackson Itikawa ; Jan Timo Essig ; Jawad Snoussi ; Jean-Paul Michel Ildephonse Brasselet ; João Carlos Ferreira Costa ; Jose Antonio Seade Kuri ; Josnei Antonio Novacoski ; Luis Florial Espinoza Sánchez ; Maria Michalska ; Matthias Zach ; Michal Farnik ; Michelle Ferreira Zanchetta Morgado ; Miriam da Silva Pereira ; Nguyen Thi Bich Thuy ; Nicolas Dutertre ; Nivaldo de Góes Grulha Júnior ; Patricia Hernandes Baptistelli ; Raúl Adrián Oset Sinha ; Roberta Godoi Wik Atique ; RODRIGO MARTINS ; Saurabh Trivedi ; Terence James Gaffney ; Thais Maria Dalbelo ; Thiago Filipe da Silva ; Victor Hugo Jorge Pérez ; Wilker Thiago Resende Fernandes
Associated grant(s):23/04839-7 - Society for Mathematical Biology Annual Meeting - SMB 2023, AR.EXT
22/03720-3 - Characteristic classes, transversality and Stiefel-Whitney currents, AV.EXT
Associated scholarship(s):22/14480-3 - Topologically equiresolvable approximations to analytic germs, BP.PD
23/17645-6 - Morse-Conley Theory, Singular Varieties, and Intersection Homology, BE.PQ
23/18276-4 - The study of periodic orbits in piecewise continuous differential systems, BP.MS
+ associated scholarships 24/03495-5 - Topics on Qualitative theory of ordinary differential equations, BP.IC
23/01649-2 - Singularities of maps, characteristic classes, and intersection homology, BE.PQ
23/07792-1 - Period function and criticality in planar differential systems, BP.MS
23/09279-0 - Networks of coupled dynamical systems, BP.IC
23/01018-2 - Determinantal varieties, Euler obstruction, and Whitney equisingularity, BE.PQ
23/04888-8 - An introduction to Milnor's theory, BP.IC
22/07822-5 - Study of crossing limit cycles in piecewise smooth systems on the plane, BP.PD
21/10198-9 - Invariant manifolds and limit periodic sets of discontinuous foliations, BP.PD
22/10020-8 - Introduction to Intersection Homology, BP.MS
22/09294-6 - Indices of vector fields in singular varieties, BP.IC
22/10965-2 - Mappings that are their own inverse, BP.IC
22/08662-1 - The Bruce-Roberts numbers and the logarithmic characteristic variety, BP.PD
22/06968-6 - Equisingularity and invariants associated to the topology of functions with non-isolated singularity, BP.PD
21/12630-5 - Cyclicity and local structural stability of piecewise vector fields, BP.PD
22/01251-6 - Algebraic graph theory and the Kuramoto model, BP.IC
21/14703-0 - Introduction to the toric varieties, BP.IC
22/02210-1 - Introduction to the center problem and cyclicity problem in the class of the polinomial differential systems, BP.IC
21/14987-8 - Bifurcation of limit cycles in smooth piecewise systems and an application in Medicine, BP.PD
21/14695-7 - Limit cycles, regularization and period function of piecewise smooth planar systems., BP.PD
21/09524-9 - Semigroups, toric actions and monomial surfaces, BP.IC
21/07192-9 - The topological degree and applications, BP.IC
21/07656-5 - Introduction to the study of differential equations: a dynamic approach, BP.IC
21/05770-5 - Physic's differential equations, BP.IC
21/02970-3 - Chow group, BP.IC
21/02598-7 - Qualitative theory of ordinary differential equations and applications, BP.IC
21/02951-9 - A study of vector fields indexes: from topology to the geometry, BP.IC
21/04961-1 - Differential equations: a dynamical approach for the Poincaré-Hopf Theorem, BP.IC
20/14442-9 - Topology of polynomial mappings and Thom polynomials, BP.DR
21/01817-7 - Differential forms and applications, BP.IC
20/16263-4 - Estudo de sistemas de equações diferenciais: bifurcações e aplicações, BP.IC
21/00851-7 - A study of singularities on deep neural networks, BP.IC
20/05978-2 - An introduction to differential geometry of curves and surfaces in Minkowski space, BP.IC
19/21230-0 - Synchrony in coupled systems: a connection between graphs and singularities, BP.DR
19/25235-7 - Obstruction theory, characteristic classes and applications, BP.MS - associated scholarships

Abstract

Singularity theory has wide applications to several areas of science such as optics, robotic and computer vision, and interacts with several areas of mathematics, among which we highlight the algebraic geometry and algebraic topology, commutative algebra, differential and affine geometry, the qualitative theory of ordinary differential equations, dynamical systems, and bifurcation theory. On the other hand, these areas enrich the singularity theory with interesting problems and relevant results. This project has as its central theme the development of methods of classification and recognition of the topology and geometry of real and complex singularities, as well as the determination of families that satisfy some equisingularity condition. The invariants of singularities are investigated in their most diverse forms, whether geometric, algebraic or topological. The research on the singularities of matrices and varieties as well as the bi-Lipschitz geometry and the classification of maps have pioneering results being obtained and allowing new lines of research in this area. We emphasize also the development of research on multiplicities of ideals and modules. Computational methods will be applied for the understanding of the invariants and the topology of singularities, and in the development of algorithms for the study of multiplicities and the local cohomology theory of modules. The project, with well defined objectives, aims at activities in the state of São Paulo and the interaction will promote the development in the following lines of research: Classification, equisingularity and invariants; geometry and topology; commutative algebra, algebraic geometry and singularities; applications to qualitative aspects in discrete and continuous dynamical systems. These lines of research are articulated to each other enabling the interaction of the various researchers involved in the project and the fulfillment of the proposed objectives. Among the researchers involved in the project we have researchers with extensive experience and whose collaboration between them has already produced key advances in the Theory of Singularities and their applications. We also have young researchers who demonstrate an excellent ability to contribute to the advancement of science in Singularities. Moreover, in this project we aim to strengthen the collaboration of the researchers of Sâo Paulo with researchers from other states such as, for example, Maranhão, Ceará, Paraíba, Piaui, Minas Gerais, Espírito Santo, Paraná, Rondônia, and also other countries such as Germany, Spain, the United States, France, Ir\~a, Japan, the United Kingdom and Portugal. (AU)

Articles published in Agência FAPESP Newsletter about the research grant:
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Scientific publications (26)
(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)
RIUL, PEDRO BENEDINI; SOARES RUAS, MARIA APARECIDA; SACRAMENTO, ANDREA DE JESUS. Singular 3-manifolds in R-5. REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS, v. 116, n. 1, . (19/00194-6, 19/21181-0)
OLIVEIRA, REGILENE; SCHLOMIUK, DANA; TRAVAGLINI, ANA MARIA. Geometry and integrability of quadratic systems with invariant hyperbolas. Electronic Journal of Qualitative Theory of Differential Equations, v. N/A, n. 6, p. 56-pg., . (19/21181-0)
ARTES, JOAN C.; MOTA, MARCOS C.; REZENDE, ALEX C.. Structurally unstable quadratic vector fields of codimension two: families possessing a finite saddle-node and an infinite saddle-node. Electronic Journal of Qualitative Theory of Differential Equations, v. N/A, n. 35, p. 89-pg., . (19/21181-0, 18/21320-7)
BALDISSERA, MAIRA DURAN; LLIBRE, JAUME; OLIVEIRA, REGILENE. Dynamics of a Generalized Rayleigh System. DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS, v. N/A, p. 9-pg., . (19/21181-0)
ARTES, JOAN C.; MOTA, MARCOS C.; REZENDE, ALEX C.. Structurally unstable quadratic vector fields of codimension two: families possessing a finite saddle-node and an infinite saddle-node. Electronic Journal of Qualitative Theory of Differential Equations, n. 35, p. 1-89, . (18/21320-7, 19/21181-0)
RIBEIRO, MAICO F.; ARAUJO DOS SANTOS, RAIMUNDO NONATO. Geometrical Conditions for the Existence of a Milnor Vector Field. BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, v. 52, n. 4, p. 771-789, . (19/21181-0, 17/20455-3)
MOTA, MARCOS COUTINHO; REZENDE, ALEX CARLUCCI; SCHLOMIUK, DANA; VULPE, NICOLAE. GEOMETRIC ANALYSIS OF QUADRATIC DIFFERENTIAL SYSTEMS WITH INVARIANT ELLIPSES. TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS, v. 59, n. 2A, p. 63-pg., . (19/21181-0)
OLIVEIRA, REGILENE; SCHLOMIUK, DANA; TRAVAGLINI, ANA MARIA; VALLS, CLAUDIA. Geometry, integrability and bifurcation diagrams of a family of quadratic differential systems as application of the Darboux theory of integrability. Electronic Journal of Qualitative Theory of Differential Equations, n. 45, p. 1-90, . (19/21181-0)
ARTES, JOAN C.; MOTA, MARCOS C.; REZENDE, ALEX C.. Quadratic Differential Systems with a Finite Saddle-Node and an Infinite Saddle-Node (1,1) SN - (B). INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, v. 31, n. 09, . (18/21320-7, 19/21181-0)
RIUL, PEDRO BENEDINI; SINHA, RAUL OSET; SOARES RUAS, MARIA APARECIDA. Curvature loci of 3-manifolds. Mathematische Nachrichten, v. N/A, p. 17-pg., . (19/21181-0)
LLIBRE, JAUME; OLIVEIRA, REGILENE D. S.; RODRIGUES, CAMILA A. B.. QUADRATIC SYSTEMS WITH AN INVARIANT ALGEBRAIC CURVE OF DEGREE 3 AND A DARBOUX INVARIANT. Electronic Journal of Differential Equations, . (19/21181-0)
OLIVEIRA, REGILENE; SCHLOMIUK, DANA; TRAVAGLINI, ANA MARIA. Geometry and integrability of quadratic systems with invariant hyperbolas. Electronic Journal of Qualitative Theory of Differential Equations, n. 6, . (19/21181-0)
BAPTISTELLI, PATRICIA H.; HERNANDES, MARIA ELENICE R.; TEREZIO, ERALCILENE MOREIRA. Normal Forms of ?-Hamiltonian Vector Fields with Symmetries. BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, v. 54, n. 3, p. 32-pg., . (19/21181-0)
GRULHA JR, NIVALDO G. JR; RUIZ, CAMILA M.; SANTANA, HELLEN. The geometrical information encoded by the Euler obstruction of a map. INTERNATIONAL JOURNAL OF MATHEMATICS, v. 33, n. 04, p. 17-pg., . (19/21181-0, 15/25191-9)
LOPES, DEBORA; SOARES RUAS, MARIA APARECIDA; SANTOS, IGOR CHAGAS. Singularities of 3-parameter line congruences in R-4. PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, v. N/A, p. 26-pg., . (19/21181-0)
OLIVEIRA, REGILENE; VALLS, CLAUDIA. GLOBAL DYNAMICS OF THE MAY-LEONARD SYSTEM WITH A DARBOUX INVARIANT. Electronic Journal of Differential Equations, . (17/20854-5, 19/21181-0)
LLIBRE, JAUME; OLIVEIRA, REGILENE. On the limit cycle of a Belousov-Zhabotinsky differential systems. MATHEMATICAL METHODS IN THE APPLIED SCIENCES, v. 45, n. 2, . (19/21181-0)
CAMARA, LEONARDO MEIRELES; SOARES RUAS, MARIA APARECIDA. On the Moduli Space of Quasi-Homogeneous Functions. BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, v. N/A, p. 14-pg., . (19/21181-0)
ITIKAWA, JACKSON; OLIVEIRA, REGILENE; TORREGROSA, JOAN. First-order perturbation for multi-parameter center families. Journal of Differential Equations, v. 309, p. 291-310, . (19/21181-0)
OLIVEIRA, REGILENE D. S.; SANCHEZ-SANCHEZ, IVAN; TORREGROSA, JOAN. Simultaneous Bifurcation of Limit Cycles and Critical Periods. Qualitative Theory of Dynamical Systems, v. 21, n. 1, . (19/21181-0)
MOTA, MARCOS C.; OLIVEIRA, REGILENE; TRAVAGLINI, ANA MARIA. The interplay among the topological bifurcation diagram, integrability and geometry for the family QSH(D). Geometriae Dedicata, v. 217, n. 6, p. 42-pg., . (19/21181-0)
DO ESPIRITO SANTO, A. A.; DREIBELBIS, D.; RIBEIRO, M. F.; ARAUJO DOS SANTOS, R. N.. A QUICK TRIP THROUGH FIBRATION STRUCTURES. JOURNAL OF SINGULARITIES, v. 22, p. 25-pg., . (17/20455-3, 19/21181-0)
DA SILVA, THIAGO F.; GRULHA JR, NIVALDO G.; PEREIRA, MIRIAM S.. REAL AND COMPLEX INTEGRAL CLOSURE, LIPSCHITZ EQUISINGULARITY AND APPLICATIONS ON SQUARE MATRICES. JOURNAL OF SINGULARITIES, v. 22, p. 12-pg., . (19/21181-0)
MEZA-SARMIENTO, INGRID S.; OLIVEIRA, REGILENE; SILVA, PAULO R. DA. Quadratic slow-fast systems on the plane. NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS, v. 60, . (19/21181-0)
C. MENDES DE JESUS; E. BOIZAN BATISTA; J. C. F. COSTA. Stable Bi-Maps on Surfaces and Their Graphs. Trends in Computational and Applied Mathematics, v. 24, n. 2, p. 337-356, . (18/25157-3, 19/21181-0)
OLIVEIRA, REGILENE; VALLS, CLAUDIA. GLOBAL DYNAMICS OF THE MAY-LEONARD SYSTEM WITH A DARBOUX INVARIANT. Electronic Journal of Differential Equations, v. N/A, p. 19-pg., . (17/20854-5, 19/21181-0)

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