Singularity theory and its applications to differential geometry, differential equations and computer vision

• Grant number: 19/07316-0
• Support Opportunities: Research Projects - Thematic Grants
• Start date: August 01, 2019
• End date: July 31, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Farid Tari
• Grantee: Farid Tari
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Pesquisadores principais: João Do Espirito Santo Batista Neto ; João Nivaldo Tomazella ; Moacir Antonelli Ponti
• Associated researchers: Ana Claudia Nabarro ; Antonio Castelo Filho ; Bruna Orefice Okamoto ; Daiane Alice Henrique Ament ; Débora Lopes da Silva ; Douglas Hilário da Cruz ; Fabio Scalco Dias ; Francisco Braun ; James William Bruce ; Jaume Llibre Salo ; Jorge Luiz Deolindo Silva ; Lakehal Belarbi ; Leandro Nery de Oliveira ; Luciana de Fátima Martins ; Luis Fernando de Osório Mello ; Marcelo Escudeiro Hernandes ; Maria Elenice Rodrigues Hernandes ; Míriam Garcia Manoel ; Mostafa Salarinoghabi ; Raúl Adrián Oset Sinha ; Ronaldo Alves Garcia
• Associated research grant(s): 24/06602-7 - 18th International Workshop on Real and Complex Singularities, AR.EXT
• Associated scholarship(s): 24/20213-3 - Extensions of the D'Ocagne-Koenderink Theorem to Singular Surfaces, BE.PQ; 24/09462-1 - Multimodal Models for Images and 3D Representations in a Unified Vision and Language Approach, BP.DR; 23/17907-0 - Curvature Estimation Using Machine Learning Algorithms, BP.IC; + associated scholarships 23/15751-3 - 3D Reconstruction of Human Faces: a Laplacian Approach, BP.IC; 23/11669-0 - Multi-local singularities of k-folding maps on curves and surfaces., BP.DR; 23/09684-1 - Topology of polynomial maps through their bifurcation locus, BP.DR; 23/09727-2 - Invariants of singular germs and Newton polyedra, BP.MS; 23/10748-4 - Gröbner Bases: A first contact, BP.IC; 23/10828-8 - Singularities of germs of smooth functions, BP.IC; 22/10370-9 - On the geometry of singular surfaces, BP.PD; 22/10661-3 - Discrete differential geometry from a singularity theory view point, BP.IC; 22/06325-8 - Geometric deformations of singularities, BP.PD; 22/00133-0 - Geometric bifurcations of singular plane curves, BP.MS; 22/01501-2 - Algebraic theory of graphs and neural networks, BP.IC; 21/08895-3 - Qualitative theory of differential equations in the plane: topological classification results, BP.MS; 21/02932-4 - On k-folding map-germs and hidden symmetries of curves in the Euclidean plane, BP.MS; 21/03830-0 - Deep multi-domain representations for analyzing social media posts, BP.IC; 21/02923-5 - Discrete differential geometry of plane curves from a singularity theory viewpoint, BP.IC; 20/16475-1 - Mapping representations between different domains and subspaces using geometric deep learning, BP.IC; 20/10888-2 - Equisingularity of families of surfaces with non-isolated singularities, BP.PD; 20/07224-5 - Image curves reconstruction by means of robust features, BP.IC; 20/04143-4 - Dynamics of piecewise smooth functions on the interval, BP.IC; 19/19714-0 - Apparent contour of surfaces in R3 and extensions of Koenderink's formula, BP.MS - associated scholarships

Abstract

Singularity theory deals with the study of singular varieties and mappings. It is a well established theory and gained wide interestdue to its applications to various areas of science and its interaction with several areas of mathematics. It has applications to, among others, optics, robotics and computer vision. The project has four central research directions: one deals with problems in singularity theory proper, and the other three consider applications of this theory to differential geometry, differential equations, computer vision and image analysis. The project will continue the work of the team in these area of research and will start two new lines of research on the study of vector fields from an infinitesimal point of view and recognition of images using a geometric approach. The problems are challenging, ambitious and innovative, both from a theoretical point of view and applications. The team, formed by internationally renowned researchers and with its extensive and varied experience, is well prepared to face the challenges of the project. It is worth emphasising that a part of the project is multi-disciplinary an envolves mathematicians as well as computer scientists. In addition to the expected scientific results, the team will contribute to the training of human resources through the supervision of undergraduate, doctoral and post-doctoral students, and dissemination of scientific knowledge. (AU)

Scientific publications

(44)

(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)

DO COUTO FERNANDES, MARCO ANTONIO; TARI, FARID. ON THE MULTIPLICITY OF UMBILIC POINTS. PACIFIC JOURNAL OF MATHEMATICS, v. 319, n. 1, p. 29-pg., 2022-07-01. (19/07316-0)

CASTELO, ANTONIO; BUENO, LUCAS MOUTINHO; GAMEIRO, MARCIO. A combinatorial marching hypercubes algorithm. COMPUTERS & GRAPHICS-UK, v. 102, p. 67-77, FEB 2022. (13/07375-0, 19/06249-7, 19/07316-0, 17/25631-4)

LIMA-PEREIRA, B. K.; NUNO-BALLESTEROS, J. J.; OREFICE-OKAMOTO, B.; TOMAZELLA, J. N.. THE RELATIVE BRUCE-ROBERTS NUMBER OF A FUNCTION ON A HYPERSURFACE. PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, v. 64, n. 3, p. 662-674, AUG 2021. (19/07316-0)

BRAUN, FRANCISCO; FERNANDES, FILIPE. Very degenerate polynomial submersions and counterexamples to the real Jacobian conjecture. Journal of Pure and Applied Algebra, v. 227, n. 8, p. 10-pg., 2023-02-17. (20/14498-4, 19/07316-0)

DOS SANTOS, FERNANDO P.; ZOR, CEMRE; KITTLER, JOSEF; PONTI, MOACIR A.. Learning image features with fewer labels using a semi-supervised deep convolutional network. NEURAL NETWORKS, v. 132, p. 131-143, DEC 2020. (19/07316-0, 18/22482-0)

CASTRO, AUGUSTO R.; GRASSI JR, VALDIR; PONTI, MOACIR A.; FARINELLA, GM; RADEVA, P; BOUATOUCH, K. Deep Depth Completion of Low-cost Sensor Indoor RGB-D using Euclidean Distance-based Weighted Loss and Edge-aware Refinement. PROCEEDINGS OF THE 17TH INTERNATIONAL JOINT CONFERENCE ON COMPUTER VISION, IMAGING AND COMPUTER GRAPHICS THEORY AND APPLICATIONS (VISAPP), VOL 4, v. N/A, p. 9-pg., 2022-01-01. (19/07316-0)

CAVALLARI, GABRIEL B.; PONTI, MOACIR A.; IEEE COMP SOC. Semi-supervised siamese network using self-supervision under scarce annotation improves class separability and robustness to attack. 2021 34TH SIBGRAPI CONFERENCE ON GRAPHICS, PATTERNS AND IMAGES (SIBGRAPI 2021), v. N/A, p. 8-pg., 2021-01-01. (19/07316-0, 19/02033-0)

BRAGA, DENIS DE CARVALHO; DA FONSECA, ALEXANDER FERNANDES; MELLO, LUIS FERNANDO; RIBEIRO, RONISIO MOISES; PESSOA, CLAUDIO GOMES. Crossing Limit Cycles Bifurcating from Two or Three Period Annuli in Discontinuous Planar Piecewise Linear Hamiltonian Differential Systems with Three Zones. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, v. 33, n. 10, p. 17-pg., 2023-08-01. (19/10269-3, 19/07316-0, 23/04061-6)

BRANDER, DAVID; TARI, FARID. Wave maps and constant curvature surfaces: singularities and bifurcations. ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA-CLASSE DI SCIENZE, v. 23, n. 1, p. 37-pg., 2022-01-01. (19/07316-0)

BRAUN, FRANCISCO; FERNANDES, FILIPE. On Reeb components of nonsingular polynomial differential systems on the real plane. Journal of Differential Equations, v. 320, p. 10-pg., 2022-05-25. (19/07316-0, 20/14498-4)

FERRAZ RIBEIRO, LEO SAMPAIO; BUI, TU; COLLOMOSSE, JOHN; PONTI, MOACIR; IEEE COMP SOC. Scene Designer: a Unified Model for Scene Search and Synthesis from Sketch. 2021 IEEE/CVF INTERNATIONAL CONFERENCE ON COMPUTER VISION WORKSHOPS (ICCVW 2021), v. N/A, p. 10-pg., 2021-01-01. (19/02808-1, 17/22366-8, 19/07316-0)

MANOEL, MIRIAM; OLIVEIRA, LEANDRO N.. EQUIVARIANT MAPPINGS AND INVARIANT SETS ON MINKOWSKI SPACE. Colloquium Mathematicum, v. 167, n. 1, MAY 2021. (19/07316-0)

IZUMIYA, SHYUICHI; NABARRO, ANA CLAUDIA; SACRAMENTO, ANDREA DE JESUS. CURVES IN A SPACELIKE HYPERSURFACE IN MINKOWSKI SPACE-TIME. OSAKA JOURNAL OF MATHEMATICS, v. 58, n. 4, p. 947-966, OCT 2021. (19/07316-0, 16/19139-7)

DOS SANTOS, FERNANDO PEREIRA; THUME, GABRIELA SALVADOR; PONTI, MOACIR ANTONELLI; IEEE COMP SOC. Data Augmentation Guidelines for Cross-Dataset Transfer Learning and Pseudo Labeling. 2021 34TH SIBGRAPI CONFERENCE ON GRAPHICS, PATTERNS AND IMAGES (SIBGRAPI 2021), v. N/A, p. 8-pg., 2021-01-01. (19/07316-0)

BRUCE, J. W.; TARI, F.. Binary Differential Equations Associated to Congruences of Lines in Euclidean 3-Space. BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, v. 54, n. 4, p. 21-pg., 2023-12-01. (19/07316-0)

HERNANDES, MARCELO ESCUDEIRO; HERNANDES, MARIA ELENICE RODRIGUES. The analytic classification of plane curves. COMPOSITIO MATHEMATICA, v. 160, n. 4, p. 31-pg., 2024-03-19. (19/07316-0)

SILVA, JOAO MARCOS C.; REZENDE, PEDRO MARTELLETO B.; PONTI, MOACIR A.. Detecting and mitigating issues in image-based COVID-19 diagnosis. NEURIPS WORKSHOPS, 2020, v. 184, p. 9-pg., 2022-01-01. (19/07316-0)

SAMPAIO FERRAZ RIBEIRO, LEO; BUI, TU; COLLOMOSSE, JOHN; PONTI, MOACIR. Scene designer: compositional sketch-based image retrieval with contrastive learning and an auxiliary synthesis task. MULTIMEDIA TOOLS AND APPLICATIONS, v. N/A, p. 23-pg., 2022-12-20. (19/02808-1, 17/22366-8, 19/07316-0)

MANCHEIN, CESAR; BERGER, HELENA F.; ALBUQUERQUE, HOLOKX A.; MELLO, LUIS FERNANDO. Arnold Tongue-Like Structures and Coexisting Attractors in the Memristive Muthuswamy-Chua-Ginoux Circuit Model. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, v. 33, n. 12, p. 20-pg., 2023-09-30. (19/07316-0)

DIAS, FABIO SCALCO; MELLO, LUIS FERNANDO; VALLS, CLAUDIA. Bounded Kukles Systems of Degree Three. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, v. 34, n. 14, p. 19-pg., 2024-10-05. (19/07316-0)

BRAUN, FRANCISCO; DA CRUZ, LEONARDO PEREIRA COSTA; TORREGROSA, JOAN. On the number of limit cycles in piecewise planar quadratic differential systems. NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS, v. 79, p. 15-pg., 2024-05-04. (22/14484-9, 19/07316-0, 21/14987-8, 20/14498-4, 23/00376-2)

BRAUN, FRANCISCO; LLIBRE, JAUME. PARALLEL VECTOR FIELDS AND GLOBAL INJECTIVITY IN TWO DIMENSIONS. PACIFIC JOURNAL OF MATHEMATICS, v. 333, n. 1, p. 18-pg., 2024-11-01. (20/14498-4, 19/07316-0, 23/00376-2)

CHAVES, FELIPE EMANOEL; MELLO, LUIS FERNANDO; TEIXEIRA, MARCO ANTONIO. Generic singularities of planar reversible Filippov vector fields. SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES, v. 19, n. 1, p. 34-pg., 2025-06-01. (18/13481-0, 19/07316-0)

ANTUNES, VINICIUS F. R.; MANCHEIN, CESAR; ALBUQUERQUE, HOLOKX A.; MELLO, LUIS FERNANDO; HOFF, ANDERSON. Exploring the dynamics of a four-dimensional Chua-like system: Numerical simulations and hardware investigations. INTERNATIONAL JOURNAL OF MODERN PHYSICS C, v. N/A, p. 22-pg., 2024-07-02. (19/07316-0)

DIAS, FABIO SCALCO; RIBEIRO, RONISIO MOISES; VALLS, CLAUDIA. Global Phase Portraits for the Kukles Systems of Degree 3 with Z(2)-Reversible Symmetries. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, v. 31, n. 06, MAY 2021. (19/07316-0)

CASTELO, SONIA; PONTI, MOACIR; MINGHIM, ROSANE. A Visual Mining Approach to Improved Multiple- Instance Learning. ALGORITHMS, v. 14, n. 12, DEC 2021. (19/07316-0, 13/25055-2)

OLIVEIRA DE RESENDE, DAMARES CRYSTINA; PONTI, MOACIR ANTONELLI. obust image features for classification and zero-shot tasks by merging visual and semantic attribute. NEURAL COMPUTING & APPLICATIONS, v. 34, n. 6, JAN 2022. (18/22482-0, 19/07316-0)

BRAUN, FRANCISCO; GONCALVES DIAS, LUIS RENATO; SANTOS, JEAN VENATO. ON GLOBAL INVERTIBILITY OF SEMI-ALGEBRAIC LOCAL DIFFEOMORPHISMS. TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS, v. 58, n. 2, p. 713-730, DEC 2021. (17/00136-0, 19/07316-0)

SANCHIS, GUILLERMO PENAFORT; TARI, FARID. On k-folding map-germs and hidden symmetries of surfaces in the Euclidean 3-space. PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, v. N/A, p. 45-pg., 2023-01-26. (19/07316-0)

BRAUN, FRANCISCO; FERNANDES, FILIPE. A counterexample to a folk real Jacobian conjecture. Israel Journal of Mathematics, v. N/A, p. 4-pg., 2025-02-03. (20/14498-4, 19/07316-0)

BRAUN, FRANCISCO; THOMAZ, RODRIGO. Revisiting Markus-Neumann theorem. SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES, v. 19, n. 1, p. 33-pg., 2025-06-01. (19/07316-0, 23/09684-1, 21/08895-3, 23/00376-2)

CASTELO, ANTONIO; NAKASSIMA, GUILHERME; BUENO, LUCAS MOUTINHO; GAMEIRO, MARCIO. A generalized combinatorial marching hypercube algorithm. COMPUTATIONAL & APPLIED MATHEMATICS, v. 43, n. 3, p. 23-pg., 2024-04-01. (19/07316-0, 19/06249-7, 13/07375-0, 17/25631-4)

SILVA, AQUISSON T. G. DA; FERNANDES, CELIO; ORGANISTA, JUNIORMAR; SOUZA, LEANDRO; CASTELO, ANTONIO. Validation of HiG-Flow Software for Simulating Two-Phase Flows with a 3D Geometric Volume of Fluid Algorithm. MATHEMATICS, v. 11, n. 18, p. 21-pg., 2023-09-01. (19/07316-0, 13/07375-0, 23/00062-8)

PONTI, MOACIR A.; DOS SANTOS, FERNANDO P.; RIBEIRO, LEO S. F.; CAVALLARI, GABRIEL B.; IEEE COMP SOC. Training Deep Networks from Zero to Hero: avoiding pitfalls and going beyond. 2021 34TH SIBGRAPI CONFERENCE ON GRAPHICS, PATTERNS AND IMAGES (SIBGRAPI 2021), v. N/A, p. 8-pg., 2021-01-01. (19/07316-0, 17/22366-8, 19/02033-0)

DIAS, FABIO SCALCO; MELLO, LUIS FERNANDO. Polynomial Vector Fields on Algebraic Surfaces of Revolution. Results in Mathematics, v. 76, n. 1, MAR 2021. (19/07316-0)

BERTOCO, JULIANA; CASTELO, ANTONIO; FERRAS, LUIS L.; FERNANDES, CELIO. Numerical Simulation of Three-Dimensional Free Surface Flows Using the K-BKZ-PSM Integral Constitutive Equation. POLYMERS, v. 15, n. 18, p. 19-pg., 2023-09-01. (19/07316-0, 13/07375-0)

DIAS, FABIO SCALCO; MELLO, LUIS FERNANDO. Globally Asymptotically Stable Equilibrium Points in Kukles Systems. Qualitative Theory of Dynamical Systems, v. 19, n. 3, p. 7-pg., 2020-10-22. (19/07316-0)

ALVES, ALEXANDRE; SALARINOGHABI, MOSTAFA. ON THE FAMILY OF CUBIC PARABOLIC POLYNOMIALS. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, v. N/A, p. 14-pg., 2021-01-21. (19/07316-0)

DIAS, FABIO FELIX; PONTI, MOACIR ANTONELLI; MINGHIM, ROSANE. classification and quantification approach to generate features in soundscape ecology using neural network. NEURAL COMPUTING & APPLICATIONS, v. 34, n. 3, SI, SEP 2021. (19/07316-0)

CANDIDO, DOUGLAS MODESTO; DIAS, FABIO SCALCO; MELLO, LUIS FERNANDO. Global Phase Portraits of Ordinary Differential Equations Modeling the Acute Promyelocytic Leukemia. DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS, v. N/A, p. 18-pg., 2023-06-29. (19/07316-0)

BRAUN, FRANCISCO; MEZA-SARMIENTO, INGRID S.. Topological equivalence of submersion functions and topological equivalence of their foliations on the plane: The linear-like case. Journal of Mathematical Analysis and Applications, v. 521, n. 2, p. 25-pg., 2022-12-20. (20/14498-4, 19/07316-0)

CASANOVA, EDRESSON; WEBER, JULIAN; SHULBY, CHRISTOPHER; CANDIDO JUNIOR, ARNALDO; GOELGE, EREN; PONTI, MOACIR ANTONELLI; CHAUDHURI, K; JEGELKA, S; SONG, L; SZEPESVARI, C; et al. YourTTS: Towards Zero-Shot Multi-Speaker TTS and Zero-Shot Voice Conversion for Everyone. INTERNATIONAL CONFERENCE ON MACHINE LEARNING, VOL 162, v. N/A, p. 12-pg., 2022-01-01. (19/07316-0)

BRAGA, DENIS DE CARVALHO; DIAS, FABIO SCALCO; LLIBRE, JAUME; MELLO, LUIS FERNANDO. The discontinuous matching of two globally asymptotically stable crossing piecewise smooth systems in the plane do not produce in general a piecewise differential system globally asymptotically stable. SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES, v. N/A, p. 8-pg., 2023-05-30. (19/07316-0)

DIAS, FABIO SCALCO; MELLO, LUIS FERNANDO; VALLS, CLAUDIA. Kukles Systems of Degree Three with Global Centers. BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, v. 54, n. 4, p. 15-pg., 2023-12-01. (19/07316-0)