Membros Principais: Marcelo José Saia, Maria Aparecida Soares Ruas, Maria Michalska, Miriam Manoel, Nivaldo de Góes Grulha Júnior, Regilene D. S. Oliveira, Raimundo Nonato Araújo dos Santos
Descrição: Singularity theory has applications in a wide range of scientific fields, such as optics, robotics, and computer vision, and interacts with several areas of mathematics, including algebraic geometry and topology, commutative algebra, differential and affine geometry, qualitative theory of differential equations, and bifurcation theory. Conversely, these areas enrich singularity theory with interesting and relevant problems and results.
This project aims to develop fundamental topics in singularity theory, and we believe it will contribute to advancing the frontiers of knowledge in this research area. Special emphasis is given to themes such as the classification, topology, and geometry of singularities of real and complex mappings, as well as determining equisingularity in families. Invariants are investigated in various forms—geometric, algebraic, and topological. Bi-Lipschitz geometry, matrix singularities, and determinantal varieties are central topics in this investigation, motivating new research directions in the field. We also highlight the development of research relating multiplicities to the theory of local cohomology of rings and modules.
Computational methods will be applied both to better understand invariants and the topology of singularities and to develop algorithms for the study of multiplicities. The project is structured around four interconnected research lines, enabling interaction among the various researchers involved and supporting the achievement of the proposed goals. The research lines are: Classification, equisingularity, and invariants; Geometry and topology; Commutative algebra, algebraic geometry, and singularities; Applications to qualitative aspects of continuous and discrete dynamical systems.
The project includes researchers with extensive experience in the relevant areas, who have already produced fundamental advances in the theory and its applications. We also emphasize the excellent capabilities of the group’s young researchers, who have made significant contributions to scientific progress in singularity theory. Another objective is to strengthen collaboration with researchers from other Brazilian states—such as Maranhão, Ceará, Paraíba, Piauí, Minas Gerais, Espírito Santo, Paraná, and Rondônia—as well as from other countries, including Germany, Spain, the United States, France, Japan, England, Iran, Mexico, Poland, and Portugal.
