Description: The theory of singularities has applications in a wide range of scientific fields, such as optics, robotics, and computer vision, and it interacts with various areas of mathematics, including algebraic geometry and topology, commutative algebra, differential and affine geometry, the qualitative theory of differential equations, and bifurcation theory. Conversely, these areas enrich singularity theory with interesting and relevant problems and results.
This project aims at the development of fundamental topics in singularity theory, and we believe it will contribute to advancing the frontiers of knowledge within this research line. Notable topics include the classification, topology, and geometry of singularities of real and complex mappings, as well as the determination of equisingularity in families. Invariants are studied in their various forms—geometric, algebraic, or topological. Bi-Lipschitz geometry and singularities of matrices and determinantal varieties are central to this investigation, with topics that inspire new research directions in the area.
We also emphasize research on the relationship between multiplicities and the local cohomology theory of rings and modules. Computational methods will be applied both to understand invariants and the topology of singularities and to develop algorithms for the study of multiplicities.
This project consists of four interconnected research lines, enabling interaction among the various researchers involved and facilitating the achievement of the proposed objectives. 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 involves researchers with extensive experience in the relevant areas, who have already made fundamental contributions to theory and applications. We also highlight the excellent capabilities of the group’s young researchers, whose contributions significantly advance the field of singularities. 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—and from other countries, including Germany, Spain, the United States, France, Japan, the United Kingdom, Iran, Mexico, Poland, and Portugal.
Grant Period: March 2020 – February 2026
Principal Investigator: Regilene D. S. Oliveira
Funding Agency: FAPESP
Key Personnel: Marcelo José Saia, Maria Aparecida Soares Ruas, Miriam Manoel, Nivaldo de Góes Grulha Júnior, Regilene D. S. Oliveira, Raimundo Nonato Araújo dos Santos
