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New frontiers in Singularity Theory

Grant number: 19/21181-0
Support type:Research Projects - Thematic Grants
Duration: March 01, 2020 - February 28, 2025
Field of knowledge:Physical Sciences and Mathematics - Mathematics
Principal Investigator:Marcelo José Saia
Grantee:Marcelo José Saia
Home Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
Co-Principal Investigators:Maria Aparecida Soares Ruas ; Míriam Garcia Manoel ; Nivaldo de Góes Grulha Júnior ; Raimundo Nonato Araújo dos Santos ; Regilene Delazari dos Santos Oliveira
Assoc. researchers:Alex Carlucci Rezende ; Eliris Cristina Rizziolli ; Grazielle Feliciani Barbosa ; João Carlos Ferreira Costa ; Josnei Antonio Novacoski ; Michelle Ferreira Zanchetta Morgado ; Nguyen Thi Bich Thuy ; Roberta Godoi Wik Atique ; Thais Maria Dalbelo ; Victor Hugo Jorge Pérez

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)