About some classic problems of the topological classification of germs and maps

• Grant number: 23/08176-2
• Support Opportunities: Regular Research Grants
• Start date: October 01, 2023
• End date: September 30, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: João Carlos Ferreira Costa
• Grantee: João Carlos Ferreira Costa
• Host Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil

Abstract

In this project we want to study some topics of the classical Singularity theory such as: problems related to the (topological) classification of singularities of maps and map-germs, topological triviality, finite determinacy and sufficiency of jets. We will try to answer some questions related to these themes and give some correlations between them. In order to study the topological triviality and questions related to topological finite determinacy of map germs, we will focus on the case where f : (R^3,0) \to (R^2,0). This type of problem, with a focus similar to what we want, is addressed in the literature only for cases in which the source dimension is less than or equal to the domain dimension. So this work has a certain pioneering spirit in that sense. With respect to sufficiency of jets problem, we will explore it in a broader context for germs from (R^n,0) to (R^p,0) from weighted point of view and with a modification in the metric used in the source. We hope to obtain better results than those given in the literature. (AU)