Curves and introduction to singularity theory

• Grant number: 18/18791-8
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: October 01, 2018
• End date: May 31, 2019
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Maria Aparecida Soares Ruas
• Grantee: Amanda Dias Falqueto
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 14/00304-2 - Singularities of differentiable mappings: theory and applications, AP.TEM

Abstract

The singularity theory allows a very rich study of the geometric properties of curves, families of curves and surfaces, which successfully completes the classic approach to differential geometry. The aim of this project is to introduce the student to the singularity theory, through the application of this theory to the study of the differential geometry of curves in Euclidean spaces. Other important aspects are to arouse the student's interest in the postgraduate course, developing essential skills such as reading and understanding texts in English, writing texts in LATEX, and elaboration of figures and complementary calculations, for example, in Maple or Wolfram software. We propose the development of the following topics:1. Vector functions and curves in Rn - elementary concepts. Reparametrization, parameterization by arc length. Local theory of flat curves - Frenet referential, curvature, Frenet formulas. Fundamental theorem of flat curves. Spatial curves local theory - Frenet referential, curvature, torsion, Frenet formulas, osculating plane, rectifying and normal. Examples and Figures.2. Basics of singularities: functions defined on curves; study of height and distance functions squared over plane and spatial curves. Contact theory. Right quivalence; Hadamard's Lemma - classification of singularities; flat functions; jets. Examples.3. Understand Wolfram or similar software to plot curves and elements studied above. It may also be necessary to work with the Mayura or similar software to improve illustrative figures. This current project corresponds to the project Introduction to Singularity Theory with the use of the Transversal Program, which is part of the scientific initiation projects approved in the Projeto Temático. The difference between the two projects is due to changing software choice. As the student is in the second year of mathematics under graduation course, Maple or Wolfram software is more appropriate for this phase.