Mappings that are their own inverse

• Grant number: 22/10965-2
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: September 01, 2022
• End date: October 31, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Míriam Garcia Manoel
• Grantee: Marcelo da Silveira Salazar Filho
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 19/21181-0 - New frontiers in Singularity Theory, AP.TEM

Abstract

An involution is a mapping that is its own inverse. The concept of involution is fundamental in group theory and algebras. The interest of this project is to consider involutions under the rich algebraic point of view as well as illustrating its use in solving geometric problems and that are related to discrete dynamical systems. Motivated by the student's interest in algebraic theory, the supervisor proposes the present project by the algebraic richness of the group generated by a finite number of involutions, which already allows permeating the subject with quite rich and surprising results. The proposal intends to go beyond, working on the construction of classes of involutions, stratified with respect to continuity and differentiability in Euclidean spaces of different dimensions. In the applied aspect, this project also proposed the study of an involution as anti-symmetry of a dicrete dynamixal system, also called reversible dynamical systems. (AU)