Fixed point theory via Algebraic Topology and Nilmanifolds

• Grant number: 24/18154-9
• Support Opportunities: Scholarships in Brazil - Master
• Start date: March 01, 2025
• End date: February 28, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Daciberg Lima Gonçalves
• Grantee: Gabriel Cassimiro Pereira
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 22/16455-6 - Algebraic, geometric, and differential topology, AP.TEM

Abstract

This project is about the interaction between topology and group theory. The project beginswith important topology concepts, including homotopy, covering spaces,and manifolds and their homeomorphisms. Then we will begin to study problemsof fixed points of homeomorphisms of a particular type of manifolds, the so-callednilmanifolds. The objective is to understand when homeomorphisms of such manifolds haveessential fixed points and, if so, calculate how many there are. The classic toolto be used is the fundamental group: the homeomorphism of the manifold will give us anautomorphism of the fundamental group that can be used to investigate propertiesdata of the original homeomorphism. The main objective of this project would be to understand thefixed points of two families of specific homogeneous spaces - namely, real matricesunitriangular of order 3 and 4, modules of their entire lattices, which are part of the Nilmanifolds- in order to apply the theoretical knowledge acquired to concrete cases. (AU)