Hyperbolic geometry in low dimensions

• Grant number: 20/07644-4
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: August 01, 2020
• End date: July 31, 2021
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: André Salles de Carvalho
• Grantee: Rodolfo Cesar Macedo Soares
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 16/25053-8 - Dynamics and geometry in low dimensions, AP.TEM

Abstract

Hyperbolic geometry differs from Euclidean geometry in that it does without the parallel postulate of Euclid. When it was discovered in the early nineteenth century, independently by Taurinus, Gauss, Lobachevski and Bolyai, hyperbolic geometry was considered a oddity with great philosophical implications. Today, it is an integral part of several areas of mathematics and physics, from the mathematical study of manifold topology and geometry to the relativistic description of the physical world. For its proper study, different areas of mathematics come into play, from algebra and complex analysis to topology and differential geometry. The project aims to study hyperbolic geometry, and the consequent development of the mathematical maturity of the student, who will need to understand how to combine these different areas, which usually taught in separate modules. As a goal, at the end of the project, the student should be able to understand the statement of Thurston's Geometrization Theorem for complements of knots and links.