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Hyperbolic geometry in low dimensions

Grant number: 20/07644-4
Support type:Scholarships in Brazil - Scientific Initiation
Effective date (Start): August 01, 2020
Effective date (End): July 31, 2021
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal researcher:André Salles de Carvalho
Grantee:Rodolfo Cesar Macedo Soares
Home 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


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.

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