The Thurston earthquakes theorem on Teichmüller spaces

• Grant number: 16/22928-3
• Support Opportunities: Scholarships in Brazil - Master
• Effective date (Start): March 01, 2017
• Effective date (End): December 31, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Acordo de Cooperação: Coordination of Improvement of Higher Education Personnel (CAPES)
• Principal Investigator: Alexandre Paiva Barreto
• Grantee: Lino Chiozzini Neto
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
• Associated research grant: 12/24454-8 - Algebraic, geometric and differential topology, AP.TEM

Abstract

The study of 2-dimensionals and 3-dimensionals manifolds begins with Henry Poincaré in the late 19th century. At the beginning of the 20th century, Poincaré conjectured that any tridimensional closed manifold with trivial homology should be homeomorphic to a sphere. This famous conjecture, as known as "Poincaré Conjecture", conducted the researches in low-dimensional topology (dimensions 2 and 3) until its verification in 2002/03, as a consequence of the Grigori Perelman work. In fact, Perelman demonstrated a much more general conjecture called "Geometrization Conjecture", formulated by William Thurston in the 80´s, and the verification of the Poincaré Conjecture is a consequence of this one. Thurston's work has radically changed low-dimensional topology studies. His innovative techniques has made clear the strength and importance of using a more geometric point of view (in Klein's sense) in the essentially topological problems studies.The aim of this project is to give the student a solid basic formation on the geometric techniques developed by Thurston and prepare him to successfully develop his PhD in low-dimensional topology area. In this way, we chose Thurston's works on surfaces as the central theme. Special attention will be given to Teichmuller spaces of hyperbolic surfaces and to Thurston Earthquake Theorem (which has new demonstrations/versions developed by F.Gardiner, N. Lakic and J. Hu between 2002 and 2012). (AU)