Geometric aspects of surface homeomorphisms

• Grant number: 11/51671-7
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: March 01, 2012
• End date: September 12, 2013
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Edson de Faria
• Grantee: Ferry Henrik Kwakkel
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

Diffeomorphisms between closed surfaces can be thought of in two different ways. One is the topological behaviour of the diffeomorphism, which only takes into account the action of a diffeomorphism up to continuous deformations and is therefore rather weak information. The second is the geometrical behaviour of a diffeomorphism, which measures precisely how distances and shapes of configurations oh the surface are distorted under the action of the diffeomorphism. What is interesting is to understand how these two actions are interrelated. We propose to study three problems that each approach this interplay in their own respect, but share the common denominator of aiming to gain a better understanding of the interplay the topological action and the geometrical action of a surface diffeomorphism. The first problem concerns diffeomorphisms of the torus and their dynamical properties, the second problem concerns how to deform a surface into another by bending portions of the surface onto the other. The third problem concerns the homogeneity of closed surfaces that is, to understand how one portion of a surface is geometrically similar to another portion of the surface. (AU)