Geometry and Dynamics of the group of Hamiltonian diffeomorphisms of a surface

点击翻译按钮获取中文摘要

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).In this proposal, the PI plans to study various problems at the intersection of low dimensional topology and low dimensional dynamics on the one hand, and symplectic geometry and topology on the other hand. The purpose is to improve our understanding of certain objects, which appeared recently in the symplectic world (such as Hofer's metric) and to apply them to attack some old problems of geometrical and dynamical nature. Specific directions include the study of actions of higher rank lattices on surfaces, following Zimmer's program, of the simplicity of certain groups of transformations, and the existence of quasi-morphisms.The last twenty-five years have witnessed the birth of a remarkable mathematical field, namely symplectic topology. This field is connected to many areas of pure mathematics and modern physics. Some historical developments are associated to the name of Gromov, who introduced the concept of pseudo-holomorphic curve, as well as to the name of Hofer who introduced a remarkable new metric invariant in the classical field of Hamiltonian dynamics. In this project the PI plans to use these new objects and techniques, which are now already unavoidable, to establish new connections with other areas of pure mathematics such as geometric group theory.