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

这个奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。在这个提议中，PI计划一方面研究低维拓扑和低维动力学的交叉点上的各种问题，另一方面研究辛几何和拓扑学的交叉问题。其目的是提高我们对最近出现在辛世界中的某些物体(如霍费尔度规)的理解，并将它们应用于解决一些具有几何和动力学性质的老问题。具体方向包括研究高阶格子在曲面上的作用，某些变换群的简单性，以及拟态射的存在。在过去的25年里，见证了一个引人注目的数学领域的诞生，即辛拓扑。这个领域与纯数学和现代物理学的许多领域有关。一些历史发展与Gromov的名字有关，他引入了伪全纯曲线的概念，也与Hofer的名字有关，他在哈密顿动力学的经典领域中引入了一个引人注目的新度量不变量。在这个项目中，PI计划使用这些现在已经不可避免的新对象和技术，与其他纯数学领域建立新的联系，如几何群论。

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.