Geometric and Algebraic Structures in the Group of Hamiltonian Diffeomorphisms

哈密​​顿微分同胚群中的几何和代数结构

• 批准号: 0706976
• 负责人: Denis Auroux
• 金额: $ 11.94万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706976&HistoricalAwards=false
• 关键词: Geometric; Algebraic; Structures; Group; Hamiltonian

This research program concentrates on three problems in symplectic geometry. First, we hope to show that Hofer's bi-invariant Finslermetric on the group of Hamiltonian diffeomorphisms of a symplectic manifold is unique. The second project seeks to adapt techniques ofasymptotic geometric analysis to the study of symplectic capacities.Previous arguments of this kind have made progress toward a symplecticisoperimetric inequality, indicating the power of the method. Third,the recently discovered Calabi quasi-morphisms of a symplectic manifoldwill be studied and applied to Lagrangian intersection theory and toa study of the dependence of quantum homology on symplectic structure.These research projects investigate symplectic geometry, the geometric structure that serves as background to the Hamiltonian approachto mechanics and quantum theory. Coordinate systems in this geometryencode both position and momentum of a moving particle, and additionalstructure automates the derivation of laws of motion from a choice ofHamiltonian function. Symplectic geometry is an old subject thatwas renewed in the mid-1980s by exceptionally useful work of M. Gromov on surfaces in symplectic manifolds, and is today one of the most activeareas of mathematics.

这个研究计划集中在辛几何中的三个问题。 首先，我们希望证明辛流形的Hamilton同态群上的霍费尔双不变Finslermetric是唯一的。第二个项目旨在适应技术的渐近几何分析的研究辛capacity.Previous参数这类取得了进展，对一个辛等周不等式，表明权力的方法。 第三，最近发现的辛流形的Calabi拟态射将被研究并应用于拉格朗日交理论和研究量子同调对辛结构的依赖性。这些研究项目研究辛几何，作为Hamilton方法的背景的几何结构到力学和量子理论。这种几何中的坐标系统对运动粒子的位置和动量进行编码，另外的结构使从选择的哈密顿函数中自动推导出运动定律。 辛几何是一门古老的学科，80年代中期，M. Gromov关于辛流形中曲面的研究，是当今数学中最活跃的领域之一.