Geometric and Algebraic Structures in the Group of Hamiltonian Diffeomorphisms

本研究计划集中于辛几何中的三个问题。首先，我们希望证明辛流形的哈密顿微分同态群上的Hofer双不变Finslermetric是唯一的。第二个项目寻求将渐近几何分析技术应用于辛能力的研究。这类先前的论证已经在辛等周期不等式方面取得了进展，表明了该方法的威力。第三，研究新近发现的辛流形的Calabi拟态，并将其应用于拉格朗日交理论和量子同调对辛结构的依赖研究。这些研究项目调查辛几何，几何结构，作为背景的哈密顿方法力学和量子理论。这个几何中的坐标系统编码了移动粒子的位置和动量，另外的结构从选择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，在20世纪80年代中期由M. Gromov关于辛流形曲面的特别有用的工作而复兴，是今天数学中最活跃的领域之一。

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.