Geometric and Algebraic Structures in the Group of Hamiltonian Diffeomorphisms

这个研究项目集中在辛几何中的三个问题上。首先，我们希望证明Hofer关于辛流形的哈密顿微分同胚群上的双不变Finsler距离是唯一的。第二个项目试图将渐近几何分析技术应用到辛电容的研究中，以前的这类论证已经取得了进展，得到了一个辛等周不等式，表明了该方法的有效性。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。这些研究项目研究辛几何，作为哈密顿方法的基础的几何结构的力学和量子理论。这种几何中的坐标系编码运动粒子的位置和动量，附加结构自动从选择的哈密顿函数推导运动定律。辛几何是一门古老的学科，在20世纪80年代中期M.Gromov关于辛流形中的曲面的特别有用的工作中得到了更新，今天是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.