Geometric and Algebraic Structures in the Group of Hamiltonian Diffeomorphisms

该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哈密顿微分同胚群上的 Hofer 双不变芬斯勒度量是唯一的。第二个项目旨在将渐近几何分析技术应用于辛容量的研究。以前的此类论证已经在辛等周不等式方面取得了进展，表明了该方法的强大功能。 第三，最近发现的辛流形的卡拉比准态射将被研究并应用于拉格朗日交集理论和量子同调性对辛结构的依赖性的研究。这些研究项目研究辛几何，这种几何结构是力学和量子理论的哈密顿方法的背景。该几何中的坐标系对运动粒子的位置和动量进行编码，并且附加结构自动从哈密尔顿函数的选择中推导运动定律。 辛几何是一门古老的学科，在 20 世纪 80 年代中期，由于 M. Gromov 在辛流形曲面上的非常有用的工作而得到更新，并且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.