Symplectic topology and equivariant geometry

辛几何以其哈密顿公式为经典力学提供了数学框架。它也是量子理论和广义相对论的现代方法的基础。在过去的30年里，对经典和量子系统之间相互作用的研究，以及强大的几何和分析技术的引入，导致了辛物体特有的新结构和现象的发现，这些结构和现象在经典微分几何和拓扑学中没有对应。辛几何的一个中心问题是理解辛空间的对称性，也称为辛形态。给定空间的所有辛对称的集合本身是一个无限维的几何对象，称为空间的辛形态群。这个组是一个非常丰富的几何对象，它编码了底层空间的所有属性。从物理学的角度来看，对称性是我们理解宇宙的核心。例如，经典物理系统的时间演化对应于相空间对称群上的连续路径。在更深的层次上，连续的对称族对应于守恒量，如能量、动量、角动量等。我们甚至可以根据对称性对基本粒子进行分类。因此，我们可以说辛变换的一般性质对应于物理系统的一般性质。此外，通过辛空间及其对称性的“量子化”过程，我们经常得到一个将经典系统的性质与量子系统的性质联系起来的字典。在这项研究中，我们特别感兴趣的是辛形态群的几何性质，以及理解这些无限维空间在相空间变化时如何相互比较。希望能更好地理解在微分几何中研究的所有其他可能的几何空间中辛空间和辛变换的特征。

Symplectic geometry provides the mathematical framework of classical mechanics in its Hamiltonian formulation. It also underlies modern approaches to quantum theories and to General Relativity. During the past 30 years, the study of the interactions between classical and quantum systems, as well as the introduction of powerful geometrical and analytical techniques, has led to the discovery of new structures and phenomena peculiar to symplectic objects that have no counterparts in classical differential geometry and topology. A central problem in symplectic geometry is to understand the symmetries, also called symplectomorphisms, of symplectic spaces. The set of all symplectic symmetries of a given space is itself an infinite dimensional geometric object called the symplectomorphism group of the space. This group is a very rich geometric object that encodes all the properties of the underlying space. From a physical point of view, symmetries are central to our understanding of the universe. For instance, the time evolution of a classical physical system correspond to a continous path on the symmetry group of the phase space. At a deeper level, continuous families of symmetries correspond to conserved quantities like energy, momentum, angular momentum, etc. We can even classify elementary particles in terms of symmetries.  Consequently, we can say that general properties of symplectic transformations correspond to general properties of physical systems. Moreover, through "quantization" procedures of symplectic spaces and of their symmetries, we often get a dictionary that relates the properties of classical systems with those of quantum systems. In this research, we are especially interested in the geometric properties of symplectomorphism groups, and in understanding how these infinite dimensional spaces compare to each other as the phase spaces change. The hope is to better understand what characterize symplectic spaces and symplectic transformations among all other possible geometric spaces studied in differential geometry.