Symplectic topology and equivariant geometry

辛拓扑和等变几何

• 批准号: RGPIN-2020-06428
• 负责人: Pinsonnault, Martin
• 金额: $ 1.31万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758560
• 关键词: Symplectic; topology; equivariant; geometry

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.

Symple毒性的几何形状在其哈密顿公式中提供了古典力学的数学框架。它还是量子理论和一般相对论的现代方法的基础。在过去的30年中，对经典和量子系统之间的相互作用的研究以及强大的几何和分析技术的引入导致发现了在经典差异几何学和拓扑中没有对应物的相关物体特有的新结构和现象。对称几何形状的一个中心问题是了解对称性的对称性（也称为对称性）。给定空间的所有对称性的集合本身就是一个无限的尺寸几何对象，称为空间的对称组。该组是一个非常丰富的几何对象，它编码了基础空间的所有属性。从物理的角度来看，对称是我们对宇宙的理解的核心。例如，经典物理系统的时间演变对应于相空间对称组上的连续路径。在更深层次的水平上，对称的连续家庭对应于能量，动量，角动量等保守量。我们甚至可以根据对称性对基本粒子进行分类。因此，我们可以说对称的一般特性对应于物理系统的一般特性。此外，通过对称空间及其对称性的“量化”程序，我们经常得到一个词典，将经典系统的属性与量子系统的属性相关联。在这项研究中，我们对符号形态组的几何特性特别感兴趣，并了解这些无限尺寸空间如何随着相位空间的变化相比。希望是更好地理解在差异几何形状中研究的所有其他可能的几何空间之间的对称空间和对称转换的特征。