Group actions, symplectic and contact geometry, and applications

群作用、辛几何和接触几何以及应用

• 批准号: RGPIN-2018-05771
• 负责人: Karshon, Yael
• 金额: $ 4.08万
• 依托单位: University of Toronto
• 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=758289
• 关键词: Group; actions; symplectic; contact; geometry

The central theme of my research program is using symmetries to learn about geometry, with a focus on symplectic and contact geometry.Symplectic geometry has its roots in classical mechanics. It is the mathematical structure that underlies the equations of motion of celestial bodies, spinning tops, and mechanical linkages. Symplectic spaces are even dimensional; contact geometry is a related odd dimensional structure that comes up, for example, in geometric quantization -- a geometric procedure for passing from a classical mechanical system to a quantum mechanical system.The field of symplectic geometry has gone through spectacular progress in recent decades, and deep connections have emerged with other fields of mathematics as well as with theoretical physics. Progress in the related field of contact geometry, specifically in dimensions greater than three, has been particularly dramatic in the last 5--10 years.A "baby example" of a symplectic space with symmetry is the two dimensional sphere with its rotational symmetry. A higher dimensional example is the complex projective plane; symplectically, it can be viewed as a four dimensional ball with a two dimensional sphere sewed along its edge. Examples of contact spaces include odd dimensional spheres, real projective spaces, and more general so-called lens spaces. The full symmetry group of a symplectic or contact space is always infinite dimensional, but one can often find inside it a compact finite dimensional subgroup (which can be thought of as rotations in multiple dimensions). My research program involves the study of symplectic and contact spaces through such finite dimensional symmetries.The proposal covers applications to geometric mechanics of multi-body systems (which model robot manipulators), applications to geometric quantization, rigidity phenomena in symplectic and contact geometry, and new classification schemes.

我的研究计划的中心主题是使用对称性来了解几何形状，重点是符号和接触几何。隔透明的几何形状源于古典力学。 是数学结构是天体，旋转顶部和机械链接的运动方程的基础。 符号空间甚至是尺寸。接触几何形状是一种相关的奇数结构，例如在几何量化中出现，这是一种从经典的机械系统到量子机械系统传递的几何过程。近几十年来，符合性几何形状的领域经历了壮观的进步，并且与其他数学领域以及与特定的物理学相关。 在过去的5--10年中，相关接触几何学的相关领域的进展，特别是在三个大于三个的维度上的进展。具有对称性的符号空间的“婴儿示例”是具有旋转对称性的两个维度球体。 一个更高的尺寸示例是复杂的投影平面。在符合性上，它可以看作是一个四维球，其边缘缝制了二维球。 接触空间的示例包括奇数球，真实的投射空间和更通用的所谓镜头空间。 符号或接触空间的完整对称组总是无限的维度，但是通常可以在其中找到一个紧凑的有限维度亚组（可以将其视为多个维度的旋转）。 我的研究计划涉及通过这种有限的维度对称性对符号和接触空间进行研究。该提案涵盖了多体系统（模拟机器人操纵器模拟的几何力学）的应用，对几何量化，在象征性和接触几何形状和新分类方案中的几何量化，刚性现象以及新的分类方案。