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: $ 2.04万
• 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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=682407
• 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利用对称性来了解几何，重点是辛几何和接触几何。******辛几何起源于经典力学。 它是天体运动方程、陀螺和机械连杆的基础数学结构。 辛空间是偶维的；接触几何是一种相关的奇数维结构，例如在几何量子化中出现，这是一种从经典力学系统过渡到量子力学系统的几何过程。*****辛几何领域在近几十年来取得了惊人的进步，并且与其他数学领域以及理论物理学建立了深刻的联系。 在过去的 5--10 年里，接触几何的相关领域，特别是大于 3 的维度，取得了特别显着的进展。****** 具有对称性的辛空间的一个“婴儿例子”是具有旋转对称性的二维球体。 一个高维的例子是复射影平面；辛，它可以被视为一个四维球，其边缘缝有一个二维球体。 接触空间的示例包括奇数维球体、实射影空间以及更一般的所谓透镜空间。 辛或接触空间的完全对称群总是无限维的，但人们经常可以在其中找到一个紧凑的有限维子群（可以将其视为多维旋转）。 我的研究计划涉及通过这种有限维对称性来研究辛和接触空间。******该提案涵盖了多体系统几何力学（对机器人机械手建模）的应用、几何量化的应用、辛和接触几何中的刚性现象以及新的分类方案。**