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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=713839
• 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来学习几何，重点是辛几何和接触几何。