Group actions in symplectic and contact topology

辛和接触拓扑中的群作用

基本信息

批准号：
261958-2013
负责人：
Karshon, Yael
金额：
$ 2.11万
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：
2015
资助国家：
加拿大
起止时间：
2015-01-01 至 2016-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=571186
关键词：
Group actions symplectic contact topology

项目摘要

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. The field has gone through spectacular progress in recent decades, and deep connections have emerged with other fields of mathematics as well as theoretical physics. Karshon's research focuses on aspects of symplectic geometry that involve symmetries. A "baby example" is the two dimensional sphere with its rotational symmetry. A higher dimensional example is the complex projective plane; symplectically, this can be viewed as a four dimensional ball with a two dimensional sphere sewed along its edge. As envisioned by Felix Klein in his 1872 "Erlanger programm", there is great benefit in studying a geometry through its group of symmetries. The full symmetry group of a symplectic space is always infinite dimensional; however, for many important spaces (such as the two-sphere or the complex projective plane), inside this infinite dimensional group one can find compact finite dimensional subgroups (which can be thought of rotations in multiple dimensions). Karshon's research programme involves the study of symplectic spaces through these finite dimensional symmetries. Current projects involve new classification schemes, recovering symmetries from their underlying "quotient diffeology", and relations with geometric quantization.

符号几何形状起源于经典力学。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。近几十年来，该领域经历了壮观的进步，并且与其他数学领域以及理论物理学领域也出现了深厚的联系。 Karshon的研究重点是涉及对称性的符合性几何形状的各个方面。一个“婴儿示例”是旋转对称性的二维球。一个更高的尺寸示例是复杂的投影平面。在符合性上，这可以看作是一个四维球，沿其边缘缝制了二维球。正如Felix Klein在1872年的“ Erlanger Programm”中所设想的那样，通过其对称性组研究几何形状有很大的好处。符号空间的完整对称群总是无限的尺寸。但是，对于许多重要的空间（例如两个球形或复杂的射射线平面），在这个无限的尺寸组内，可以找到紧凑的有限尺寸亚组（可以认为这是多个维度的旋转）。 Karshon的研究计划涉及通过这些有限维度对称性对符号空间进行研究。当前的项目涉及新的分类方案，从其基本的“商差异学”中恢复对称性以及与几何量化的关系。