Group actions, symplectic and contact geometry, and applications

* 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。辛几何起源于经典力学。 它是构成天体、旋转陀螺和机械连杆运动方程基础的数学结构。 辛空间是偶维的;接触几何是一种相关的奇维结构，例如，在几何量子化中出现-从经典力学系统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展，与其他数学领域以及理论物理学都有着深刻的联系。 在接触几何学的相关领域，特别是在大于3的维度方面，在过去5- 10年中取得了特别引人注目的进展。具有对称性的辛空间的一个“小例子”是具有旋转对称性的二维球面。 一个更高维的例子是复射影平面;从符号上讲，它可以被看作是一个四维球，沿着它的边缘沿着缝着一个二维球体。 接触空间的例子包括奇数维球面、真实的射影空间和更一般的所谓透镜空间。 辛或接触空间的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致的有限维子群（可以认为是多维的旋转）。 我的研究计划包括通过这种有限维对称来研究辛和接触空间。该提案涵盖了多体系统（机器人操纵器模型）几何力学的应用，几何量化的应用，辛几何和接触几何中的刚性现象，以及新的分类方案。

***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.**