Group actions in symplectic and contact topology

辛几何起源于经典力学。正是这种数学结构构成了天体、自转顶端和机械连杆的运动方程。近几十年来，该领域取得了令人瞩目的进步，与数学的其他领域以及理论物理也出现了深刻的联系。 卡尔松的研究集中在涉及对称性的辛几何的各个方面。一个“小例子”是具有旋转对称性的二维球体。一个更高维的例子是复射影平面；从辛的角度看，这可以被视为一个四维球，其边缘缝合着一个二维球。正如Felix Klein在他1872年的《Erlanger程序》中所设想的那样，通过它的对称性群来研究几何有很大的好处。辛空间的完全对称群总是无限维的，然而，对于许多重要的空间(如两个球面或复射影平面)，在这个无限维群中可以找到紧致的有限维子群(可以认为是多维旋转)。Karshon的研究计划涉及通过这些有限维对称来研究辛空间。 目前的项目涉及到新的分类方案，从基本的“商微分学”中恢复对称性，以及与几何量子化的关系。

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.