Deformations and Rigidity in Poisson Geometry

经典力学的哈密顿描述通过将运动粒子的位置和动量记录为两个坐标，并通过哈密顿函数H引入这些坐标之间的关系，哈密顿函数H是总能量的一种形式，起源于动能和势能的总和。然后，守恒能量的物理性质在数学上被报告为这样的性质，即哈密顿函数H的值不随着系统的演化而改变，但H通过等同于牛顿定律F=ma的微分方程组来确定系统的演化。该提案标题的泊松几何是哈密顿力学基础上的几何学的一个版本，它很好地适应了量子力学的需要。该奖项支持的项目集中在泊松结构的变形和刚性特性上。主要研究者在以前的工作中建立的刚性结果是新工作的组成部分之一；另一个组成部分是局部群胚的显式构造，它可用于证明结果，如Poisson断面周围的局部范式。

AbstractAward: DMS 1405671, Principal Investigator: Ioan-Tiberiu MarcutThe Hamiltonian description of classical mechanics describes the state of a moving particle by recording its position and momentum as two coordinates, and by introducing a relationship between those coordinates through the Hamiltonian function H, a form of total energy that originated as the sum of kinetic energy and potential energy. The physical property of the conservation energy is then reported mathematically as the property that the value of the Hamiltonian function H does not change as the system evolves, but H determines that evolution through a system of differential equations that is equivalent to Newton's law that F = ma. The Poisson geometry of this proposal's title is a version of the geometry underlying Hamiltonian mechanics that is well-adapted to the needs of quantum mechanics.Projects supported by this award are focused on deformation and rigidity properties of Poisson structures. A rigidity result established by the principal investigator in earlier work is one of the ingredients for new work; another ingredient is an explicit construction of local groupoids which can be applied to prove results such as a local normal form around Poisson transversals.