New Application of Equivariant Index Theory

在原子物理的尺度上，由牛顿开创的经典物理定律必须与海森堡、薛定谔、狄拉克等人发现的量子力学定律相适应。这个过程被称为量子化。在量子力学中，正如在经典力学中一样，描述力学系统的定律通常表现出高度的对称性。这反映了19世纪的发现，即对称性对应于能量和动量等守恒量。与动量守恒定律有关的对称性表现在这样一个事实中：物理定律被假定在宇宙的任何地方都是一样的。在这个项目中，首席研究员将使用非交换几何工具，如等变指标理论来研究涉及各种对称性的量化的几个有趣问题。主要研究员将在几何量化和等变指标理论方面进行三个研究项目：(1)将指标理论应用于哈密顿环群空间和对数辛流形的各种推广的几何量化；(2)几何上实现了约化李群的不可约表示为相应协伴轨道上的等变解析指标，并研究了其对其最大紧子群的限制；(3)将等变指标理论推广到非紧流形或群；基于卡斯帕罗夫提出的kk理论。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

At the scale of atomic physics, the classical laws of physics pioneered by Newton must be adjusted to the laws of quantum mechanics discovered by Heisenberg, Schrodinger, Dirac and others. This process is called quantization. Within quantum mechanics, as within classical mechanics, the laws describing the mechanical systems usually display a high degree of symmetry. This is a reflection of the nineteenth century discovery that symmetries correspond to the conserved quantities like energy and momentum. The symmetry related to the law of conservation of momentum expresses itself in the fact that the laws of physics are presumed to be the same everywhere in the universe. In this project the principal investigator will use noncommutative geometry tools such as equivariant index theory to study several interesting problems about quantization involving various symmetries. The principal investigator will pursue three research projects in geometry quantization and equivariant index theory: (1) applying index theory to geometric quantization of various generalization of symplectic manifolds such as Hamiltonian loop group spaces and log sympelctic manifolds (2) geometrically realizing an irreducible representation of a reductive Lie group as an equivariant analytic index on corresponding coadjoint orbit, and studying its restriction to its maximal compact subgroup (3) generalizing equivariant index theory to noncompact manifolds or groups, based on the KK-theory developed by Kasparov.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.