New Application of Equivariant Index Theory

在原子物理学的尺度上，牛顿开创的经典物理定律必须根据海森伯格、薛定谔、狄拉克等人发现的量子力学定律进行调整。这个过程被称为量子化。在量子力学中，就像在经典力学中一样，描述机械系统的定律通常表现出高度的对称性。这反映了十九世纪的发现，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上，即假设物理定律在宇宙中的任何地方都是相同的。在这个项目中，主要的研究者将使用非对易几何工具，如等变指数理论，来研究涉及各种对称性的量子化的几个有趣的问题。主要研究人员将在几何量子化和等变指数理论方面进行三个研究项目：(1)将指数理论应用于辛流形的各种推广，如哈密顿环群空间和对数渐近流形的几何量子化；(2)几何上实现约化李群的不可约表示为相应余共轭轨道上的等变解析指数，并研究其对其最大紧子群的限制；(3)基于Kasparov发展的KK-理论，将等变指数理论推广到非紧流形或群。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.