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New Application of Equivariant Index Theory

等变指数理论的新应用

基本信息

批准号：
1800667
负责人：
Yanli Song
金额：
$ 16.08万
依托单位：
Washington University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800667&HistoricalAwards=false
关键词：
New Application Equivariant Index Theory

项目摘要

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.

在原子物理学的尺度上，牛顿开创的经典物理定律必须与海森堡、薛定谔、狄拉克和其他人发现的量子力学定律相适应。这个过程称为量化。在量子力学中，就像在经典力学中一样，描述力学系统的定律通常表现出高度的对称性。这反映了世纪发现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，即物理定律被假定在宇宙中的任何地方都是相同的。在这个项目中，主要研究者将使用非交换几何工具，如等变指数理论，研究涉及各种对称性的量子化的几个有趣的问题。首席研究员将从事几何量化和等变指数理论的三个研究项目：（1）将指标理论应用于辛流形的各种推广如Hamilton圈群空间和对数辛流形的几何量子化（2）在几何上实现约化李群的不可约表示作为相应余伴随轨道上的等变解析指标，并研究其限制其最大紧子群（3）推广等变指数理论的非紧流形或群体，基于KK理论开发的卡斯帕罗夫。这个奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。