New Application of Equivariant Index Theory

等变指数理论的新应用

• 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

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.

根据原子物理学的规模，牛顿开创的物理学定律必须调整为海森伯格，施罗德格，迪拉克等发现的量子力学定律。此过程称为量化。在量子力学中，与经典力学一样，描述机械系统的定律通常显示高度的对称性。这是十九世纪发现的反映，即对称性对应于能量和动量等保守数量。与动量保护定律有关的对称性表达了以下事实：物理定律被认为是宇宙中到处都是相同的。在该项目中，主要研究者将使用非交通性的几何工具（例如均值索引理论）来研究有关涉及各种对称性的量化的几个有趣的问题。首席研究者将在几何量化和模棱两可的指数理论方面追求三个研究项目：（1）将指数理论应用于符号歧管的各种概括（例如，哈密顿循环群体和对数循环流形流形）的各种泛化的几何量化（2）地理位置分析和对等效的分析或对等效的分析或对等效的分析或对等效性的表达方式根据Kasparov开发的KK理论，将其限制在其最大的平台亚组中（3）将概括性指数理论概括为非庞大的歧管或群体。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子和更广泛的影响来评估的支持，并被认为是值得的。

项目成果

A KK-theoretic perspective on deformed Dirac operators

变形狄拉克算子的 KK 理论视角

• DOI: 10.1016/j.aim.2021.107604
• 发表时间: 2021
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Loizides, Yiannis;Rodsphon, Rudy;Song, Yanli
• 通讯作者: Song, Yanli

Spinor modules for Hamiltonian loop group spaces

哈密​​顿环群空间的旋量模

• DOI: 10.4310/jsg.2020.v18.n3.a10
• 发表时间: 2020
• 期刊: Journal of Symplectic Geometry
• 影响因子: 0.7
• 作者: Loizides, Yiannis;Meinrenken, Eckhard;Song, Yanli
• 通讯作者: Song, Yanli

A geometric realisation of tempered representations restricted to maximal compact subgroups

限制于最大紧子群的调节表示的几何实现

• DOI: 10.1007/s00208-020-02006-4
• 发表时间: 2020
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Hochs, Peter;Song, Yanli;Yu, Shilin
• 通讯作者: Yu, Shilin

Norm-square localization and the quantization of Hamiltonian loop group spaces

哈密​​顿循环群空间的范数平方局部化和量化

• DOI: 10.1016/j.jfa.2019.108445
• 发表时间: 2020
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: Loizides, Yiannis;Song, Yanli
• 通讯作者: Song, Yanli

A geometric formula for multiplicities of 𝐾-types of tempered representations

• DOI: 10.1090/tran/7857
• 发表时间: 2018-05
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: P. Hochs;Yanli Song;Shilin Yu