Differential geometric methods in mathematical physics

数学物理中的微分几何方法

• 批准号: 7855-2011
• 负责人: McLenaghan, Raymond
• 金额: $ 0.73万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568544
• 关键词: Differential; geometric; methods; mathematical; physics

The main theme of my research is the study of differential equations arising in mathematical physics by the methods of differential geometry and Lie group theory. The work is being conducted in three interrelated areas, the common threads of which are the geometrical and group invariant methods used. The first area is the study of completely integrable Hamiltonian systems. Such systems include a large number of important physical models described in general by nonlinear systems of differential equations that can be integrated by quadratures. To solve the problem of integrability for such systems defined on spaces of constant curvature of low dimension, the method of moving frames and an analogue of classical invariant theory called the Invariant Theory of Killing Tensors is being used. The second area, which is closely related to the first, concerns separability theory for Dirac's equation on pseudo-Riemannian background spaces. This study is important for the elaboration of particle creation mechanisms in black hole physics. The goal is the development of a comprehensive theory of separation of variables for the Dirac equation. The third area of study is the problem of the validity of Huygens' principle in the strict sense for second order linear partial differential equations of normal hyperbolic type. The physical significance of the Huygens' property is that the wave phenomena governed by the equation propagate sharply without the presence wave tails. Work is being undertaken, in the physically important case of four-dimensions, on Hadamard's problem which consists in the determination, up to equivalence, of all the Huygens' equations.

我研究的主题是通过微分几何和李群理论的方法研究数学物理中出现的微分方程。 这项工作在三个相互关联的领域进行，其共同点是所使用的几何和群不变方法。 第一个领域是完全可积哈密顿系统的研究。 此类系统包括大量重要的物理模型，通常由可通过求积积分的微分方程的非线性系统来描述。 为了解决在低维常曲率空间上定义的此类系统的可积性问题，正在使用移动框架的方法和经典不变理论的类似物，称为“杀死张量不变理论”。 第二个领域与第一个领域密切相关，涉及伪黎曼背景空间上狄拉克方程的可分离性理论。 这项研究对于阐述黑洞物理学中的粒子产生机制具有重要意义。 目标是发展狄拉克方程的综合变量分离理论。 第三个研究领域是惠更斯原理在严格意义上对于正双曲型二阶线性偏微分方程的有效性问题。 惠更斯性质的物理意义在于，由该方程控制的波动现象在不存在波尾的情况下急剧传播。 在物理上重要的四维情况下，人们正在研究阿达玛问题，该问题包括确定所有惠更斯方程直至等价。