Differential geometry methods in mathematical physics

数学物理中的微分几何方法

• 批准号: RGPIN-2016-04133
• 负责人: McLenaghan, Raymond
• 金额: $ 1.09万
• 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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=614857
• 关键词: Differential; geometry; 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 the theory of warped products and concircular tensors (a special type of conformal Killing tensor) is being used to give an invariant characterization of all inequivalent separable coordinate systems for the Hamilton-Jacobi equation for the geodesics. 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.

我的研究主题是用微分几何和李群理论研究数学物理中出现的微分方程式。这项工作是在三个相互关联的领域进行的，它们的共同主线是所使用的几何和群不变量方法。 第一个领域是对完全可积哈密顿系统的研究。这类系统包括大量重要的物理模型，这些模型一般由可用求积积分的非线性微分方程组描述。为了解决定义在常曲率空间上的这类系统的可积性问题，利用翘曲积和共圆张量(一种特殊的共形Killing张量)理论，给出了测地线的Hamilton-Jacobi方程的所有不等价可分坐标系的不变刻画. 第二个领域与第一个领域密切相关，它涉及伪黎曼背景空间上狄拉克方程的可分性理论。这一研究对于阐明黑洞物理中的粒子产生机制具有重要意义。我们的目标是为狄拉克方程发展一个全面的分离变量理论。 第三个研究领域是正规双曲型二阶线性偏微分方程惠更斯原理在严格意义下的有效性问题。惠更斯性质的物理意义在于，由该方程控制的波动现象在没有波尾的情况下会急剧传播。在物理上重要的四维情况下，阿达玛问题的工作正在进行，这一问题在于确定所有惠更斯方程的等价性。