Differential geometry methods in mathematical physics

我的研究主题是用微分几何和李群论的方法研究数学物理中出现的微分方程。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方法和群不变方法。第一个领域是完全可积哈密顿系统的研究。这样的系统包括大量重要的物理模型，这些模型通常由非线性微分方程组描述，可以用正交积分来积分。为了解决这类系统在常曲率空间上的可积性问题，利用弯曲积和共圆张量（一种特殊类型的共形杀张量）理论给出了测地学上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，涉及伪黎曼背景空间上狄拉克方程的可分性理论。该研究对阐明黑洞物理中粒子产生机制具有重要意义。目标是为狄拉克方程发展一个综合的分离变量理论。***第三个研究领域是正规双曲型二阶线性偏微分方程严格意义上的惠更斯原理的有效性问题。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存在的情况下急剧传播。在物理上重要的四维情况下，人们正在研究哈达玛尔问题，该问题包括确定所有惠更斯方程的等价性。**

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. **