Differential geometric methods in mathematical physics

点击翻译按钮获取中文摘要

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.