Differential geometry methods in mathematical physics

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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. **