  This dissertation reports research about the phase space perspective for
solving wave problems, with particular emphasis on the phenomenon of mode
conversion in multicomponent wave systems, and the mathematics which underlie
the phase space perspective. Part I of this dissertation gives a review of the
phase space theory of resonant mode conversion. We describe how the WKB
approximation is related to geometrical structures in phase space, and how in
particular ray-tracing algorithms can be used to construct the WKB solution. We
also present new higher order corrections to the local solution for the mode
conversion problem which allow better asymptotic matching to the WKB solutions.
The phase space tools used in Part I rely on the Weyl symbol calculus, which
gives a way to relate operators to functions on phase space. In Part II of this
dissertation, we explore the mathematical foundations of the theory of symbols.
We go on to present the group theoretical formulation of symbols, as developed
recently by Zobin. This defines the symbol of an operator in terms of a double
Fourier transform on a non-commutative group. We then show how the path
integral arises when calculating the symbol of a function of an operator. We
conclude with a survey of ideas for future research and describe several
potential applications of this group theoretical perspective to problems in
mode conversion.
