X PREFACE distributions in such a way as to permit maximum interplay between these two functors. In § 1 we indicate the basic properties of these two functors, and use them to define a special class of distributions called 8-distributions and derive a fixed point formula. In §2 we give a few applications of the ideas discussed in §1 in particular we prove a few theorems about characters of induced representations (in the theory of group representations) and sketch a proof of the Atiyah-Bott fixed point formula. In §3 we define the wave front set of a distribution. It turns out that from our functorial point of view it is convenient to do this using the Radon transform rather than the Fourier transform as Hormander does. (This is because the Radon transform can be defined as a composition of a "push- forward" and a "pull-back".) Therefore, we have included in this section a discussion of the basic properties of the Radon transform. The idea of using the Radon transform was pointed out to us by Dave Schaeffer in a lecture in which he expounded some fundamental papers by Ludwig who had developed the theory of singularities from this point of view. Ludwig's papers have served as a guide to us in constructing much of the theory of this chapter. In addition we discuss a few other Radon-like transforms which are defined by push-pull operations and show that the only "elliptic" examples have properties very similar to the classical Radon transforms on rank one homogeneous spaces. In §4 we discuss a larger class of distributions than those of §1. This class is obtained by starting with the 8-function on the real line and applying the operations of pull-back, push-forward, differentiation and multiplication by smooth functions. What one gets is approximately the class of distributions discussed by Hormander in his basic paper on Fourier integral operators (though his way of defining these distributions is quite different from ours.) In §5 we develop a symbol calculus for these distributions, again making heavy use of the two functorialities. Our symbols differ a little from those of Hormander in that instead of using the Maslov line bundle, as Hormander does, to handle phase adjustments we use the metaplectic structure discussed in Chapter V. In particular, our symbols are "half-forms" instead of half-densities. We get a slightly less general theory than that of Hormander. (The manifolds we consider have to satisfy wY(X) 0, w^X) being the first Stiefel-Whitney class.) Howev- er, we feel this disadvantage is outweighed by the advantage that the symbols are less complicated. We indicate how our theory needs to be modified to cover the more general situation. In §6 we discuss the calculus of Fourier integral operators and derive some applications to partial differential equations. In §7 we describe how our distributions behave under composition with differential operators and show that on the symbol level o(Pfx) is given by applying the transport equation to a(/x) just as we did in Chapter II in a somewhat more elementary setting. As an application we describe the progressing and the regressing fundamental solutions of the wave equation
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