Preface There has been a great deal of active research recently in the fields of symplectic geometry and the theory of Fourier integral operators. In symplectic geometry there has been much progress in understanding the geometry of dynamical systems, and the process of "quantization" as applied not only to the theory of dynamical systems, but also as a key tool in the analysis of group representations. Fourier integral operators have made possible a much more systematic analysis of the singularities of solutions of linear partial differential equations than existed heretofore, together with a good deal of geometric information associated with the spectra of such operators. These two subjects are modern manifestations of themes that have occupied a central position in mathematical thought for the past three hundred years—the relations between the wave and the corpuscular theories of light. The purpose of this book is to develop these themes, and present some of the recent advances, using the language of differential geometry as a unifying influence. We cannot pretend to do justice to the history of our subject, either in the short space of this preface, or in the body of the book. Yet some very brief mention of the early history is in order. The contributions of Newton and Huygens to the theory of light are known to all students. (Huygens' "envelope construction"— wherein the superposition of singularities distributed along a family of surfaces accumulates as singularity along the envelope—makes its appearance in the modern setting as a formula for the behaviour of wave front sets under functorial operations cf. Chapter VI, equation (3.6).) In Fresnel's prize memoir of 1818, he combined Huygens' envelope construction with Young's "principle of interfer- ence" to explain not only the rectilinear propagation of light but also diffraction effects. Thus Fresnel applied Huygens' method to the superposition of oscilla- tions instead of disturbances and was able to calculate the diffraction caused by straight edges, small aperatures and screens. In Chapter VI we study the geometry of the superposition of "disturbances" while in Chapter VII we study the geometry of the superposition of (high frequency) oscillations. In 1828 v
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