xii Preface The auditors included many at the beginnings of their careers, and I would like to thank in particular R. Carles, E. Dumas, J. Bronski, J. Col- liander, M. Keel, L. Miller, K. McLaughlin, R. McLaughlin, H. Zag, G. Crippa, A. Figalli, and N. Visciglia for many interesting questions and com- ments. The book is aimed at the level of graduate students who have studied one hard course in partial differential equations. Following the lead of the book of Guillemin and Pollack (1974), there are exercises scattered throughout the text. The reader is encouraged to read with paper and pencil in hand, filling in and verifying as they go. There is a big difference between passive reading and active acquisition. In a classroom setting, correcting students’ exercises offers the opportunity to teach the writing of mathematics. To shorten the treatment and to avoid repetition with a solid partial differential equations course, basic material such as the fundamental solution of the wave equation in low dimensions is not presented. Naturally, I like the treatment of that material in my book Partial Differential Equations [Rauch, 1991]. The choice of subject matter is guided by several principles. By restrict- ing to symmetric hyperbolic systems, the basic energy estimates come from integration by parts. The majority of examples from applications fall under this umbrella. The treatment of constant coeﬃcient problems does not follow the usual path of describing classes of operators for which the Cauchy problem is weakly well posed. Such results are described in Appendix 2.I along with the Kreiss matrix theorem. Rather, the Fourier transform is used to analyse the dispersive properties of constant coeﬃcient symmetric hyperbolic equations including Brenner’s theorem and Strichartz estimates. Pseudodifferential operators are neither presented nor used. This is not because they are in any sense vile, but to get to the core without too many pauses to develop machinery. There are several good sources on pseudo- differential operators and the reader is encouraged to consult them to get alternate viewpoints on some of the material. In a sense, the expansions of geometric optics are a natural replacement for that machinery. Lax’s parametrix and H¨ ormander’s microlocal propagation of singularities theo- rem require the analysis of oscillatory integrals as in the theory of Fourier integral operators. The results require only the method of nonstationary phase and are included. The topic of caustics and caustic crossing is not treated. The sharp linear results use more microlocal machinery and the nonlinear analogues are topics of current research. The same is true for supercritical nonlinear geometric optics which is not discussed. The subjects of dispersive and

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