Near the Horizon starts out by considering several optical phenomena that can occur when the sun is near the horizon. One can sometimes see objects that are actually below the horizon. Sometimes there seems to be a dark strip in the middle of the solar disk. These are a result of the way that the atmosphere affects the geometry of light rays. Broer starts his book with the Fermat principle (rays of light take least-time paths) and deduces from it laws for refraction and reflection; by expressing these as conservation laws, he can handle both the case of inhomogeneous layers of air and the case of continuous variation in the refraction index. A surprising application is the brachistochrone problem, in which the path of fastest descent is determined by studying how a light ray would behave in a “flat earth” atmosphere whose refraction index is determined by the gravitational potential. This leads to a very interesting chapter on the cycloid and its properties. The final chapters move from the elementary theory to a more sophisticated version in which the Fermat Principle leads to a Riemannian metric whose geodesics are the paths of light rays. This gives us an optics which is geometric in a new sense, and serves as a nice demonstration of the physical applicability of Riemannian geometry.

It is by no means easy for the applied mathematician to decide how much importance he should attach to the more abstract and aesthetic side of his work ... To all appearances, Sir William Rowan Hamilton (1850-1865) attached little importance to the practical applications of his method, and it was only with the publication of his Mathematical Papers that it was possible to form a more correct and balanced judgement of Hamilton as an applied mathematician.

This book contains fourteen research papers which are expanded versions of conferences given at a meeting held in September 1996 in Cortona, Italy. The topics include blowup questions for quasilinear equations in two dimensions, time decay of waves in LP, uniqueness results for systems of conservation laws in one dimension, concentra tion effects for critical nonlinear wave equations, diffraction of nonlin ear waves, propagation of singularities in scattering theory, caustics for semi-linear oscillations. Other topics linked to microlocal analysis are Sobolev embedding theorems in Weyl-Hormander calculus, local solv ability for pseudodifferential equations, hypoellipticity for highly degen erate operators. The book also contains a result on uniqueness for the Cauchy problem under partial analyticity assumptions and an article on the regularity of solutions for characteristic initial-boundary value problems. On each topic listed above, one will find new results as well as a description of the state of the art. Various methods related to nonlinear geometrical optics are a transversal theme of several articles. Pseu dodifferential techniques are used to tackle classical PDE problems like Cauchy uniqueness. We are pleased to thank the speakers for their contributions to the meeting: Serge Alinhac, Mike Beals, Alberto Bressan, Jean-Yves Chemin, Christophe Cheverry, Daniele Del Santo, Nils Dencker, Patrick Gerard, Lars Hormander, John Hunter, Richard Melrose, Guy Metivier, Yoshinori Morimoto, and Tatsuo Nishitani. The meeting was made possible in part by the financial support of a European commission pro gram, "Human capital and mobility CHRX-CT94-044."