This monograph is the first to develop a mathematical theory of gravitational lensing. The theory applies to any finite number of deflector planes and highlights the distinctions between single and multiple plane lensing. Introductory material in Parts I and II present historical highlights and the astrophysical aspects of the subject. Part III employs the ideas and results of singularity theory to put gravitational lensing on a rigorous mathematical foundation.

The observation, in 1919 by A.S. Eddington and collaborators, of the gra- tational de?ection of light by the Sun proved one of the many predictions of Einstein's Theory of General Relativity: The Sun was the ?rst example of a gravitational lens. In 1936, Albert Einstein published an article in which he suggested - ing stars as gravitational lenses. A year later, Fritz Zwicky pointed out that galaxies would act as lenses much more likely than stars, and also gave a list of possible applications, as a means to determine the dark matter content of galaxies and clusters of galaxies. It was only in 1979 that the ?rst example of an extragalactic gravitational lens was provided by the observation of the distant quasar QSO 0957+0561, by D. Walsh, R.F. Carswell, and R.J. Weymann. A few years later, the ?rst lens showing images in the form of arcs was detected. The theory, observations, and applications of gravitational lensing cons- tute one of the most rapidly growing branches of astrophysics. The gravi- tional de?ection of light generated by mass concentrations along a light path producesmagni?cation,multiplicity,anddistortionofimages,anddelaysp- ton propagation from one line of sight relative to another. The huge amount of scienti?c work produced over the last decade on gravitational lensing has clearly revealed its already substantial and wide impact, and its potential for future astrophysical applications.

The observation, in 1919 by A.S. Eddington and collaborators, of the gra- tational de?ection of light by the Sun proved one of the many predictions of Einstein's Theory of General Relativity: The Sun was the ?rst example of a gravitational lens. In 1936, Albert Einstein published an article in which he suggested - ing stars as gravitational lenses. A year later, Fritz Zwicky pointed out that galaxies would act as lenses much more likely than stars, and also gave a list of possible applications, as a means to determine the dark matter content of galaxies and clusters of galaxies. It was only in 1979 that the ?rst example of an extragalactic gravitational lens was provided by the observation of the distant quasar QSO 0957+0561, by D. Walsh, R.F. Carswell, and R.J. Weymann. A few years later, the ?rst lens showing images in the form of arcs was detected. The theory, observations, and applications of gravitational lensing cons- tute one of the most rapidly growing branches of astrophysics. The gravi- tional de?ection of light generated by mass concentrations along a light path producesmagni?cation,multiplicity,anddistortionofimages,anddelaysp- ton propagation from one line of sight relative to another. The huge amount of scienti?c work produced over the last decade on gravitational lensing has clearly revealed its already substantial and wide impact, and its potential for future astrophysical applications.

This monograph, unique in the literature, is the first to develop a mathematical theory of gravitational lensing. The theory applies to any finite number of deflector planes and highlights the distinctions between single and multiple plane lensing. Introductory material in Parts I and II present historical highlights and the astrophysical aspects of the subject. Among the lensing topics discussed are multiple quasars, giant luminous arcs, Einstein rings, the detection of dark matter and planets with lensing, time delays and the age of the universe (Hubble's constant), microlensing of stars and quasars. The main part of the book---Part III---employs the ideas and results of singularity theory to put gravitational lensing on a rigorous mathematical foundation and solve certain key lensing problems. Results are published here for the first time. Mathematical topics discussed: Morse theory, Whitney singularity theory, Thom catastrophe theory, Mather stability theory, Arnold singularity theory, and the Euler characteristic via projectivized rotation numbers. These tools are applied to the study of stable lens systems, local and global geometry of caustics, caustic metamorphoses, multiple lensed images, lensed image magnification, magnification cross sections, and lensing by singular and nonsingular deflectors. Examples, illustrations, bibliography and index make this a suitable text for an undergraduate/graduate course, seminar, or independent thesis project on gravitational lensing. The book is also an excellent reference text for professional mathematicians, mathematical physicists, astrophysicists, and physicists.