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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.
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.
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