Ebook: Singularity Theory and Gravitational Lensing

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 lens 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 these project on gravitational lensing. The book is also an excellent reference text for professional mathematicians, mathematical physicists, astrophysicists, and physicists.

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Content:
Front Matter....Pages i-xxv
Front Matter....Pages 1-1
Historical Highlights....Pages 3-14
Central Problems....Pages 15-21
Front Matter....Pages 23-23
Basic Physical Concepts....Pages 25-117
Physical Applications....Pages 119-141
Observations of Gravitational Lensing....Pages 143-168
Front Matter....Pages 169-169
Time Delay and Lensing Maps....Pages 171-208
Critical Points and Stability....Pages 209-286
Classification and Genericity of Stable Lens Systems....Pages 287-325
Local Lensing Geometry....Pages 327-392
Morse Inequalities....Pages 393-418
Counting Lensed Images: Single-Plane Case....Pages 419-444
Counting Lensed Images: Multiplane Case....Pages 445-465
Total Magnification....Pages 467-485
Computing the Euler Characteristic....Pages 487-501
Global Geometry of Caustics....Pages 503-559
Back Matter....Pages 561-603

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Content:
Front Matter....Pages i-xxv
Front Matter....Pages 1-1
Historical Highlights....Pages 3-14
Central Problems....Pages 15-21
Front Matter....Pages 23-23
Basic Physical Concepts....Pages 25-117
Physical Applications....Pages 119-141
Observations of Gravitational Lensing....Pages 143-168
Front Matter....Pages 169-169
Time Delay and Lensing Maps....Pages 171-208
Critical Points and Stability....Pages 209-286
Classification and Genericity of Stable Lens Systems....Pages 287-325
Local Lensing Geometry....Pages 327-392
Morse Inequalities....Pages 393-418
Counting Lensed Images: Single-Plane Case....Pages 419-444
Counting Lensed Images: Multiplane Case....Pages 445-465
Total Magnification....Pages 467-485
Computing the Euler Characteristic....Pages 487-501
Global Geometry of Caustics....Pages 503-559
Back Matter....Pages 561-603
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