Singularity Theory I (Encyclopaedia of Mathematical Sciences, 6)

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9783540637110: Singularity Theory I (Encyclopaedia of Mathematical Sciences, 6)

This is a compact guide to the principles and main applications of Singularity Theory by one of the world’s top research groups. It includes a number of new results as well as a carefully prepared and extensive bibliography that makes it easy to find the necessary details. It’s ideal for any mathematician or physicist interested in modern mathematical analysis.

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From the reviews of the first printing of this book, published as volume 6 of the Encyclopaedia of Mathematical Sciences: "... My general impression is of a particularly nice book, with a well-balanced bibliography, recommended!"
Medelingen van Het Wiskundig Genootschap, 1995

"... The authors offer here an up to date guide to the topic and its main applications, including a number of new results. It is very convenient for the reader, a carefully prepared and extensive bibliography ... makes it easy to find the necessary details when needed. The books (EMS 6 and EMS 39) describe a lot of interesting topics. ... Both volumes are a very valuable addition to the library of any mathematician or physicist interested in modern mathematical analysis."
European Mathematical Society Newsletter, 1994
"...The authors are recognized experts in their fields and so are ideal choices to write such a survey. ...The text of the book is liberally sprinkled with illustrative examples and so the style is not heavy going or turgid... The bibliography is very good and extremely large ..."

IMS Bulletin, 1995

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From the reviews: " My general impression is of a particularly nice book, with a well-balanced bibliography.  Recommended!"  Mededelingen van Het Wiskundig Genootschap, 1995  "The authors offer here an up- to- date guide to the topic and its main applications, including a number of new results. It is very convenient for the reader... a carefully prepared and extensive bibliography makes it easy to find the necessary details when needed. The books (EMS 6 and EMS 39) describe a lot of interesting topics.  Both volumes are a very valuable addition to the library of any mathematician or physicist interested in modern mathematical analysis." European Mathematical Society Newsletter, 1994

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V.I. Arnold (author), V.V. Goryunov (author), O.V. Lyashko (author), V.A. Vasil'ev (author), A. Iacob (translator)
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Book Description Book Condition: New. Publisher/Verlag: Springer, Berlin | This is a compact guide to the principles and main applications of Singularity Theory by one of the world's top research groups. It includes a number of new results as well as a carefully prepared and extensive bibliography that makes it easy to find the necessary details. It's ideal for any mathematician or physicist interested in modern mathematical analysis. | 1. Critical Points of Functions.- 1. Invariants of Critical Points.- 1.1. Degenerate and Nondegenerate Critical Points.- 1.2. Equivalence of Critical Points.- 1.3. Stable Equivalence.- 1.4. The Local Algebra and the Multiplicity of a Singularity.- 1.5. Finite Determinacy of an Isolated Singularity.- 1.6. Lie Group Actions on Manifolds.- 1.7. Versal Deformations of a Critical Point.- 1.8. Infinitesimal Versality.- 1.9. The Modality of a Critical Point.- 1.10. The Level Bifurcation Set.- 1.11. Truncated Versal Deformations and the Function Bifurcation Set.- 2. The Classification of Critical Points.- 2.1. Normal Forms.- 2.2. Classes of Low Modality.- 2.3. Singularities of Modality ? 2.- 2.4. Simple Singularities and Klein Singularities.- 2.5. Resolution of Simple Singularities.- 2.6. Unimodal and Bimodal Singularities.- 2.7. Adjacency of Singularities.- 2.8. Real Singularities.- 3. Reduction to Normal Forms.- 3.1. The Newton Diagram.- 3.2. Quasihomogeneous Functions and Filtrations.- 3.3. The Multiplicity and the Generators of the Local Algebra of a Semi-Quasihomogeneous Function.- 3.4. Quasihomogeneous Maps.- 3.5. Quasihomogeneous Diffeomorphisms and Vector Fields.- 3.6. The Normal Form of a Semi-Quasihomogeneous Function.- 3.7. The Normal Form of a Quasihomogeneous Function.- 3.8. The Newton Filtration.- 3.9. The Spectral Sequence.- 3.10. Theorems on Normal Forms for the Spectral Sequence.- 2. Monodromy Groups of Critical Points.- 1. The Picard-Lefschetz Theory.- 1.1. Topology of the Nonsingular Level Manifold.- 1.2. The Classical Monodromy and the Variation Operator.- 1.3. The Monodromy of a Morse Singularity.- 1.4. The Monodromy Group of an Isolated Singularity.- 1.5. Vanishing Cycles and Distinguished Bases.- 1.6. The Intersection Matrix of a Singularity.- 1.7. Stabilization of Singularities.- 1.8. Dynkin Diagrams.- 1.9. Transformations of a Basis and of its Dynkin Diagram.- 1.10. The Milnor Fibration over the Complement of the Level Bifurcation Set.- 1.11. The Topological Type of a Singularity Along the ?-Constant Stratum.- 2. Dynkin Diagrams and Monodromy Groups.- 2.1. Intersection Matrices of Singularities of Functions of Two Variables.- 2.2. The Intersection Matrix of a Direct Sum of Singularities.- 2.3. Pham Singularities.- 2.4. The Polar Curve and the Intersection Matrix.- 2.5. Modality and Quadratic Forms of Singularities.- 2.6. The Monodromy Group and the Intersection Form.- 2.7. The Monodromy Group in the Skew-Symmetric Case.- 3. Complex Monodromy and Period Maps.- 3.1. The Cohomology Bundle and the Gauss-Manin Connection.- 3.2. Sections of the Cohomology Bundle.- 3.3. The Vanishing Cohomology Bundle.- 3.4. The Period Map.- 3.5. The Residue Form.- 3.6. Trivializations of the Cohomology Bundle.- 3.7. The Classical Complex Monodromy.- 3.8. Differential Equations and Asymptotics of Integrals.- 3.9. Nondegenerate Period Maps.- 3.10. Stability of Period Maps.- 3.11. Period Maps and Intersection Forms.- 3.12. The Characteristic Polynomial and the Zeta Function of the Monodromy Operator.- 4. The Mixed Hodge Structure in the Vanishing Cohomology.- 4.1. The Pure Hodge Structure.- 4.2. The Mixed Hodge Structure.- 4.3. The Asymptotic Hodge Filtration in the Fibres of the Cohomology Bundle.- 4.4. The Weight Filtration.- 4.5. The Asymptotic Mixed Hodge Structure.- 4.6. The Hodge Numbers and the Spectrum of a Singularity.- 4.7. Computing the Spectrum.- 4.8. Semicontinuity of the Spectrum.- 4.9. The Spectrum and the Geometric Genus.- 4.10. The Mixed Hodge Structure and the Intersection Form.- 4.11. The Number of Singular Poin. Bookseller Inventory # K9783540637110

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V.I. Arnold; V.V. Goryunov; O.V. Lyashko; V.A. Vasil'ev
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Book Description Springer Mrz 1998, 1998. Taschenbuch. Book Condition: Neu. Neuware - This is a compact guide to the principles and main applications of Singularity Theory by one of the world's top research groups. It includes a number of new results as well as a carefully prepared and extensive bibliography that makes it easy to find the necessary details. It's ideal for any mathematician or physicist interested in modern mathematical analysis. 252 pp. Englisch. Bookseller Inventory # 9783540637110

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