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This volume of the Encyclopaedia is devoted to applications of singularity theory in mathematics and physics. The authors Arnold,Vasilev, Goryunov and Lyashko study bifurcation sets arising in various contexts such as the stability of singular points of dynamical systems, boundaries of the domains of ellipticity and hyperbolicity of partial differential equations, boundaries of spaces of oscillating linear equations with variable coefficients and boundaries of fundamental systems of solutions.
The book also treats applications of the following topics: functions on manifolds with boundary, projections of complete intersections,caustics, wave fronts, evolvents, maximum functions, shock waves,Petrovskij lacunas and generalizations of Newtons topological proof that Abelian integrals are transcendental.
The book contains a list of open problems, conjectures and directions for future research.
It will be of great interest for mathematicians and physicists as a reference and research aid.
The book also treats applications of the following topics: functions on manifolds with boundary, projections of complete intersections,caustics, wave fronts, evolvents, maximum functions, shock waves,Petrovskij lacunas and generalizations of Newtons topological proof that Abelian integrals are transcendental.
The book contains a list of open problems, conjectures and directions for future research.
It will be of great interest for mathematicians and physicists as a reference and research aid.
目次
Foreword
Chapter 1. Classification of Functions and Mappings
1. Functions on a Manifold with Boundary
1.1. Classification of Functions on a Manifold with a Smooth Boundary
1.2. Versal Deformations and Bifurcation Diagrams
1.3. Relative Homology Basis
1.4. Intersection Form
1.5. Duality of Boundary Singularities
1.6. Functions on a Manifold with a Singular Boundary
2. Complete Intersections
2.1. Start of the Classification
2.2. Critical and Discriminant Sets
2.3. The Nonsingular Fiber
2.4. Relations Between theTyurina and Milnor Numbers
2.5. Adding a Power of a New Variable
2.6. Relative Monodromy
2.7. Dynkin Diagrams
2.8. Parabolic and Hyperbolic Singularities
2.9. Vector Fields on a Quasihomogeneous Complete Intersection
2.10. The Space of a Miniversal Deformation of a Quasihomogeneous Singularity
2.11. Topological Triviality of Versal Deformations
3. Projections and Left-Right Equivalence
3.1. Projections of Space Curves onto the Plane
3.2. Singularities of Projections of Surfaces onto the Plane
3.3. Projections of Complete Intersections
3.4. Projections onto the Line
3.5. Mappings of the Line into the Plane
3.6. Mappings of the Plane into Three-Space
4. Nonisolated Singularities of Functions
4.1. Transversal Type of a Singularity
4.2. Realization
4.3. Topology of the Nonsingular Fiber
4.4. Series of Isolated Singularities
4.5. The Number of Indices of a Series
4.6. Functions with a One-Dimensional Complete Intersection as Critical Set and with Transversal Type A1
5. Vector Fields Tangent to Bifurcation Varieties
5.1. Functions on Smooth Manifolds
5.2. Projections onto the Line
5.3. Isolated Singularities of Complete Intersections
5.4. The Equation of a Free Divisor
6. Divergent and Cyclic Diagrams of Mappings
6.1. Germs of Smooth Functions
6.2. Envelopes
6.3. Holomorphic Diagrams
Chapter 2. Applications of the Classification of Critical Points of Functions
1. Legendre Singularities
1.1. Equidistants
1.2. Projective Duality
1.3. Legendre Transformation
1.4. Singularities of Pedals and Primitives
1.5. The Higher-Dimensional Case
2. Lagrangian Singularities
2.1. Caustics
2.2. The Manifold of Centers
2.3. Caustics of Systems of Rays
2.4. The Gauss Map
2.5. Caustics of Potential Systems of Noninteracting Particles
2.6. Coexistence of Singularities
3. Singularities of Maxwell Sets
3.1. Maxwell Sets
3.2. Metamorphoses of Maxwell Sets
3.3. Extended Maxwell Sets
3.4. Complete Maxwell Set Close to the Singularity A5
3.5. The Structure of Maxwell Sets Close to the Metamorphosis As
3.6. Enumeration of the Connected Components of Spaces of
Nondegenerate Polynomials
4. Bifurcations of Singular Points of Gradient Dynamical Systems
4.1. Thorns Conjecture
4.2. Singularities of Corank One
4.3. Guckenheimers Counterexample
4.4. Three-Parameter Families of Gradients
4.5. Normal Forms of Gradient Systems D4
4.6. Bifurcation Diagrams and Phase Portaits of Standard Families.
4.7. Multiparameter Families
Chapter 3. Singularities of the Boundaries of Domains of Function Spaces
1. Boundary of Stability
1.1. Domains of Stability
1.2. Singularities of the Boundary of Stability in Low-Dimensional Spaces
1.3. Stabilization Theorem
1.4. Finiteness Theorem
2. Boundary of Ellipticity
2.1. Domains of Ellipticity
2.2. Stabilization Theorems
2.3. Boundaries of Ellipticity and Minimum Functions
2.4. Singularities of the Boundary of Ellipticity in Low-Dimensional Spaces
3. Boundary of Hyperbolicity
3.1. Domain of Hyperbolicity
3.2. Stabilization Theorems
3.3. Local Hyperbolicity
3.4. Local Properties of Domains of Hyperbolicity
4. Boundary of the Domain of Fundamental Systems
4.1. Domain of Fundamental Systems and the Bifurcation Set
4.2. Singularities of Bifurcation Sets of Generic Three-Parameter Families
4.3. Bifurcation Sets and Schubert Cells
4.4. Normal Forms
4.5. Duality
4.6. Bifurcation Sets and Tangential Singularities
4.7. The Group of Transformations of Sets and Finite Determinacy
4.8. Bifurcation Diagrams of Flattenings of Projective Curves
5. Linear Differential Equations and Complete Flag Manifolds
Chapter 4. Applications of Ramified Integrals and Generalized Picard- Lefschetz Theories
1. Newtons Theorem on Nonintegrability
1.1. Newtons Theorem and Archimedess Example
1.2. Multi-dimensional Newton Theorem (Even Case)
1.3. Obstructions to Inegrability in the Odd-Dimensional Case
1.4. Newtons Theorem for Nonconvex Domains
1.5. The Case of Nonsmooth Domains
1.6. Homological Formulation and the General Statement ofthe Problem
1.7. Localization and Lowering the Dimension in the Calculation of Monondromy
1.8. General Construction of the Variation Operators
1.9. The Cap Element
1.10. Ramification of Cycles Close to Nonsingular Points
1.11. Ramification Close to Individual Singularities
1.12. Stabilization of Monodromy Close to Strata of Positive Dimension
1.13. Ramification Around the Asymptotic Directions and Monodromy of Boundary Singularities
1.14. Phams Formulas
1.15. Problems, Conjectures, Complements
2. Ramification of Solutions of Hyperbolic Equations
2.1. Hyperbolic Operators and Hyperbolic Polynomials
2.2. Wave Front of a Hyperbolic Operator
2.3. Singularities of Wave Fronts and Generating Functions
2.4. Lacunas, Sharpness, Diffusion
2.5. Sharpness and Diffusion Close to the Simplest Singularities of Wave Fronts
2.6. The Herglotz-Petrovskii-Leray Integral Formula
2.7. The Petrovskii Criterion
2.8. Local Petrovskii Criterion
2.9. Local Petrovskii Cycle
2.10. C∞-Inversion of the Petrovskii Criterion, Stable Singularities of Fronts and Sneaky Diffusion
2.11. Normal Forms of Nonsharpness Close to Singularities of Wave Fronts
2.12. Construction of Leray and Petrovskii Cycles for Strictly Hyperbolic Polynomials
2.13. Problems
3. Integrals of Ramified Forms and Monodromy of Homology with Nontrivial Coefficients
3.1. The Hypergeometric Function of Gauss
3.2. Homology of Local Systems
3.3. Meromorphy of the Integral of the Function Pλ
3.4. The Integral of the Function pλ as a Function of P
3.5. Monodromy and Linear Independence of Hypergeometric Functions
3.6. Twisted Picard-Lefschetz Theory of Isolated Singularities of Smooth Functions and Representations of Hecke Algebras
Chapter 5. Deformations of Real Singularities and Local Petrovskii Lacunas
1. Local Petrovskii Cycles and their Properties
1.1. Definition of Local Petrovskii Cycles
1.2. Complex Conjugation
1.3. Boundary of the Petrovskii Class
1.4. Computation of Petrovskii Cocycles in Terms of Vanishing Cycles
1.5. Stabilization
2. Local Lacunas for Concrete Singularities
2.1. Local Lacunas for Singularities that are Stably Equivalent to Extrema
2.2. The Number of Local Lacunas for the Tabulated Singularit
3. Complements of Discriminants of Real Singularities
3.1. Components of the Complement of the Discriminant of Simple Singularities
3.2. A Regular Search Algorithm for Morse Decompositions of Singularities
3.3. Remarks on the Realization of the Algorithm
3.4. Problems and Perspectives
Bibliography
Author Index
Subject Index
Chapter 1. Classification of Functions and Mappings
1. Functions on a Manifold with Boundary
1.1. Classification of Functions on a Manifold with a Smooth Boundary
1.2. Versal Deformations and Bifurcation Diagrams
1.3. Relative Homology Basis
1.4. Intersection Form
1.5. Duality of Boundary Singularities
1.6. Functions on a Manifold with a Singular Boundary
2. Complete Intersections
2.1. Start of the Classification
2.2. Critical and Discriminant Sets
2.3. The Nonsingular Fiber
2.4. Relations Between theTyurina and Milnor Numbers
2.5. Adding a Power of a New Variable
2.6. Relative Monodromy
2.7. Dynkin Diagrams
2.8. Parabolic and Hyperbolic Singularities
2.9. Vector Fields on a Quasihomogeneous Complete Intersection
2.10. The Space of a Miniversal Deformation of a Quasihomogeneous Singularity
2.11. Topological Triviality of Versal Deformations
3. Projections and Left-Right Equivalence
3.1. Projections of Space Curves onto the Plane
3.2. Singularities of Projections of Surfaces onto the Plane
3.3. Projections of Complete Intersections
3.4. Projections onto the Line
3.5. Mappings of the Line into the Plane
3.6. Mappings of the Plane into Three-Space
4. Nonisolated Singularities of Functions
4.1. Transversal Type of a Singularity
4.2. Realization
4.3. Topology of the Nonsingular Fiber
4.4. Series of Isolated Singularities
4.5. The Number of Indices of a Series
4.6. Functions with a One-Dimensional Complete Intersection as Critical Set and with Transversal Type A1
5. Vector Fields Tangent to Bifurcation Varieties
5.1. Functions on Smooth Manifolds
5.2. Projections onto the Line
5.3. Isolated Singularities of Complete Intersections
5.4. The Equation of a Free Divisor
6. Divergent and Cyclic Diagrams of Mappings
6.1. Germs of Smooth Functions
6.2. Envelopes
6.3. Holomorphic Diagrams
Chapter 2. Applications of the Classification of Critical Points of Functions
1. Legendre Singularities
1.1. Equidistants
1.2. Projective Duality
1.3. Legendre Transformation
1.4. Singularities of Pedals and Primitives
1.5. The Higher-Dimensional Case
2. Lagrangian Singularities
2.1. Caustics
2.2. The Manifold of Centers
2.3. Caustics of Systems of Rays
2.4. The Gauss Map
2.5. Caustics of Potential Systems of Noninteracting Particles
2.6. Coexistence of Singularities
3. Singularities of Maxwell Sets
3.1. Maxwell Sets
3.2. Metamorphoses of Maxwell Sets
3.3. Extended Maxwell Sets
3.4. Complete Maxwell Set Close to the Singularity A5
3.5. The Structure of Maxwell Sets Close to the Metamorphosis As
3.6. Enumeration of the Connected Components of Spaces of
Nondegenerate Polynomials
4. Bifurcations of Singular Points of Gradient Dynamical Systems
4.1. Thorns Conjecture
4.2. Singularities of Corank One
4.3. Guckenheimers Counterexample
4.4. Three-Parameter Families of Gradients
4.5. Normal Forms of Gradient Systems D4
4.6. Bifurcation Diagrams and Phase Portaits of Standard Families.
4.7. Multiparameter Families
Chapter 3. Singularities of the Boundaries of Domains of Function Spaces
1. Boundary of Stability
1.1. Domains of Stability
1.2. Singularities of the Boundary of Stability in Low-Dimensional Spaces
1.3. Stabilization Theorem
1.4. Finiteness Theorem
2. Boundary of Ellipticity
2.1. Domains of Ellipticity
2.2. Stabilization Theorems
2.3. Boundaries of Ellipticity and Minimum Functions
2.4. Singularities of the Boundary of Ellipticity in Low-Dimensional Spaces
3. Boundary of Hyperbolicity
3.1. Domain of Hyperbolicity
3.2. Stabilization Theorems
3.3. Local Hyperbolicity
3.4. Local Properties of Domains of Hyperbolicity
4. Boundary of the Domain of Fundamental Systems
4.1. Domain of Fundamental Systems and the Bifurcation Set
4.2. Singularities of Bifurcation Sets of Generic Three-Parameter Families
4.3. Bifurcation Sets and Schubert Cells
4.4. Normal Forms
4.5. Duality
4.6. Bifurcation Sets and Tangential Singularities
4.7. The Group of Transformations of Sets and Finite Determinacy
4.8. Bifurcation Diagrams of Flattenings of Projective Curves
5. Linear Differential Equations and Complete Flag Manifolds
Chapter 4. Applications of Ramified Integrals and Generalized Picard- Lefschetz Theories
1. Newtons Theorem on Nonintegrability
1.1. Newtons Theorem and Archimedess Example
1.2. Multi-dimensional Newton Theorem (Even Case)
1.3. Obstructions to Inegrability in the Odd-Dimensional Case
1.4. Newtons Theorem for Nonconvex Domains
1.5. The Case of Nonsmooth Domains
1.6. Homological Formulation and the General Statement ofthe Problem
1.7. Localization and Lowering the Dimension in the Calculation of Monondromy
1.8. General Construction of the Variation Operators
1.9. The Cap Element
1.10. Ramification of Cycles Close to Nonsingular Points
1.11. Ramification Close to Individual Singularities
1.12. Stabilization of Monodromy Close to Strata of Positive Dimension
1.13. Ramification Around the Asymptotic Directions and Monodromy of Boundary Singularities
1.14. Phams Formulas
1.15. Problems, Conjectures, Complements
2. Ramification of Solutions of Hyperbolic Equations
2.1. Hyperbolic Operators and Hyperbolic Polynomials
2.2. Wave Front of a Hyperbolic Operator
2.3. Singularities of Wave Fronts and Generating Functions
2.4. Lacunas, Sharpness, Diffusion
2.5. Sharpness and Diffusion Close to the Simplest Singularities of Wave Fronts
2.6. The Herglotz-Petrovskii-Leray Integral Formula
2.7. The Petrovskii Criterion
2.8. Local Petrovskii Criterion
2.9. Local Petrovskii Cycle
2.10. C∞-Inversion of the Petrovskii Criterion, Stable Singularities of Fronts and Sneaky Diffusion
2.11. Normal Forms of Nonsharpness Close to Singularities of Wave Fronts
2.12. Construction of Leray and Petrovskii Cycles for Strictly Hyperbolic Polynomials
2.13. Problems
3. Integrals of Ramified Forms and Monodromy of Homology with Nontrivial Coefficients
3.1. The Hypergeometric Function of Gauss
3.2. Homology of Local Systems
3.3. Meromorphy of the Integral of the Function Pλ
3.4. The Integral of the Function pλ as a Function of P
3.5. Monodromy and Linear Independence of Hypergeometric Functions
3.6. Twisted Picard-Lefschetz Theory of Isolated Singularities of Smooth Functions and Representations of Hecke Algebras
Chapter 5. Deformations of Real Singularities and Local Petrovskii Lacunas
1. Local Petrovskii Cycles and their Properties
1.1. Definition of Local Petrovskii Cycles
1.2. Complex Conjugation
1.3. Boundary of the Petrovskii Class
1.4. Computation of Petrovskii Cocycles in Terms of Vanishing Cycles
1.5. Stabilization
2. Local Lacunas for Concrete Singularities
2.1. Local Lacunas for Singularities that are Stably Equivalent to Extrema
2.2. The Number of Local Lacunas for the Tabulated Singularit
3. Complements of Discriminants of Real Singularities
3.1. Components of the Complement of the Discriminant of Simple Singularities
3.2. A Regular Search Algorithm for Morse Decompositions of Singularities
3.3. Remarks on the Realization of the Algorithm
3.4. Problems and Perspectives
Bibliography
Author Index
Subject Index
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