Table of Contents
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 Points of a Complex Projective Hypersurface.- 4.12. The Generalized Petrovski?-Ole?nik Inequalities.- 5. Simple Singularities.- 5.1. Reflection Groups.- 5.2. The Swallowtail of a Reflection Group.- 5.3. The Artin-Brieskorn Braid Group.- 5.4. Convolution of Invariants of a Coxeter Group.- 5.5. Root Systems and Weyl Groups.- 5.6. Simple Singularities and Weyl Groups.- 5.7. Vector Fields Tangent to the Level Bifurcation Set.- 5.8. The Complement of the Function Bifurcation Set.- 5.9. Adjacency and Decomposition of Simple Singularities.- 5.10. Finite Subgroups of SU2, Simple Singularities, and Weyl Groups.- 5.11. Parabolic Singularities.- 6. Topology of Complements of Discriminants of Singularities.- 6.1. Complements of Discriminants and Braid Groups.- 6.2. The mod-2 Cohomology of Braid Groups.- 6.3. An Application: Superposition of Algebraic Functions.- 6.4. The Integer Cohomology of Braid Groups.- 6.5. The Cohomology of Braid Groups with Twisted Coefficients.- 6.6. Genus of Coverings Associated with an Algebraic Function, and Complexity of Algorithms for Computing Roots of Polynomials.- 6.7. The Cohomology of Brieskorn Braid Groups and Complements of the Discriminants of Singularities of the Series C and D.- 6.8. The Stable Cohomology of Complements of Level Bifurcation Sets.- 6.9. Characteristic Classes of Milnor Cohomology Bundles.- 6.10. Stable Irreducibility of Strata of Discriminants.- 3. Basic Properties of Maps.- 1. Stable Maps and Maps of Finite Multiplicity.- 1.1. The Left-Right Equivalence.- 1.2. Stability.- 1.3. Transversality.- 1.4. The Thom-Boardman Classes.- 1.5. Infinitesimal Stability.- 1.6. The Groups l and K.- 1.7. Normal Forms of Stable Germs.- 1.8. Examples.- 1.9. Nice and Semi-Nice Dimensions.- 1.10. Maps of Finite Multiplicity.- 1.11. The Number of Roots of a System of Equations.- 1.12. The Index of a Singular Point of a Real Germ, and Polynomial Vector Fields.- 2. Finite Determinacy of Map-Germs, and Their Versal Deformations.- 2.1. Tangent Spaces and Codimensions.- 2.2. Finite Determinacy.- 2.3. Versal Deformations.- 2.4. Examples.- 2.5. Geometric Subgroups.- 2.6. The Order of a Sufficient Jet.- 2.7. Determinacy with Respect to Transformations of Finite Smoothness.- 3. The Topological Equivalence.- 3.1. The Topologically Stable Maps are Dense.- 3.2. Whitney Stratifications.- 3.3. The Topological Classification of Smooth Map-Germs.- 3.4. Topological Invariants.- 3.5. Topological Triviality and Topological Versality of Deformations of Semi-Quasihomogeneous Maps.- 4. The Global Theory of Singularities.- 1. Thom Polynomials for Maps of Smooth Manifolds.- 1.1. Cycles of Singularities and Topological Invariants of Maps.- 1.2. Thom’s Theorem on the Existence of Thom Polynomials.- 1.3. Resolution of the Singularities of the Closures of the Thom-Boardman Classes.- 1.4. Thorn Polynomials for Singularities of First Order.- 1.5. Survey of Results on Thom Polynomials for Singularities of Higher Order.- 2. Integer Characteristic Classes and Universal Complexes of Singularities.- 2.1. Examples: the Maslov Index and the First Pontryagin Class.- 2.2. The Universal Complex of Singularities of Smooth Functions.- 2.3. Cohomology of the Complexes of R0-Invariant Singularities, and Invariants of Foliations.- 2.4. Computations in Complexes of Singularities of Functions. Geometric Consequences.- 2.5. Universal Complexes of Lagrangian and Legendrian Singularities.- 2.6. On Universal Complexes of General Maps of Manifolds.- 3. Multiple Points and Multisingularities.- 3.1. A Formula for Multiple Points of Immersions, and Embedding Obstructions for Manifolds.- 3.2. Triple Points of Singular Surfaces.- 3.3. Multiple Points of Complex Maps.- 3.4. Self-Intersections of Lagrangian Manifolds.- 3.5. Complexes of Multisingularities.- 3.6. Multisingularities and Multiplication in the Cohomology of the Target Space of a Map.- 4. Spaces of Functions with Critical Points of Mild Complexity.- 4.1. Functions with Singularities Simpler than A3.- 4.2. The Group of Curves Without Horizontal Inflexional Tangents.- 4.3. Homotopy Properties of the Complements of Unfurled Swallowtails.- 5. Elimination of Singularities and Solution of Differential Conditions.- 5.1. Cancellation of Whitney Umbrellas and Cusps. The Immersion Problem.- 5.2. The Smale-Hirsch Theorem.- 5.3. The w.h.e.- and h-Principles.- 5.4. The Gromov-Lees Theorem on Lagrangian Immersions.- 5.5. Elimination of Thom-Boardman Singularities.- 5.6. The Space of Functions with no A3 Singularities.- 6. Tangential Singularities and Vanishing Inflexions.- 6.1. The Calculus of Tangential Singularities.- 6.2. Vanishing Inflexions: The Case of Plane Curves.- 6.3. Inflexions that Vanish at a Morse Singular Point.- 6.4. Integration with Respect to the Euler Characteristic, and its Applications.- References.- Author Index.
