دانلود کتاب Introduction to Lipschitz Geometry of Singularities: Lecture Notes of the International School on Singularity Theory and Lipschitz Geometry, Cuernavaca, June 2018 (Lecture Notes in Mathematics)

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Preface
Contents
Contributors
1 Geometric Viewpoint of Milnor\'s Fibration Theorem
1.1 Introduction
1.2 Preliminaries
1.2.1 Transversality
1.2.2 Vector Fields and Integral Curves
1.2.3 Fibre Bundles
1.2.4 Complex and Real Gradients
1.2.4.1 Level Hypersurfaces of theta
1.2.4.2 Level Hypersurfaces of log|f|
1.3 Conical Structure
1.3.1 Whitney Stratifications
1.4 Fibration Theorems
1.4.1 Milnor Fibration
1.4.2 Monodromy
1.4.3 Open books
1.4.4 Thom af-condition
1.4.5 Milnor-Lê Fibration
1.4.6 Equivalence of the Fibrations
1.5 Resolution and Monodromy
1.5.1 Plumbing
1.5.2 Resolution and Blowup
1.5.3 Description of the Milnor Fibre in the Resolution
References
2 A Quick Trip into Local Singularities of Complex Curves and Surfaces
2.1 Introduction
2.2 General Settings
2.2.1 Multiplicity and Tangent Cone
2.2.2 Blowups
2.2.3 Normality and Normalization
2.2.4 Resolution of Singularities
2.3 Complex Curve Singularities
2.3.1 Normality and Normalization
2.3.2 Tangent Cone and Blowups
2.3.3 Resolution and Dual Graph
2.3.4 Newton-Puiseux Parametrization, Characteristic Exponents and Topological Type
2.3.5 Some Invariants: δ-Number and Milnor Number
2.4 Complex Surface Singularities
2.4.1 Normality and Normalization
2.4.2 Blowups and Resolution
2.4.3 Nash Modification, Exceptional Tangents and Polar Curves
2.4.4 Jung\'s Method
References
3 3-Manifolds and Links of Singularities
3.1 Introduction
3.2 Basics of 3-Manifold Topology
3.2.1 Basics
3.3 JSJ Decomposition
3.4 Seifert Fibered Manifolds
3.4.1 Seifert Manifolds with Boundary
3.5 Plumbing, Plumbing Graphs, Dual Resolution Graphs
3.5.1 Plumbing
3.5.2 Constructing S1-Bundles over Surfaces
3.5.3 Seifert Manifolds via Plumbing
3.5.4 General Plumbing
3.6 Resolution and Plumbing Graphs
3.7 Relationship with Thurston Geometries
3.7.1 The Thurston Geometries
3.8 The Panorama of Classical Singularities
3.8.1 ADE
3.8.2 Hirzebruch-Jung Singularities
3.8.3 Quasihomogeneous Surface Singularities
3.8.4 Hirzebruch Cusp Singularities
3.8.5 Further Comments
References
4 Stratifications, Equisingularity and Triangulation
4.1 Stratifications
4.1.1 Whitney\'s Conditions (a) and (b)
4.1.2 The Kuo-Verdier Condition (w)
4.1.3 Mostowski\'s Lipschitz Stratifications
4.1.4 Applications of Whitney (a)-regularity
4.2 Equisingularity
4.2.1 Topological Equisingularity
4.2.2 Some Complex Equisingularity and Real Analogues
4.2.2.1 Zariski\'s Problem
4.3 Triangulation of Stratified Sets and Maps
4.3.1 Triangulation of Sets
4.3.2 Thom Maps and the (af) Condition
4.3.2.1 (c)-regularity
References
5 Basics on Lipschitz Geometry
5.1 Introduction
5.2 Semialgebraic Sets and Mappings
5.2.1 Definitions and Basic Properties
5.2.2 Semialgebraic Arcs and Curve Selection Lemma
5.2.3 The Conic Structure Theorem
5.3 Basic Concepts on Lipschitz Geometry
5.3.1 Inner and Outer Metrics: Lipschitz Normal Embeddings
5.3.2 Existence of Extension of Lipschitz Mappings
5.4 Lipschitz Geometry of Real Curves and Surfaces
5.4.1 Lipschitz Geometry of Semialgebraic Curves
5.4.2 Lipschitz Geometry of Real Surfaces
5.4.2.1 The General Case
5.5 Bi-Lipschitz Invariance of the Tangent Cone
5.5.1 Properties of the Tangent Cone
5.5.2 Applications
5.6 Lipschitz Theory of Singularities
5.6.1 Basic Definitions and Results
5.6.2 Finiteness of Bi-Lipschitz K-Classification
5.6.2.1 Proof of Theorem 5.6.7
5.6.3 Invariants of Bi-Lipschitz R-Equivalence
5.6.4 Henry-Parusiński\'s Example
5.7 o-Minimal Structures (by Nhan Nguyen)
References
6 Surface Singularities in R4: First Steps Towards Lipschitz Knot Theory
6.1 Introduction
6.2 Examples in R3 and R4 Based on Sampaio\'s Theorem
6.3 Bridge Construction
References
7 An Introduction to Lipschitz Geometry of Complex Singularities
7.1 Introduction
7.2 Preliminaries
7.2.1 What is Lipschitz Geometry of Singular Spaces?
7.2.2 Independence of the Embedding and Motivations
7.3 The Lipschitz Geometry of a Complex Curve Singularity
7.3.1 Complex Curves Have Trivial Inner Lipschitz Geometry
7.3.2 The Outer Lipschitz Geometry of a Complex Curve
7.3.3 The Bubble Trick with Jumps
7.4 The Thick-Thin Decomposition of a Surface Singularity
7.4.1 Fast Loops as Obstructions to Metric Conicalness
7.4.2 Thick-Thin Decomposition
7.4.3 The Thick-Thin Decomposition in a Resolution
7.4.4 Generic Projection and Inner Metric: A Key Lemma
7.4.4.1 Polar Curves and Generic Projections
7.4.4.2 Nash Modification
7.4.4.3 Application 1
7.4.4.4 Application 2
7.4.5 Fast Loops in the Thin Pieces
7.5 Geometric Decompositions of a Surface Singularity
7.5.1 The Standard Pieces
7.5.1.1 The B-Pieces
7.5.1.2 The A-Pieces
7.5.1.3 Conical Pieces (or B(1)-Pieces)
7.5.2 Geometric Decompositions of C2
7.5.3 The Polar Wedge Lemma
7.5.4 The Geometric Decomposition and the Complete Lipschitz Classification for the Inner Metric
7.5.5 The Outer Lipschitz Decomposition
7.6 Appendix: The Resolution of the E8 Surface Singularity
7.6.1 Hirzebruch–Jung Algorithm
7.6.2 Laufer\'s Method
References
8 The biLipschitz Geometry of Complex Curves: An Algebraic Approach
8.1 Introduction
8.1.1 What Is a Germ of Complex Analytic Curve?
8.1.2 Structuring a Parametrization
8.2 Normalization
8.3 Fitting Ideals: A Good Structure for the Image of a Finite Map
8.3.1 Equations Versus Parametrizations
8.3.2 Deformations of Equations vs. Deformations of Parametrizations
8.4 General Projections
8.4.1 The Case of Dimension 1
8.5 Main Result
8.5.1 Integral Closure of Ideals
8.5.2 Lipschitz Saturation
8.5.3 The Case of Dimension 1
8.5.4 Application to Local Polar Curves
References
9 Ultrametrics and Surface Singularities
9.1 Introduction
9.2 Multiplicity and Intersection Numbers for Plane Curve Singularities
9.3 The Statement of Płoski\'s Theorem
9.4 Ultrametrics and Rooted Trees
9.5 A Proof of Płoski\'s Theorem Using Eggers-Wall Trees
9.6 An Ultrametric Characterization of Arborescent Singularities
9.7 The Brick-Vertex Tree of a Connected Graph
9.8 Our Strongest Generalization of Płoski\'s Theorem
9.9 Mumford\'s Intersection Theory
9.10 A Reformulation of the Ultrametric Inequality
9.11 A Theorem of Graph Theory
References
10 Lipschitz Fractions of a Complex Analytic Algebra and Zariski Saturation
10.1 Introduction
10.2 Preliminaries
10.2.1 Conventions
10.2.2 Universal Property of the Normalisation
10.2.3 Universal Property of the Blowing-up (see Hir64-a)
10.2.4 Universal Property of the Normalized Blowup
10.2.5 Normalized Blowup and Integral Closure of an Ideal
10.2.6 Majoration Theorems
10.3 Algebraic Characterization of Lipschitz Fractions
10.4 Geometric Interpretation of the Exceptional Divisor DX: Pairs of Infinitely Near Points on X
10.5 Lipschitz Fractions Relative to a Parametrization
10.6 The Particular Case of Plane Curves
10.7 Lipschitz Saturation and Zariski Saturation
10.8 Equisaturation and Lipschitz Equisingularity
Relative Equisaturation
Speculation on Equisingularity
Appendix: Stratification, Whitney\'s (a)-property and Transversality
References
List of Participants to the International School on Singularity Theory and Lipschitz Geometry
Index