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Singularities of Mappings: The Local Behaviour of Smooth and Complex Analytic Mappings 2020 ed. [Kõva köide]

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  • Formaat: Hardback, 567 pages, kõrgus x laius: 235x155 mm, kaal: 1300 g, XV, 567 p., 1 Hardback
  • Sari: Grundlehren der mathematischen Wissenschaften 357
  • Ilmumisaeg: 24-Jan-2020
  • Kirjastus: Springer Nature Switzerland AG
  • ISBN-10: 3030344398
  • ISBN-13: 9783030344399
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Singularities of Mappings: The Local Behaviour of Smooth and Complex Analytic Mappings 2020 ed.Suurem pilt
  • Formaat: Hardback, 567 pages, kõrgus x laius: 235x155 mm, kaal: 1300 g, XV, 567 p., 1 Hardback
  • Sari: Grundlehren der mathematischen Wissenschaften 357
  • Ilmumisaeg: 24-Jan-2020
  • Kirjastus: Springer Nature Switzerland AG
  • ISBN-10: 3030344398
  • ISBN-13: 9783030344399
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783030344399.html

The first monograph on singularities of mappings for many years, this book provides an introduction to the subject and an account of recent developments concerning the local structure of complex analytic mappings.

Part I of the book develops the now classical real C8 and complex analytic theories jointly. Standard topics such as stability, deformation theory and finite determinacy, are covered in this part. In Part II of the book, the authors focus on the complex case. The treatment is centred around the idea of the "nearby stable object" associated to an unstable map-germ, which includes in particular the images and discriminants of stable perturbations of unstable singularities. This part includes recent research results, bringing the reader up to date on the topic.

By focusing on singularities of mappings, rather than spaces, this book provides a necessary addition to the literature. Many examples and exercises, as well as appendices on background material, make it an invaluable guide for graduate students and a key reference for researchers. A number of graduate level courses on singularities of mappings could be based on the material it contains.

Arvustused

Exercises at the end of each section are intended to help the interested reader to study the material in depth. In general, this book is a nice supplement to the classical and modern monographs devoted to various sides of singularity theory. It comprises a lot of material scattered throughout numerous papers and presents it in a systematic and rigorous way. No doubt, it will become a common reference for various issues covered in this book. (Eugenii Shustin, Mathematical Reviews, January, 2023)

The book is written in a clear pedagogical style; it contains many examples, exercises, comments, remarks, nice pictures, very useful instructive and systematic references, computational algorithms with implementation in the computer algebra software systems Macaulay 2 and Mathematica, etc. Without a doubt, the book is understandable, interesting and useful for graduate students and can serve as a good starting point for those who are interested in various aspects of both pure and applied mathematics. (Aleksandr G. Aleksandrov, zbMATH 1448.58032, 2020)

1 Introduction
1(12)
1.1 Real or Complex?
2(1)
1.2 Structure of the Book
3(3)
1.3 The Nearby Stable Object
6(4)
1.4 Exercises and Open Questions
10(1)
1.5 Notation
10(3)
Part I Thorn-Mather Theory: Right-Left Equivalence, Stability, Versal Unfoldings, Finite Determinacy
2 Manifolds and Smooth Mappings
13(32)
2.1 Germs
13(2)
2.2 Manifolds and Their Tangent Spaces
15(9)
2.3 Inverse Mapping Theorem and Consequences
24(5)
2.4 Submanifolds
29(3)
2.5 Vector Fields and Flows
32(8)
2.6 Transversality
40(4)
2.7 Local Conical Structure
44(1)
3 Left-Right Equivalence and Stability
45(52)
3.1 Classification of Functions by Right Equivalence
46(12)
3.2 Left-Right Equivalence and Stability
58(16)
3.2.1 Right Equivalence and Left Equivalence
71(3)
3.3 First Calculations
74(7)
3.4 Multi-Germs
81(7)
3.4.1 Notation
81(7)
3.5 Infinitesimal Stability Implies Stability
88(4)
3.6 Stability of Multi-Germs
92(5)
4 Contact Equivalence
97(44)
4.1 The Contact Tangent Space
97(5)
4.2 Using T Ke f to Calculate T Ae f
102(3)
4.3 Construction of Stable Germs as Unfoldings
105(3)
4.4 Contact Equivalence
108(8)
4.5 Geometric Criterion for Finite Ae-Codimension
116(6)
4.5.1 Sheafification
117(5)
4.6 Transversality
122(6)
4.7 Thom-Boardman Singularities
128(13)
5 Versal Unfoldings
141(40)
5.1 Versality
142(14)
5.2 Global Stability of C∞ Mappings
156(6)
5.2.1 Stable Maps Are Not Always Dense
158(2)
5.2.2 Mather's Nice Dimensions
160(2)
5.3 Topological Stability
162(1)
5.4 Bifurcation Sets
163(13)
5.5 The Notion of Stable Perturbation of a Map-Germ
176(5)
6 Finite Determinacy
181(36)
6.1 Proof of the Finite Determinacy Theorem
184(7)
6.2 Estimates for the Determinacy Degree
191(9)
6.3 Determinacy and Unipotency
200(10)
6.3.1 Unipotent Affine Algebraic Groups
204(2)
6.3.2 Unipotent Groups of k-Jets of Diffeomorphisms
206(3)
6.3.3 When Is a Closed Affine Space of Germs Contained in a A-Orbit?
209(1)
6.3.4 Complexification and Determinacy Degrees
209(1)
6.3.5 Notes
209(1)
6.4 Complete Transversals
210(5)
6.5 Notes and Further Developments
215(2)
7 Classification of Stable Germs by Their Local Algebras
217(36)
7.1 Stable Germs Are Classified by Their Local Algebras
217(6)
7.2 Construction of Stable Germs as Unfoldings
223(4)
7.3 The Isosingular Locus
227(5)
7.3.1 Weighted Homogeneity and Local Quasihomogeneity
231(1)
7.4 Quasihomogeneity and the Nice Dimensions
232(21)
7.4.1 Multi-Germs
235(1)
7.4.2 The Case n ≥ p
236(2)
7.4.3 The Case n < p
238(15)
Part II Images and Discriminants: The Topology of Stable Perturbations
8 Stable Images and Discriminants
253(50)
8.1 Introduction
253(6)
8.1.1 Complex Not Real
258(1)
8.2 Review of the Milnor Fibre
259(2)
8.3 The Homotopy Type of the Discriminant of a Stable Perturbation: Discriminant and Image Milnor Numbers
261(8)
8.4 Finding TtAe f in the Geometry of f: Maps from n-Space ton + 1-Space
269(10)
8.4.1 The Conductor Ideal
275(4)
8.5 Finding T1,Ae f in the Geometry of f: Sections of Stable Discriminants and Images
279(10)
8.5.1 Critical Space and Discriminant
283(6)
8.6 Bifurcation Sets
289(3)
8.7 Calculating the Discriminant Milnor Number
292(6)
8.8 Image Milnor Number and Ae-Codimension
298(1)
8.9 Further Developments
299(4)
8.9.1 Almost Free Divisors
299(1)
8.9.2 Thorn Polynomial Techniques
300(1)
8.9.3 Does nA Constant Imply Topological Triviality?
300(1)
8.9.4 The Milnor-Tjurina Relation
300(1)
8.9.5 Augmentation and Concatenation: New Germs from Old
301(2)
9 Multiple Points
303(66)
9.1 Introduction
303(1)
9.2 Choosing the Right Definition
304(11)
9.2.1 Semi-Simplicial Spaces
311(1)
9.2.2 When Is Dkd(f) Reduced?
312(1)
9.2.3 Irritating Notation, Occasionally Necessary
312(3)
9.2.4 Equations or Procedures?
315(1)
9.3 Expected Dimension
315(5)
9.4 Equations for D2(f)
320(7)
9.5 Equations for Dk(f) When f Is a Corank 1 Germ
327(1)
9.5.1 Generalities on Functions of One Variable
327(6)
9.5.2 Application to Multiple Points
333(8)
9.6 Bifurcation Sets for Germs of Corank
341(4)
9.7 Disentangling a Singularity: The Geometry of a Stable Perturbation
345(6)
9.8 Blowing-Up Multiple Points
351(15)
9.8.1 Construction of an Ambient Space for Kk
352(3)
9.8.2 Construction of Kk(f) as Subspace of Bk(X)
355(11)
9.9 What Remains To Be Done
366(3)
10 Calculating the Homology of the Image
369(44)
10.1 The Alternating Chain Complex
370(10)
10.1.1 Motivation
373(7)
10.2 The Image Computing Spectral Sequence
380(9)
10.2.1 Towards the ICSS
384(1)
10.2.2 The Filtrations
385(1)
10.2.3 The Spectral Sequence of a Filtered Complex
385(1)
10.2.4 The Spectral Sequences Arising from the Two Filtrations on the Total Complex of the Double Complex
386(3)
10.3 Finite Simplicial Maps
389(9)
10.3.1 Triangulating Dk(f)
391(3)
10.3.2 (CAkt (D'(f)). Is a Resolution of Cn(Y)
394(4)
10.4 Finite Complex Maps Are Triangulable
398(1)
10.5 Other Proofs
399(1)
10.6 Cohomology
399(3)
10.7 Examples and Applications of the ICSS
402(10)
10.7.1 The Reidemeister Moves
403(1)
10.7.2 Reidemeister I
403(1)
10.7.3 Reidemeister II
404(1)
10.7.4 Reidemeister III
405(1)
10.7.5 Map-Germs of Multiplicity 2
406(3)
10.7.6 Codimension 1 Corank 1 Germs
409(1)
10.7.7 Generalised Mayer-Vietoris
410(1)
10.7.8 Relation Between A H* and H*
411(1)
10.7.9 Exercises for Sect. 10.7
411(1)
10.8 Open Questions
412(1)
11 Multiple Points in the Target: The Case of Parameterised Hypersurfaces
413(56)
11.1 Finding a Presentation
414(7)
11.1.1 Using Macaulay2 to Find a Presentation
418(3)
11.2 Fitting Ideals and Multiple Points in the Target
421(10)
11.2.1 Are the Fitting Ideal Spaces M*(f) Cohen-Macaulay?
428(3)
11.3 Double Points in the Target
431(5)
11.4 Ae-Codimension and Image Milnor Number of Map-Germs(Cn, S) → (Cn+1, 0)
436(6)
11.5 The Rank Condition
442(5)
11.6 Corank 1 Mappings: Cyclic Extensions
447(4)
11.7 Duality and Symmetric Presentations
451(11)
11.7.1 Gorenstein Rings and Symmetric Presentations
455(3)
11.7.2 Geometrical Interpretation of the Trace Homomorphism
458(4)
11.8 Triple Points in the Target
462(7)
A Jet Spaces and Jet Bundles
469(8)
B Stratifications
477(12)
B.1 Stratification of Sets
477(4)
B.2 Stratification of Mappings
481(2)
B.3 Semialgebraic Sets
483(6)
C Background in Commutative Algebra
489(28)
C.1 Spaces and Functions on Spaces
489(4)
C.2 Associated Primes
493(2)
C.3 Dimension, Depth and Cohen-Macaulay Modules
495(8)
C.3.1 Krull Dimension
495(1)
C.3.2 Slicing Dimension
496(2)
C.3.3 Hilbert-Samuel Dimension
498(1)
C.3.4 Weierstrass Dimension
498(1)
C.3.5 The Hauptidealsatz
498(2)
C.3.6 Depth and Cohen-Macaulay Modules
500(3)
C.4 Free Resolutions
503(5)
C.4.1 Cohen-Macaulay Modules and Freeness
504(1)
C.4.2 Examples of Cohen-Macaulay Spaces
505(3)
C.5 Pulling Back Algebraic Structures
508(5)
C.6 Samuel Multiplicity
513(4)
D Local Analytic Geometry
517(20)
D.1 The Preparation Theorem
517(4)
D.2 Local Properties of Analytic Sets and Finite Mappings
521(4)
D.3 Degree and Multiplicity
525(3)
D.4 Normalisation of Analytic Set-Germs
528(9)
D.4.1 Extension Theorems
531(1)
D.4.2 Normalisation
532(5)
E Sheaves
537(16)
E.1 Presheaves and Sheaves
537(4)
E.2 Coherence
541(4)
E.3 Conservation of Multiplicity
545(4)
E.3.1 Representatives
545(4)
E.4 Conservation of Multiplicity II
549(4)
References 553(10)
Index 563
David Mond received his PhD at the University of Liverpool and has held positions at the National University of Colombia, Bogotá, and at the University of Warwick, where he is a full professor. His main field of research is the theory of singularities of mappings, especially the geometry and topology of images and discriminants, and their relation to the deformation theory of unstable germs map-germs.

Juan J. Nuño-Ballesteros is Professor at the University of Valencia. His research is in the field of singularities of real and complex mappings and his main contributions to the subject include the topological classification of real analytic map-germs and also the Whitney equisingularity of complex analytic families of map-germs. He has also other important contributions in generic geometry (applications of Singularity Theory to the differential geometry of submanifolds in Euclidean spaces) and about global aspects of singularities of smooth mappings.