A panorama of pure mathematics: as seen by N. Bourbaki
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English French |
| Veröffentlicht: |
New York [u.a.]
Acad. Press
1982
|
| Schriftenreihe: | Pure and applied mathematics
97 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | X, 289 S. graph. Darst. |
| ISBN: | 0122155602 |
Internformat
MARC
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| 100 | 1 | |a Dieudonné, Jean Alexandre |d 1906-1992 |e Verfasser |0 (DE-588)119320738 |4 aut | |
| 240 | 1 | 0 | |a Panorama des mathématiques pures |
| 245 | 1 | 0 | |a A panorama of pure mathematics |b as seen by N. Bourbaki |c Jean Dieudonné |
| 264 | 1 | |a New York [u.a.] |b Acad. Press |c 1982 | |
| 300 | |a X, 289 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Pure and applied mathematics |v 97 | |
| 650 | 7 | |a Matematica |2 larpcal | |
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| 650 | 4 | |a Mathematics | |
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Datensatz im Suchindex
| _version_ | 1804114627623124992 |
|---|---|
| adam_text | Contents
Introduction 1
A I. Algebraic and differential topology
1. Techniques. Homotopy. Homotopy groups. Homotopy and cohomology.
Homology and cohomology. Cohomology and homology rings. Fibrations 7
2. Results. The different sorts of manifolds. The Poincare conjecture.
Cobordism. Immersions, embeddings, and knot theory. Fixed points;
spaces with group action 16
3. Connections with the natural sciences 21
4. The originators 21
References 23
A II. Differential manifolds. Differential geometry
1. The general theory. Singularities of differentiable mappings. Vector fields on
differential manifolds 25
2. G structures. Riemannian manifolds 28
3. The topology of differential manifolds 31
4. Infinite dimensional differential manifolds 31
5. Connections with the natural sciences 32
6. The originators 32
References 33
A III. Ordinary differential equations
1. The algebraic theory 35
2. Ordinary differential equations in the complex domain 35
3. The qualitative study of ordinary differential equations 36
4. The classification problem 38
5. Boundary value problems 39
6. Connections with the natural sciences 39
7. The originators 40
References 40
V
Vi CONTENTS
A IV. Ergodic theory
1. The ergodic theorem 43
2. Classification problems 44
3. Connections with the natural sciences 46
4. The originators 46
References 47
A V. Partial differential equations
1. The local study of differential systems 49
2. Completely integrable systems and foliations 51
3. Linear partial differential equations: general theory. Problems. Techniques.
Results 52
4. Equations with constant coefficients. Invariant operators on homogeneous
spaces 57
5. Boundary value problems for linear equations: I. General theory 58
6. Boundary value problems for linear equations: II. Spectral theory of elliptic
operators. Second order elliptic operators and potential theory 59
7. Boundary value problems for linear equations: III. Equations of evolution.
Strictly hyperbolic equations. Parabolic equations 62
8. Pseudodifferential operators on compact manifolds 66
9. Nonlinear partial differential equations 69
10. Connections with the natural sciences 69
11. The originators 69
References 71
A VI. Noncommutative harmonic analysis
1. Elementary cases: compact groups and abelian groups 74
2. The fundamental problems 75
3. Harmonic analysis on real reductive Lie groups 76
4. Harmonic analysis on reductive p adic groups 79
5. Harmonic analysis on nilpotent and solvable Lie groups 81
6. Representations of group extensions 82
7. Connections with the natural sciences S3
8. The originators 83
References 84
A VII. Automorphic forms and modular forms
1. The analytic aspect 87
2. The intervention of Lie groups 88
3. The intervention of adele groups 90
4. Applications to number theory, (a) Extensions of abelian class field theory.
(b) Elliptic curves and modular forms, (c) The Ramanujan Petersson
conjecture, (d) Congruences and modular forms 92
5. Automorphic forms, abelian varieties, and class fields 93
6. Relations with the arithmetic theory of quadratic forms 94
CONTENTS Vii
7. Connections with the natural sciences 94
8. The originators 95
References 95
A VIII. Analytic geometry
1. Functions of several complex variables and analytic spaces. Domains of
holomorphy and Stein spaces. Analytic subspaces and coherent sheaves.
Globalization problems. Continuation problems. Properties of morphisms and
automorphisms. Singularities of analytic spaces. Singularities of analytic
functions; residues 97
2. Compact analytic spaces; Kahler manifolds. Classification problems 106
3. Variations of complex structures and infinite dimensional manifolds 109
4. Real and p adic analytic spaces 110
5. Connections with the natural sciences 110
6. The originators 110
References 111
A IX. Algebraic geometry
1. The modern framework of algebraic geometry 113
2. The fundamental notions of the theory of schemes, (a) Local properties and
global properties, (b) Quasi coherent Modules and subschemes.
(c) Relativization and base change, (d) The various types of morphisms.
(e) Techniques of construction and representable functors 116
3. The study of singularities 121
4. The transcendental theory of algebraic varieties. Monodromy. Topology of
subvarieties. Algebraic cycles. Divisors and abelian varieties. Divisors and
vector bundles. Ample divisors and projective embeddings 122
5. Cohomology of schemes. The various cohomologies. Intersection multiplicities
and homology. The fundamental group and monodromy 130
6. Classification problems. The classification of surfaces. Moduli problems 134
7. Algebraic groups. Abelian varieties. Linear algebraic groups. Invariant theory 136
8. Formal schemes and formal groups 141
9. Connections with the natural sciences 143
10. The originators 143
References 144
A X. Theory of numbers
1. The modern viewpoint in number theory. Local fields, adeles and ideles. Zeta
functions and L functions. Local fields and global fields 147
2. Class field theory. Particular class fields. Galois extensions of local and global
fields 151
3. Diophantine approximations and transcendental numbers 154
4. Diophantine geometry. Diophantine geometry over a finite field. Abelian
varieties defined over local and global fields. Diophantine geometry over a
ring of algebraic integers 157
5. Arithmetic linear groups. The arithmetic theory of quadratic forms 160
Viii CONTENTS
6. Connections with the natural sciences 163
7. The originators 163
References 164
B I. Homotogical algebra
1. Derived functors in abelian categories. Examples of derived functors 167
2. Cohomology of groups. Variants. Galois cohomology 172
3. Cohomology of associative algebras 175
4. Cohomology of Lie algebras 176
5. Simplkial structures 177
6. K theory 178
7. Connections with the natural sciences ISO
8. The originators 180
References 181
B II. Lie groups
1. Structure theorems 185
2. Lie groups and transformation groups 186
3. Topology of Lie groups and homogeneous spaces 188
4. Connections with the natural sciences 18X
5. The originators l88
References lg9
B III. Abstract groups
1. Generators and relations 191
2. Chevalley groups and Tits systems 191
3. Linear representations and characters. The classical theory. The modular
theory. Characters of particular groups 193
4. The search for finite simple groups 195
5. Connections with the natural sciences 196
6. The originators 196
References 197
B IV. Commutative harmonic analysis
1. Convergence problems 202
2. Normed algebras in harmonic analysis. Homomorphisms and idempotent
measures. Sets of uniqueness and pseudofunctions. The algebras A(E) and
harmonic synthesis. Functions acting on algebras 202
3. Symmetric perfect sets in harmonic analysis: connections with number theory 204
4. Almost periodic functions and mean periodic functions 205
5. Applications of commutative harmonic analysis 205
6. Connections with the natural sciences 206
7. The originators 206
References 207
CONTENTS IX
B V. Von Neumann algebras
1. Tomita theory and the Connes invariants 211
2. Applications to C* algebras 212
3. Connections with the natural sciences 213
4. The originators 213
References 213
B VI. Mathematical logic
1. Noncontradiction and undecidability 216
2. Uniform effective procedures and recursive relations 218
3. The technique of ultraproducts 219
4. Connections with the natural sciences 220
5. The originators 220
References 220
B VII. Probability theory
1. Fluctuations in sequences of independent random variables 224
2. Inequalities for martingales 224
3. Trajectories of processes 225
4. Generalized processes 226
5. Random variables with values in locally compact groups 226
6. Connections with the natural sciences 227
7. The originators 227
References 228
C I. Categories and sheaves
1. Categories and functors. Opposite categories; contravariant functors.
Functorial morphisms 232
2. Representable functors. Examples: final objects, products, kernels, inverse
limits. Dual notions. Adjoint functors. Algebraic structures on categories 235
3. Abelian categories 240
4. Sheaves and ringed spaces. Direct and inverse images of sheaves. Morphisms of
ringed spaces 241
5. Sites and topoi 244
6. Connections with the natural sciences 246
7. The originators 246
References 246
C II. Commutative algebra
1. The principal notions. Localization and globalization. Finiteness conditions.
Linear algebra over rings. Graduations and filiations. Topologies and
completions. Dimension. Integral closure. Excellent rings. Henselian rings.
Valuations and absolute values. Structure of complete Noetherian local rings 249
X CONTENTS
2. Problems of field theory. Quasi algebraically closed fields. Subextensions of a
pure transcendental extension. Hilbert s 14th problem 257
3. Connections with the natural sciences 259
4. The originators 259
References 259
C III. Spectral theory of operators
1. Riesz Fredhoun theory. Refinements and generalizations 263
2. Banach algebras 265
3. Hilbert von Neumann spectral theory 269
4. Connections with the natural sciences 272
5. The originators 272
References 272
Bibliography 273
Index 283
|
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| author | Dieudonné, Jean Alexandre 1906-1992 |
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| discipline | Mathematik |
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| indexdate | 2024-07-09T15:09:44Z |
| institution | BVB |
| isbn | 0122155602 |
| language | English French |
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| spelling | Dieudonné, Jean Alexandre 1906-1992 Verfasser (DE-588)119320738 aut Panorama des mathématiques pures A panorama of pure mathematics as seen by N. Bourbaki Jean Dieudonné New York [u.a.] Acad. Press 1982 X, 289 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 97 Matematica larpcal Mathématiques Wiskunde gtt Mathematik Mathematics Literaturrecherche (DE-588)4036029-5 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Bibliografie (DE-588)4006432-3 gnd rswk-swf Mathematik (DE-588)4037944-9 s Bibliografie (DE-588)4006432-3 s DE-604 Literaturrecherche (DE-588)4036029-5 s Pure and applied mathematics 97 (DE-604)BV010177228 97 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000095817&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dieudonné, Jean Alexandre 1906-1992 A panorama of pure mathematics as seen by N. Bourbaki Pure and applied mathematics Matematica larpcal Mathématiques Wiskunde gtt Mathematik Mathematics Literaturrecherche (DE-588)4036029-5 gnd Mathematik (DE-588)4037944-9 gnd Bibliografie (DE-588)4006432-3 gnd |
| subject_GND | (DE-588)4036029-5 (DE-588)4037944-9 (DE-588)4006432-3 |
| title | A panorama of pure mathematics as seen by N. Bourbaki |
| title_alt | Panorama des mathématiques pures |
| title_auth | A panorama of pure mathematics as seen by N. Bourbaki |
| title_exact_search | A panorama of pure mathematics as seen by N. Bourbaki |
| title_full | A panorama of pure mathematics as seen by N. Bourbaki Jean Dieudonné |
| title_fullStr | A panorama of pure mathematics as seen by N. Bourbaki Jean Dieudonné |
| title_full_unstemmed | A panorama of pure mathematics as seen by N. Bourbaki Jean Dieudonné |
| title_short | A panorama of pure mathematics |
| title_sort | a panorama of pure mathematics as seen by n bourbaki |
| title_sub | as seen by N. Bourbaki |
| topic | Matematica larpcal Mathématiques Wiskunde gtt Mathematik Mathematics Literaturrecherche (DE-588)4036029-5 gnd Mathematik (DE-588)4037944-9 gnd Bibliografie (DE-588)4006432-3 gnd |
| topic_facet | Matematica Mathématiques Wiskunde Mathematik Mathematics Literaturrecherche Bibliografie |
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