Topology and analysis: the Atiyah-Singer index formula and gauge-theoretic physics
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
1985
|
| Schriftenreihe: | Universitext
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 451 S. graph. Darst. |
| ISBN: | 0387961127 3540961127 9780387961125 |
Internformat
MARC
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| 100 | 1 | |a Booss-Bavnbek, Bernhelm |d 1941- |e Verfasser |0 (DE-588)108280470 |4 aut | |
| 240 | 1 | 0 | |a Topologie und Analysis |
| 245 | 1 | 0 | |a Topology and analysis |b the Atiyah-Singer index formula and gauge-theoretic physics |c B. Booss ; D. D. Bleecker |
| 264 | 1 | |a New York [u.a.] |b Springer |c 1985 | |
| 300 | |a XVI, 451 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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| 700 | 1 | |a Bleecker, David |d 1948- |e Verfasser |0 (DE-588)1017094063 |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-4684-0627-6 |
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Datensatz im Suchindex
| _version_ | 1804116493362790400 |
|---|---|
| adam_text | Contents
PART I. OPERATORS WITH INDEX
1. Fredholm Operators 1
A. Hierarchy of Mathematical Objects 1
B. Concept of Fredholm Operator 3
2. Algebraic Properties. Operators of Finite Rank 5
A. The Snake Lemma 6
B. Operators of Finite Rank and Fredholm Integral Equations 10
3. Analytic Methods. Compact Operators 12
A. Analytic Methods 12
B. The Adjoint Operator 15
C. Compact Operators 17
D. The Classical Integral Operators 23
4. The Fredholm Alternative 26
A. The Riesz Lemma 26
B. Sturm Liouville Boundary Value Problem 27
5. The Main Theorems 34
A. The Calkin Algebra 34
B. Perturbation Theory 37
C. Homotopy Invariance of the Index 42
6. Families of Invertible Operators. Kuiper s Theorem 47
A. Homotopies of Operator Valued Functions 47
B. The Theorem of Kuiper 54
7. Families of Fredholm Operators. Index Bundles 60
A. The Topology of F 60
B. The Construction of Index Bundles 62
C. The Theorem of Atiyah Janich 71
D. Homotopy and Unitary Equivalence 75
8. Fourier Series and Integrals (Fundamental Principles) 79
A. Fourier Series 79
B. The Fourier Integral 82
C. Higher Dimensional Fourier Integrals 85
9. Wiener Hopf Operators 85
A. The Reservoir of Examples of Fredholm Operators 85
B. Origin and Fundamental Significance of Wiener Hopf Operators 87
C. The Characteristic Curve of a Wiener Hopf Operator 88
D. Wiener Hopf Operators and Harmonic Analysis 89
E. The Discrete Index Formula 92
F. The Case of Systems 95
G. The Continuous Analogue 97
xiii
xiv
PART II. ANALYSIS ON MANIFOLDS
1. Partial Differential Equations 103
A. Linear Partial Differential Equations 103
B. Elliptic Differential Equations 107
C. Where Do Elliptic Differential Operators Arise? 109
D. Boundary Value Conditions 112
E. Main Problems of Analysis and the Index Problem 114
F. Numerical Aspects 115
G. Elementary Examples 116
2. Differential Operators over Manifolds 126
A. Motivation 126
B. Differentiable Manifolds Foundations 126
C. Geometry of C Mappings 130
D. Integration on Manifolds 134
E. Differential Operators on Manifolds 136
F. Manifolds with Boundary 141
3. Pseudo Differential Operators 144
A. Motivation 144
B. Canonical Pseudo Differential Operators 148
C. Pseudo Differential Operators on Manifolds 152
D. Approximation Theory for Pseudo Differential Operators 168
4. Sobolev Spaces (Crash Course) 172
A. Motivation 172
B. Definition 173
C. The Main Theorems on Sobolev Spaces 177
D. Case Studies 178
5. Elliptic Operators over Closed Manifolds 182
A. Continuity of Pseudo Differential Operators 182
B. Elliptic Operators 184
6. Elliptic Boundary Value Systems I (Differential Operators) 189
A. Differential Equations with Constant Coefficients 189
B. Systems of Differential Equations with Constant
Coefficients 194
C. Variable Coefficients 196
7. Elliptic Differential Operators of First Order with
Boundary Conditions 199
A. The Topological Interpretation of Boundary Value Conditions
(Case Study) 199
B. Generalizations (Heuristic) 203
8. Elliptic Boundary Value Systems II (Survey) 208
A. The Poisson Principle 208
B. The Green Algebra 210
C. The Elliptic Case 213
XV
PART III. THE ATIYAH SINGER INDEX FORMULA
1. Introduction to Algebraic Topology 218
A. Winding Numbers 219
B. The Topology of the General Linear Group 224
C. The Ring of Vector Bundles 230
D. K Theory with Compact Support 236
E. Proof of the Periodicity Theorem of R. Bott 239
2. The Index Formula in the Euclidean Case 246
A. Index Formula and Bott Periodicity 246
B. The Difference Bundle of an Elliptic Operator 247
C. The Index Formula 252
3. The Index Theorem for Closed Manifolds 256
A. The Index Formula 256
B. Comparison of the Proofs: The Cobordism Proof 259
C. Comparison of the Proofs: The Imbedding Proof 262
D. Comparison of the Proofs: The Heat Equation Proof 262
4. Applications (Survey) 269
A. Cohomological Formulation of the Index Formula 272
B. The Case of Systems (Trivial Bundles) 274
C. Examples of Vanishing Index 275
D. Euler Number and Signature 277
E. Vector Fields on Manifolds 281
F. Abelian Integrals and Riemann Surfaces 284
G. The Theorem of Riemann Roch Hirzebruch 289
H. The Index of Elliptic Boundary Value Problems 293
J. Real Operators 300
K. The Lefschetz Fixed Point Formula 300
L. Analysis on Symmetric Spaces 303
M. Further Applications 304
PART IV. THE INDEX FORMULA AND GAUGE THEORETICAL PHYSICS
1. Physical Motivation and Overview 305
A. Classical Field Theory 306
B. Quantum Theory 313
2. Geometric Preliminaries 331
A. Principal G Bundles 332
B. Connections and Curvature 333
C. Equivariant Forms and Associated Bundles 334
D. Gauge Transformations 339
E. Curvature in Riemannian Geometry 342
F. Bochner Weitzenbock Formulas 348
G. Chern Classes as Curvature Forms 355
H. Holonomy 358
3. Gauge Theoretic Instantons 359
A. The Yang Mills Functional 360
B. Instantons on Euclidean 4 Space 363
C. Linearization of the Manifold of Moduli of Self Dual
Connections 373
xvi
D. Manifold Structure for Moduli of Self Dual Connections 380
E. Gauge Theoretic Topology in Dimension Four 390
Appendix: What are Vector Bundles? 402
Literature 417
Index of Notation
Parts I, II, III 428
Part IV 433
Index of Names/Authors 436
Subject Index 441
|
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| classification_tum | MAT 580f MAT 470f PHY 014f PHY 013f |
| ctrlnum | (OCoLC)471826279 (DE-599)BVBBV002042967 |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV002042967 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T15:39:23Z |
| institution | BVB |
| isbn | 0387961127 3540961127 9780387961125 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001336099 |
| oclc_num | 471826279 |
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| owner_facet | DE-91G DE-BY-TUM DE-384 DE-355 DE-BY-UBR DE-634 DE-83 DE-11 DE-188 DE-19 DE-BY-UBM DE-20 |
| physical | XVI, 451 S. graph. Darst. |
| publishDate | 1985 |
| publishDateSearch | 1985 |
| publishDateSort | 1985 |
| publisher | Springer |
| record_format | marc |
| series2 | Universitext |
| spelling | Booss-Bavnbek, Bernhelm 1941- Verfasser (DE-588)108280470 aut Topologie und Analysis Topology and analysis the Atiyah-Singer index formula and gauge-theoretic physics B. Booss ; D. D. Bleecker New York [u.a.] Springer 1985 XVI, 451 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Indextheorem (DE-588)4140055-0 gnd rswk-swf Indextheorem (DE-588)4140055-0 s DE-604 Bleecker, David 1948- Verfasser (DE-588)1017094063 aut Erscheint auch als Online-Ausgabe 978-1-4684-0627-6 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001336099&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Booss-Bavnbek, Bernhelm 1941- Bleecker, David 1948- Topology and analysis the Atiyah-Singer index formula and gauge-theoretic physics Indextheorem (DE-588)4140055-0 gnd |
| subject_GND | (DE-588)4140055-0 |
| title | Topology and analysis the Atiyah-Singer index formula and gauge-theoretic physics |
| title_alt | Topologie und Analysis |
| title_auth | Topology and analysis the Atiyah-Singer index formula and gauge-theoretic physics |
| title_exact_search | Topology and analysis the Atiyah-Singer index formula and gauge-theoretic physics |
| title_full | Topology and analysis the Atiyah-Singer index formula and gauge-theoretic physics B. Booss ; D. D. Bleecker |
| title_fullStr | Topology and analysis the Atiyah-Singer index formula and gauge-theoretic physics B. Booss ; D. D. Bleecker |
| title_full_unstemmed | Topology and analysis the Atiyah-Singer index formula and gauge-theoretic physics B. Booss ; D. D. Bleecker |
| title_short | Topology and analysis |
| title_sort | topology and analysis the atiyah singer index formula and gauge theoretic physics |
| title_sub | the Atiyah-Singer index formula and gauge-theoretic physics |
| topic | Indextheorem (DE-588)4140055-0 gnd |
| topic_facet | Indextheorem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001336099&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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