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Topology and analysis : the Atiyah-Singer index formula and gauge-theoretic physics

Author: Bernhelm Booss; David D Bleecker; Adolf Mader
Publisher: New York ; Berlin ; Heidelberg : Springer, cop. 1985.
Series: Universitext
Edition/Format:   Print book : EnglishView all editions and formats
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With intensified use of mathematical ideas, the methods and techniques of the various sciences and those for the solution of practical problems demand of the mathematician not only greater readi-  Read more...

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Document Type: Book
All Authors / Contributors: Bernhelm Booss; David D Bleecker; Adolf Mader
ISBN: 3540961127 9783540961123 0387961127 9780387961125
OCLC Number: 467937301
Notes: Bibliogr. p. 417-427. Index.
Description: XVI-451 p. : graph. ; 24 cm.
Contents: I. Operators with Index.- 1. Fredholm Operators.- A. Hierarchy of Mathematical Objects.- B. Concept of Fredholm Operator.- 2. Algebraic Properties. Operators of Finite Rank.- A. The Snake Lemma.- B. Operators of Finite Rank and Fredholm Integral Equations.- 3. Analytic Methods. Compact Operators.- A. Analytic Methods.- B. The Adjoint Operator.- C. Compact Operators.- D. The Classical Integral Operators.- 4. The Fredholm Alternative.- A. The Riesz Lemma.- B. Sturm-Liouville Boundary-Value Problem.- 5. The Main Theorems.- A. The Calkin Algebra.- B. Perturbation Theory.- C. Homotopy-Invariance of the Index.- 6. Families of Invertible Operators. Kuiper's Theorem.- A. Homotopies of Operator-Valued Functions.- B. The Theorem of Kuiper.- 7. Families of Fredholm Operators. Index Bundles.- A. The Topology of F.- B. The Construction of Index Bundles.- C. The Theorem of Atiyah-Janich.- D. Homotopy and Unitary Equivalence.- 8. Fourier Series and Integrals (Fundamental Principles).- A. Fourier Series.- B. The Fourier Integral.- C. Higher Dimensional Fourier Integrals.- 9. Wiener-Hopf Operators.- A. The Reservoir of Examples of Fredholm Operators.- B. Origin and Fundamental Significance of Wiener-Hopf Operators.- C. The Characteristic Curve of a Wiener-Hopf Operator.- D. Wiener-Hopf Operators and Harmonic Analysis.- E. The Discrete Index Formula.- F. The Case of Systems.- G. The Continuous Analogue.- II. Analysis on Manifolds.- 1. Partial Differential Equations.- A. Linear Partial Differential Equations.- B. Elliptic Differential Equations.- C. Where Do Elliptic Differential Operators Arise?.- D. Boundary-Value Conditions.- E. Main Problems of Analysis and the Index Problem.- F. Numerical Aspects.- G. Elementary Examples.- 2. Differential Operators over Manifolds.- A. Motivation.- B. Differentiable Manifolds - Foundations.- C. Geometry of C? Mappings.- D. Integration on Manifolds.- E. Differential Operators on Manifolds.- F. Manifolds with Boundary.- 3. Pseudo-Differential Operators.- A. Motivation.- B. "Canonical" Pseudo-Differential Operators.- C. Pseudo-Differential Operators on Manifolds.- D. Approximation Theory for Pseudo-Differential Operators.- 4. Sobolev Spaces (Crash Course).- A. Motivation.- B. Definition.- C. The Main Theorems on Sobolev Spaces.- D. Case Studies.- 5. Elliptic Operators over Closed Manifolds.- A. Continuity of Pseudo-Differential Operators.- B. Elliptic Operators.- 6. Elliptic Boundary-Value Systems I (Differential Operators).- A. Differential Equations with Constant Coefficients.- B. Systems of Differential Equations with Constant Coefficients.- C. Variable Coefficients.- 7. Elliptic Differential Operators of First Order with Boundary Conditions.- A. The Topological Interpretation of Boundary-Value Conditions (Case Study).- B. Generalizations (Heuristic).- 8. Elliptic Boundary-Value Systems II (Survey).- A. The Poisson Principle.- B. The Green Algebra.- C. The Elliptic Case.- III. The Atiyah-Singer Index Formula.- 1. Introduction to Algebraic Topology.- A. Winding Numbers.- B. The Topology of the General Linear Group.- C. The Ring of Vector Bundles.- D. K-Theory with Compact Support.- E. Proof of the Periodicity Theorem of R. Bott.- 2. The Index Formula in the Euclidean Case.- A. Index Formula and Bott Periodicity.- B. The Difference Bundle of an Elliptic Operator.- C. The Index Formula.- 3. The Index Theorem for Closed Manifolds.- A. The Index Formula.- B. Comparison of the Proofs: The Cobordism Proof.- C. Comparison of the Proofs: The Imbedding Proof.- D. Comparison of the Proofs: The Heat Equation Proof.- 4. Applications (Survey).- A. Cohomological Formulation of the Index Formula.- B. The Case of Systems (Trivial Bundles).- C. Examples of Vanishing Index.- D. Euler Number and Signature.- E. Vector Fields on Manifolds.- F. Abelian Integrals and Riemann Surfaces.- G. The Theorem of Riemann-Roch-Hirzebruch.- H. The Index of Elliptic Boundary-Value Problems.- J. Real Operators.- K. The Lefsehetz Fixed-Point Formula.- L. Analysis on Symmetric Spaces.- M. Further Applications.- IV. The Index Formula and Gauge-Theoretical Physics.- 1. Physical Motivation and Overview.- A. Classical Field Theory.- B. Quantum Theory.- 2. Geometric Preliminaries.- A. Principal G-Bundles.- B. Connections and Curvature.- C. Equivariant Forms and Associated Bundles.- D. Gauge Transformations.- E. Curvature in Riemannian Geometry.- F. Bochner-Weitzenbock Formulas.- G. Chern Classes as Curvature Forms.- H. Holonomy.- 3. Gauge-Theoretic Instantons.- A. The Yang-Mills Functional.- B. Instantons on Euclidean 4-Space.- C. Linearization of the "Manifold" of Moduli of Self-Dual Connections.- D. Manifold Structure for Moduli of Self-Dual Connections.- E. Gauge-Theoretic Topology in Dimension Four.- Appendix: What are Vector Bundles?.- Literature.- Index of Notation Parts I, II, III.- IV.- Index of Names/Authors.
Series Title: Universitext
Responsibility: B. Booss, D.D. Bleecker ; transl. by D.D. Bleecker and A. Mader.
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