The Motivation. 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 ness for extra-mathematical applications but also more comprehensive orientations within mathematics. In applications, it is frequently less important to draw the most far-reaching conclusions from a single mathe­ matical idea than to cover a subject or problem area tentatively by a proper "variety" of mathematical theories. To do this the mathematician must be familiar with the shared as weIl as specific features of differ­ ent mathematical approaches, and must have experience with their inter­ connections. The Atiyah-Singer Index Formula, "one of the deepest and hardest results in mathematics", "probably has wider ramifications in topology and analysis than any other single result" (F. Hirzebruch) and offers perhaps a particularly fitting example for such an introduction to "Mathematics": In spi te of i ts difficulty and immensely rich interrela­ tions, the realm of the Index Formula can be delimited, and thus its ideas and methods can be made accessible to students in their middle * semesters. In fact, the Atiyah-Singer Index Formula has become progressively "easier" and "more transparent" over the years. The discovery of deeper and more comprehensive applications (see Chapter 111. 4) brought with it, not only a vigorous exploration of its methods particularly in the many­ facetted and always new presentations of the material by M. F.

Inhalt

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-Jänich.- 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-Weitzenböck 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.