Topology and Analysis: The Atiyah-Singer Index Formula and Gauge-Theoretic Physics

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.