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 readiยญ 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 matheยญ 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 differยญ ent mathematical approaches, and must have experience with their interยญ 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 interrelaยญ 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 manyยญ facetted and always new presentations of the material by M. F
CONTENT
I. Operators with Index -- 1. Fredholm Operators -- 2. Algebraic Properties. Operators of Finite Rank -- 3. Analytic Methods. Compact Operators -- 4. The Fredholm Alternative -- 5. The Main Theorems -- 6. Families of Invertible Operators. Kuiperโ{128}{153}s Theorem -- 7. Families of Fredholm Operators. Index Bundles -- 8. Fourier Series and Integrals (Fundamental Principles) -- 9. Wiener-Hopf Operators -- II. Analysis on Manifolds -- 1. Partial Differential Equations -- 2. Differential Operators over Manifolds -- 3. Pseudo-Differential Operators -- 4. Sobolev Spaces (Crash Course) -- 5. Elliptic Operators over Closed Manifolds -- 6. Elliptic Boundary-Value Systems I (Differential Operators) -- 7. Elliptic Differential Operators of First Order with Boundary Conditions -- 8. Elliptic Boundary-Value Systems II (Survey) -- III. The Atiyah-Singer Index Formula -- 1. Introduction to Algebraic Topology -- 2. The Index Formula in the Euclidean Case -- 3. The Index Theorem for Closed Manifolds -- 4. Applications (Survey) -- IV. The Index Formula and Gauge-Theoretical Physics -- 1. Physical Motivation and Overview -- 2. Geometric Preliminaries -- 3. Gauge-Theoretic Instantons -- Appendix: What are Vector Bundles? -- Literature -- Index of Notation Parts I, II, III -- IV -- Index of Names/Authors
