Elliptic Boundary Problems for Dirac Operators

Elliptic boundary problems have enjoyed interest recently, espe­ cially among C* -algebraists and mathematical physicists who want to understand single aspects of the theory, such as the behaviour of Dirac operators and their solution spaces in the case of a non-trivial boundary. However, the theory of elliptic boundary problems by far has not achieved the same status as the theory of elliptic operators on closed (compact, without boundary) manifolds. The latter is nowadays rec­ ognized by many as a mathematical work of art and a very useful technical tool with applications to a multitude of mathematical con­ texts. Therefore, the theory of elliptic operators on closed manifolds is well-known not only to a small group of specialists in partial dif­ ferential equations, but also to a broad range of researchers who have specialized in other mathematical topics. Why is the theory of elliptic boundary problems, compared to that on closed manifolds, still lagging behind in popularity? Admittedly, from an analytical point of view, it is a jigsaw puzzle which has more pieces than does the elliptic theory on closed manifolds. But that is not the only reason.

I. Clifford Algebras and Dirac Operators.- 1. Clifford Algebras and Clifford Modules.- 2. Clifford Bundles and Compatible Connections.- 3. Dirac Operators.- 4. Dirac Laplacian and Connection Laplacian.- 5. Euclidean Examples.- 6. The Classical Dirac (Atiyah-Singer) Operators on Spin Manifolds.- 7. Dirac Operators and Chirality.- 8. Unique Continuation Property for Dirac Operators.- 9. Invertible Doubles.- 10. Glueing Constructions. Relative Index Theorem.- II. Analytical and Topological Tools.- 11. Sobolev Spaces on Manifolds with Boundary.- 12. Calderón Projector for Dirac Operators.- 13. Existence of Traces of Null Space Elements.- 14. Spectral Projections of Dirac Operators.- 15. Pseudo-Differential Grassmannians.- 16. The Homotopy Groups of the Space of Self-Adjoint Fredholm Operators.- 17. The Spectral Flow of Families of Self-Adjoint Operators.- III. Applications.- 18. Elliptic Boundary Problems and Pseudo-Differential Projections.- 19. Regularity of Solutions of Elliptic Boundary Problems.- 20. Fredholm Property of the Operator AR.- 21. Exchanges on the Boundary: Agranovi?-Dynin Type Formulas and the Cobordism Theorem for Dirac Operators.- 22. The Index Theorem for Atiyah-Patodi-Singer Problems.- 23. Some Remarks on the Index of Generalized Atiyah-Patodi-Singer Problems.- 24. Bojarski’s Theorem. General Linear Conjugation Problems.- 25. Cutting and Pasting of Elliptic Operators.- 26. Dirac Operators on the Two-Sphere.