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The book deals with the localization approach to the index problem for elliptic operators. Localization ideas have been widely used for solving various specific index problems for a long time, but the fact that there is actually a fundamental localization principle underlying all these solutions has mostly passed unnoticed. The ignorance of this general principle has often necessitated using various artificial tricks and hindered the solution of new important problems in index theory. So far, the localization principle has been only scarcely covered in journal papers and not covered at all in monographs. The suggested book is intended to fill the gap. So far, it is the first and only monograph dealing with the topic. Both the general localization principle and its applications to specific problems, existing and new, are covered. The book will be of interest to working mathematicians as well as graduate and postgraduate university students specializing in differential equations and related topics.
List of contents
Preface.- Introduction.- 0.1 Basics of Elliptic Theory.- 0.2 Surgery and the Superposition Principle.- 0.3 Examples and Applications.- 0.4 Bibliographical Remarks.- Part I: Superposition Principle.- 1 Superposition Principle for the Relative Index.- 1.1 Collar Spaces.- 1.2 Proper Operators and Fredholm Operators.- 1.3 Superposition Principle.- 2 Superposition Principle for K-Homology.- 2.1 Preliminaries.- 2.2 Fredholm Modules and K-Homology.- 2.3 Superposition Principle.- 2.4 Fredholm Modules and Elliptic Operators.- 3 Superposition Principle for KK-Theory.- 3.1 Preliminaries.- 3.2 Hilbert Modules, Kasparov Modules, and KK.- 3.3 Superposition Principle.- Part II: Examples.- 4 Elliptic Operators on Noncompact Manifolds.- 4.1 Gromov-Lawson Theorem.- 4.2 Bunke Theorem.- 4.3 Roe's Relative Index Construction.- 5 Applications to Boundary Value Problems.- 5.1 Preliminaries.- 5.2 Agranovich-Dynin Theorem.- 5.3 Agranovich Theorem.- 5.4 Bojarski Theorem and Its Generalizations.- 5.5 Boundary Value Problems with Symmetric Conormal Symbol.- 6 Spectral Flow for Families of Dirac Type Operators.- 6.1 Statement of the Problem.- 6.2 Simple Example.- 6.3 Formula for the Spectral Flow.- 6.4 Computation of the Spectral Flow for a Graphene Sheet.- Bibliography.
About the author
Bert-Wolfgang Schulze ist emeritierter Professor am Institut für Mathematik an der Universität Potsdam, Deutschland. Vor der politischen Wende war er Professor am Karl-Weierstrass-Institut in Berlin, 1984 Euler-Medaille der Akademie der Wisenschaften in Berlin. 1992-96 war er Leiter der Max-Planck-Arbeitsgruppe 'Partielle Differentialgleichungen und Komplexe Analysis' in Potsdam. Nach anfänglichem Studium in Geophysik erhielt er sein Universitätsdiplom in Mathematik in Leipzig 1968. Die Promotion zum Dr. rer.nat. 1970 und die Habilitation in Mathematik 1974 erfolgten an der Universität Rostock. Seine wissenschaftlichen Aktivitäten umfassen Potentialtheorie, Randwert-Probleme, pseudo-differentielle Algebren und Index-Theorie auf berandeten Mannigfaltgikeiten und Räumen mit Singularitäten, darunterTransmissions- und Riss Probleme, Asymptotik von Lösungen, Randwert-Theorie mit globalen Projektionsbedingungen.
Summary
The book deals with the localization approach to the index problem for elliptic operators. Localization ideas have been widely used for solving various specific index problems for a long time, but the fact that there is actually a fundamental localization principle underlying all these solutions has mostly passed unnoticed. The ignorance of this general principle has often necessitated using various artificial tricks and hindered the solution of new important problems in index theory. So far, the localization principle has been only scarcely covered in journal papers and not covered at all in monographs. The suggested book is intended to fill the gap. So far, it is the first and only monograph dealing with the topic. Both the general localization principle and its applications to specific problems, existing and new, are covered. The book will be of interest to working mathematicians as well as graduate and postgraduate university students specializing in differential equations and related topics.
