The Localization Problem in Index Theory of Elliptic Operators

The book deals with the localization approach to the index problem for elliptic operators. It covers both the general localization principle and its applications to specific problems, existing and new.

The book is concisely and clearly written and contains numerous figures helping the reader to visualize the material The book uses simple language to explain the main ideas and avoids unnecessary technicalities without sacrificing mathematical rigor The applications described in the book include a carefully selected variety of results, old as well as new, each of which is described in a self-contained way and can be read independently Includes supplementary material: sn.pub/extras

Autorentext
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.

Inhalt
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 GromovLawson Theorem.- 4.2 Bunke Theorem.- 4.3 Roe's Relative Index Construction.- 5 Applications to Boundary Value Problems.- 5.1 Preliminaries.- 5.2 AgranovichDynin 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.