Elliptic theory on singular manifolds. (English) Zbl 1084.58007

Nazaikinskii, V. E.; Savin, A. Yu.; Schulze, Bert-Wolfgang; Sternin, B. Yu.

Differential and Integral Equations and Their Applications 7. Boca Raton, FL: Chapman & Hall/CRC (ISBN 1-58488-520-3/hbk). xiv, 356 p. (2006).

The book under review is an excellent, well-organized, and well-thought-out book full of outstanding theorems, beautiful pictures, and clear explanations. The presentation begins from an elementary level and continues to more recent and sophisticated results. In particular, the book is suitable for advanced undergraduate students and certainly for beginning graduate students. The book can also be used for a course on elliptic theory for singular manifolds that can equip students with methods to solve research problems. This manuscript is about the analysis and topology of elliptic operators on manifolds with singularities (for example, conical or edge singularities), which is a subject of great research interest in a variety of mathematical and physical areas. The study of operators on manifolds with singularities is more difficult than the smooth case as it requires, in addition to a usual techniques from the smooth case, new methods and ideas to handle the singularities. Unfortunately, many of the recent ideas and techniques, many of which are due to the authors, are found scattered throughout the literature and are written to a research audience. This book combines many of the ideas into one source, presents it in a systematic way, and provides all the necessary background to understand the techniques and use them for further research. The presentation treats both analytical and topological aspects of elliptic operators on manifolds with singularities and is divided as follows:
(1) The first part of the book, Chapters 1 and 2, reviews the geometry of manifolds with singularities and the operators on such manifolds.
(2) The second part of the book, Chapters 3 and 4, is on analytical tools. Here, pseudodifferential operators on smooth manifolds are reviewed and such operators are introduced on manifolds with singularities. Also the locality principle in index theory on smooth manifolds and on manifolds with singularities is studied.
(3) The next focal point, Chapters 5 and 6, is topological problems. In these Chapters the “stratified” index problem on manifolds with singularities is discussed. Here some K-theory is assumed from the reader.
(4) In the last part of the book, Chapters 7–11, applications and related topics are considered. These applications include Fourier integral operators, relative elliptic theory, index theory on manifolds with cylindrical ends, homotopy classifications of elliptic operators, and theorems of Atiyah-Bott-Lefschetz type.
Finally, the appendices cover spectral flow, eta invariants, and the index of parameter-dependent elliptic families.

Reviewer: Paul Loya (Binghamton)

MSC:

58J05	Elliptic equations on manifolds, general theory
58-02	Research exposition (monographs, survey articles) pertaining to global analysis
58J20	Index theory and related fixed-point theorems on manifolds