Elliptic Theory and Noncommutative Geometry

Vladimir E. Nazaykinskiy

ISBN 10: 3764387742 / ISBN 13: 9783764387747

Published by Springer, Basel, Birkhäuser Basel, Birkhäuser Mai 2008, 2008

About this Item

This item is printed on demand - it takes 3-4 days longer - Neuware -Noncommutative geometry, which can rightfully claim the role of a philosophy in mathematicalstudies,undertakesto replacegoodoldnotionsofclassicalgeometry (suchas manifolds,vectorbundles, metrics, di erentiable structures,etc. ) by their abstract operator-algebraic analogs and then to study the latter by methods of the theory of operator algebras. At rst sight, this pursuit of maximum possible generality harbors the danger of completely forgetting the classical beginnings, so that not only the answers but also the questions would defy stating in traditional terms. Noncommutative geometry itself would become not only a method but also the main subject of investigation according to the capacious but not too practical formula: 'Know thyself. ' Fortunately, this is not completely true (or even is completely untrue) in reality: there are numerous problems that are quite classical in their statement (or at least admit an equivalent classical statement) but can be solved only in the framework of noncommutative geometry. One of such problems is the subject of the present book. The classical elliptic theory developed in the well-known work of Atiyah and Singer on the index problem relates an analytic invariant of an elliptic pseud- i erential operator on a smooth compact manifold, namely, its index, to topol- ical invariants of the manifold itself. The index problem for nonlocal (and hence nonpseudodi erential) elliptic operators is much more complicated and requires the use of substantially more powerful methods than those used in the classical case. 224 pp. Englisch.

Seller Inventory # 9783764387747

Bibliographic Details

• Title: Elliptic Theory and Noncommutative Geometry
• Publisher: Springer, Basel, Birkhäuser Basel, Birkhäuser Mai 2008
• Publication Date: 2008
• Language: English
• ISBN 10: 3764387742
• ISBN 13: 9783764387747
• Binding: Buch
• Condition: Neu
• Dimensions: 243W x 155H x 15D millimeters 243 millimeters width by 155 millimeters height by 15 millimeters depth
• Weight: 576 grams

About this title

Synopsis

This comprehensive yet concise book deals with nonlocal elliptic differential operators. These are operators whose coefficients involve shifts generated by diffeomorphisms of the manifold on which the operators are defined. This is the first book featuring a consistent application of methods of noncommutative geometry to the index problem in the theory of nonlocal elliptic operators. To make the book self-contained, the authors have included necessary geometric material.

From the Back Cover

This comprehensive yet concise book deals with nonlocal elliptic differential operators, whose coefficients involve shifts generated by diffeomorophisms of the manifold on which the operators are defined. The main goal of the study is to relate analytical invariants (in particular, the index) of such elliptic operators to topological invariants of the manifold itself. This problem can be solved by modern methods of noncommutative geometry.

This is the first and so far the only book featuring a consistent application of methods of noncommutative geometry to the index problem in the theory of nonlocal elliptic operators. Although the book provides important results, which are in a sense definitive, on the above-mentioned topic, it contains all the necessary preliminary material, such as C*-algebras and their K-theory or cyclic homology. Thus the material is accessible for undergraduate students of mathematics (third year and beyond). It is also undoubtedly of interest for post-graduate students and scientists specializing in geometry, the theory of differential equations, functional analysis, etc.

The book can serve as a good introduction to noncommutative geometry, which is one of the most powerful modern tools for studying a wide range of problems in mathematics and theoretical physics.