Elliptic Boundary Problems for Dirac Operators

Elliptic Boundary Problems for Dirac Operators PDF Author: Bernhelm Booß-Bavnbek
Publisher: Springer Science & Business Media
ISBN: 1461203376
Category : Mathematics
Languages : en
Pages : 322

Book Description
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.

Dirac Operators in Riemannian Geometry

Dirac Operators in Riemannian Geometry PDF Author: Thomas Friedrich
Publisher: American Mathematical Soc.
ISBN: 0821820559
Category : Mathematics
Languages : en
Pages : 213

Book Description
For a Riemannian manifold M, the geometry, topology and analysis are interrelated in ways that have become widely explored in modern mathematics. Bounds on the curvature can have significant implications for the topology of the manifold. The eigenvalues of the Laplacian are naturally linked to the geometry of the manifold. For manifolds that admit spin structures, one obtains further information from equations involving Dirac operators and spinor fields. In the case of four-manifolds, for example, one has the remarkable Seiberg-Witten invariants. In this text, Friedrich examines the Dirac operator on Riemannian manifolds, especially its connection with the underlying geometry and topology of the manifold. The presentation includes a review of Clifford algebras, spin groups and the spin representation, as well as a review of spin structures and $\textrm{spin}mathbb{C}$ structures. With this foundation established, the Dirac operator is defined and studied, with special attention to the cases of Hermitian manifolds and symmetric spaces. Then, certain analytic properties are established, including self-adjointness and the Fredholm property. An important link between the geometry and the analysis is provided by estimates for the eigenvalues of the Dirac operator in terms of the scalar curvature and the sectional curvature. Considerations of Killing spinors and solutions of the twistor equation on M lead to results about whether M is an Einstein manifold or conformally equivalent to one. Finally, in an appendix, Friedrich gives a concise introduction to the Seiberg-Witten invariants, which are a powerful tool for the study of four-manifolds. There is also an appendix reviewing principal bundles and connections. This detailed book with elegant proofs is suitable as a text for courses in advanced differential geometry and global analysis, and can serve as an introduction for further study in these areas. This edition is translated from the German edition published by Vieweg Verlag.

An Introduction to Dirac Operators on Manifolds

An Introduction to Dirac Operators on Manifolds PDF Author: Jan Cnops
Publisher: Springer Science & Business Media
ISBN: 1461200652
Category : Mathematics
Languages : en
Pages : 219

Book Description
The chapters on Clifford algebra and differential geometry can be used as an introduction to the topics, and are suitable for senior undergraduates and graduates. The other chapters are also accessible at this level.; This self-contained book requires very little previous knowledge of the domains covered, although the reader will benefit from knowledge of complex analysis, which gives the basic example of a Dirac operator.; The more advanced reader will appreciate the fresh approach to the theory, as well as the new results on boundary value theory.; Concise, but self-contained text at the introductory grad level. Systematic exposition.; Clusters well with other Birkhäuser titles in mathematical physics.; Appendix. General Manifolds * List of Symbols * Bibliography * Index

Aspects of Boundary Problems in Analysis and Geometry

Aspects of Boundary Problems in Analysis and Geometry PDF Author: Juan Gil
Publisher: Birkhäuser
ISBN: 3034878508
Category : Mathematics
Languages : en
Pages : 574

Book Description
Boundary problems constitute an essential field of common mathematical interest, they lie in the center of research activities both in analysis and geometry. This book encompasses material from both disciplines, and focuses on their interactions which are particularly apparent in this field. Moreover, the survey style of the contributions makes the topics accessible to a broad audience with a background in analysis or geometry, and enables the reader to get a quick overview.

Analysis, Geometry and Topology of Elliptic Operators

Analysis, Geometry and Topology of Elliptic Operators PDF Author: Bernhelm Booss
Publisher: World Scientific
ISBN: 9812568050
Category : Science
Languages : en
Pages : 553

Book Description
Modern theory of elliptic operators, or simply elliptic theory, has been shaped by the Atiyah-Singer Index Theorem created 40 years ago. Reviewing elliptic theory over a broad range, 32 leading scientists from 14 different countries present recent developments in topology; heat kernel techniques; spectral invariants and cutting and pasting; noncommutative geometry; and theoretical particle, string and membrane physics, and Hamiltonian dynamics.The first of its kind, this volume is ideally suited to graduate students and researchers interested in careful expositions of newly-evolved achievements and perspectives in elliptic theory. The contributions are based on lectures presented at a workshop acknowledging Krzysztof P Wojciechowski's work in the theory of elliptic operators.

The Atiyah-Patodi-Singer Index Theorem

The Atiyah-Patodi-Singer Index Theorem PDF Author: Richard Melrose
Publisher: CRC Press
ISBN: 1439864608
Category : Mathematics
Languages : en
Pages : 392

Book Description
Based on the lecture notes of a graduate course given at MIT, this sophisticated treatment leads to a variety of current research topics and will undoubtedly serve as a guide to further studies.

Analysis, Geometry And Topology Of Elliptic Operators: Papers In Honor Of Krzysztof P Wojciechowski

Analysis, Geometry And Topology Of Elliptic Operators: Papers In Honor Of Krzysztof P Wojciechowski PDF Author: Matthias Lesch
Publisher: World Scientific
ISBN: 9814478024
Category : Mathematics
Languages : en
Pages : 553

Book Description
Modern theory of elliptic operators, or simply elliptic theory, has been shaped by the Atiyah-Singer Index Theorem created 40 years ago. Reviewing elliptic theory over a broad range, 32 leading scientists from 14 different countries present recent developments in topology; heat kernel techniques; spectral invariants and cutting and pasting; noncommutative geometry; and theoretical particle, string and membrane physics, and Hamiltonian dynamics.The first of its kind, this volume is ideally suited to graduate students and researchers interested in careful expositions of newly-evolved achievements and perspectives in elliptic theory. The contributions are based on lectures presented at a workshop acknowledging Krzysztof P Wojciechowski's work in the theory of elliptic operators.

Dirac Operators and Spectral Geometry

Dirac Operators and Spectral Geometry PDF Author: Giampiero Esposito
Publisher: Cambridge University Press
ISBN: 0521648629
Category : Mathematics
Languages : en
Pages : 227

Book Description
A clear, concise and up-to-date introduction to the theory of the Dirac operator and its wide range of applications in theoretical physics for graduate students and researchers.

Dirac Operators in Analysis

Dirac Operators in Analysis PDF Author: John Ryan
Publisher: CRC Press
ISBN: 9780582356818
Category : Mathematics
Languages : en
Pages : 260

Book Description
Clifford analysis has blossomed into an increasingly relevant and fashionable area of research in mathematical analysis-it fits conveniently at the crossroads of many fundamental areas of research, including classical harmonic analysis, operator theory, and boundary behavior. This book presents a state-of-the-art account of the most recent developments in the field of Clifford analysis with contributions by many of the field's leading researchers.

Operator Theory and Harmonic Analysis

Operator Theory and Harmonic Analysis PDF Author: Alexey N. Karapetyants
Publisher: Springer Nature
ISBN: 3030774937
Category : Mathematics
Languages : en
Pages : 585

Book Description
This volume is part of the collaboration agreement between Springer and the ISAAC society. This is the first in the two-volume series originating from the 2020 activities within the international scientific conference "Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis" (OTHA), Southern Federal University in Rostov-on-Don, Russia. This volume is focused on general harmonic analysis and its numerous applications. The two volumes cover new trends and advances in several very important fields of mathematics, developed intensively over the last decade. The relevance of this topic is related to the study of complex multiparameter objects required when considering operators and objects with variable parameters.