Clifford Algebras and Dirac Operators in Harmonic Analysis (Cambridge Studies in Advanced Mathematics)

Book Description

The aim of this book is to unite the seemingly disparate topics of Clifford algebras, analysis on manifolds, and harmonic analysis. The authors show how algebra, geometry, and differential equations play a more fundamental role in Euclidean Fourier analysis. They then link their presentation of the Euclidean theory naturally to the representation theory of semi-simple Lie groups.

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Full of interesting results........2000-05-17

This book is helping me a lot with my Ph.D. dissertation. It includes a lot of important results on hypercomplex analysis not usually found in the standard monographs on the subject (Brackx, Gürlebeck, Shapiro,...).

Its contents are: Clifford algebras, Dirac operators and Clifford analyticity, representations of Spin(V,Q), constant coefficient operators of Dirac type, Dirac operators and manifolds.

Presents motivation for each section and extensive references. A must-reading to become a speciallist in this area. Suitable for graduate students and researchers.

Book Description

This monograph presents a comprehensive treatment of important new ideas on Dirac operators and Dirac cohomology. Dirac operators are widely used in physics, differential geometry, and group-theoretic settings (particularly, the geometric construction of discrete series representations). The related concept of Dirac cohomology, which is defined using Dirac operators, is a far-reaching generalization that connects index theory in differential geometry to representation theory. Using Dirac operators as a unifying theme, the authors demonstrate how some of the most important results in representation theory fit together when viewed from this perspective.

Key topics covered include:

* Proof of Vogan's conjecture on Dirac cohomology

* Simple proofs of many classical theorems, such as the Bott–Borel–Weil theorem and the Atiyah–Schmid theorem

* Dirac cohomology, defined by Kostant's cubic Dirac operator, along with other closely related kinds of cohomology, such as n-cohomology and (g,K)-cohomology

* Cohomological parabolic induction and $A_q(\lambda)$ modules

* Discrete series theory, characters, existence and exhaustion

* Sharpening of the Langlands formula on multiplicity of automorphic forms, with applications

* Dirac cohomology for Lie superalgebras

An excellent contribution to the mathematical literature of representation theory, this self-contained exposition offers a systematic examination and panoramic view of the subject. The material will be of interest to researchers and graduate students in representation theory, differential geometry, and physics.

An Introduction to Dirac Operators on Manifolds

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An Introduction to Dirac Operators on Manifolds
Jan Cnops
Manufacturer: Birkhäuser Boston
ProductGroup: Book
Binding: Hardcover

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ASIN: 0817642986

Book Description

Dirac operators play an important role in several domains of mathematics and physics, for example: index theory, elliptic pseudodifferential operators, electromagnetism, particle physics, and the representation theory of Lie groups. In this essentially self-contained work, the basic ideas underlying the concept of Dirac operators are explored. Starting with Clifford algebras and the fundamentals of differential geometry, the text focuses on two main properties, namely, conformal invariance, which determines the local behavior of the operator, and the unique continuation property dominating its global behavior. Spin groups and spinor bundles are covered, as well as the relations with their classical counterparts, orthogonal groups and Clifford bundles. The chapters on Clifford algebras and the fundamentals of differential geometry can be used as an introduction to the above topics, and are suitable for senior undergraduate and graduate students. The other chapters are also accessible at this level so that this text requires very little previous knowledge of the domains covered. The reader will benefit, however, from some knowledge of complex analysis, which gives the simplest example of a Dirac operator. More advanced readers---mathematical physicists, physicists and mathematicians from diverse areas---will appreciate the fresh approach to the theory as well as the new results on boundary value theory.

Dirac Operators in Riemannian Geometry (Graduate Studies in Mathematics)

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Dirac Operators in Riemannian Geometry (Graduate Studies in Mathematics)
Thomas Friedrich
Manufacturer: American Mathematical Society
ProductGroup: Book
Binding: Hardcover

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ASIN: 0821820559

Book Description

For a Riemannian manifold $M$, the geometry, topology and analysis are interrelated in ways that are 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 (or $\textrm{spin}^\mathbb{C}$) 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.

Book Description

The first edition of this book presented simple proofs of the Atiyah-Singer Index Theorem for Dirac operators on compact Riemannian manifolds and its generalizations (due to the authors and J.-M. Bismut), using an explicit geometric construction of the heat kernel of a generalized Dirac operator; the new edition makes this popular book available to students and researchers in an attractive softcover. The first four chapters could be used as the text for a graduate course on the applications of linear elliptic operators in differential geometry and the only prerequisites are a familiarity with basic differential geometry. The next four chapters discuss the equivariant index theorem, and include a useful introduction to equivariant differential forms. The last two chapters give a proof, in the spirit of the book, of Bismut's Local Family Index Theorem for Dirac operators.

Customer Reviews:

Highly advanced treatise on global analysis........2000-07-14

Dirac operators on Riemannian manifolds are of fundamental importance in differential geometry: they occur in situations such as Hodge theory, gauge theory, and geometric quantization.

The book is based on a simple principle: Dirac operators are a quantization of the theory of connections, and the supertrace of the heat kernel of the square of a Dirac operator is the quantization of the Chern character of the corresponding connection. From this point of view, the index theorem for Dirac operators is a statement about the relationship between the heat kernel of the square of a Dirac operator and the Chern character of the associated connection. This relationship holds at the level of differential forms and not just in cohomology, and leads to think of index theory and heat kernels as a quantization of Chern-Weil theory. The importance of the heat kernel is that it interpolates between the identity operator and the projection onto the kernel of the Dirac operator. However, the authors study the heat kernel, and more particularly its restriction to the diagonal, in its own right, and not only as a tool in understanding the kernel of the Dirac operator.

The authors attempt to express allof their constructions in such a way that they generalize easily to the equivariant setting, in which a compact Lie group acts on the manifold and leaves the Dirac operator invariant. They consider the most general type of Dirac operators, associated to a Clifford module over a manifold, to avoid restricting to manifolds with spin connections. They also work within Quillen's theory of superconnections.

The book is not necessarily meant to be read sequentially, and consists of four groups of chapters: (1) Chapters 1 and 7, the former giving various preliminary results in differential geometry and the latter on equivariant differential forms; they do not depend on any other chapters. (2) Chapters 2, 3, and 4 introduce the main ideas of the book, and take the reader through the main properties of Dirac operators, culminating in the local index theorem. (3) Chapters 5, 6, and 8 are on the equivariant index theorem, and may be read after the first four chapters, although Chapter 7 is needed in Chapter 8. (4) Chapters 9 and 10 are on the family index theorem, and can be read after the first four chapters, except sections 9.4 and 10.7 which have Chapter 8 as a prerequisite.

The book is intended for researchers and advanced graduate students; you need a very strong background in differntial geometry, algebraic topology, harmonic analysis, and hypercomplex analysis to read it. The style is definitely French, so if you have had trouble with Bourbaki be prepared. The list of references is adequately long. Very nice printing and binding quality.

difficult but worth the effort.......2000-07-02

This monograph treats the modern theory of the heat equation on manifolds, with special emphasis on recent work related to the Atiyah-Singer index theorem, and in particular Bismut's contributions. Much of this work has been concerned with finding explicit differential-form representatives of the cohomology classes that arise in the index formula and its generalizations.

The authors have completely eliminated the probability theory that figures so notoriously in Bismut's papers, replacing it by the more classical asymptotic expansions. (However, I must say that my study of Bismut's papers goaded me into learning the probability theory, and I'm glad I did; for apart from being interesting in itself, it also proved very useful in my thesis). Moreover, they have also managed to eliminate almost all of the analysis. What remains is a fair amount of differential geometry and a great deal of algebra.

In those parts of the book that are written very concisely, readers will have trouble supplying full details. An example is Sec. 1.6 on the Euler and Thom classes; even my thesis adviser did not understand the algebra of differential forms, which becomes confusing given all the various pullback bundles; here understanding the algebra in Lemma 1.51 is the crux of the matter. Another example of extreme conciseness is Sec. 3.6, which sketches the standard Clifford modules in the important cases -- De Rham, signature, spin, Kahler. In general, just enough information is provided to enable a well-motivated graduate student to fill in the details and/or acquire the necessary background. Doing this took me the better part of two years.

On the other hand, the parts of the book dealing directly with heat kernels are written less concisely and are therefore more readable. The book gives a nice construction of the heat kernel for generalized Laplacians (Thm. 2.30) in Chap. 2, which is devoted to the asymptotic expansion of the heat kernel, essentially following Hadamard's classical approach; however the treatment is highly algebraic. Chapter 6, based on work of Berline and Vergne, re-covers much of the same ground from the viewpoint of equivariant vector bundles; it has a more overtly differential-geometric flavor. Chapters 9 and 10 on the index bundle and Bismut's version of the index theorem for families are again quite readable and again highly algebraic.

Seeley's work on pseudodifferential operators, which played such an important role in the original proof of the Atiyah-Singer index theorem, of course has its counterpart here in the asymptotic expansion, but the treatment makes it seems rather innocuous if not quite trivial. In general, the lack of "hard analysis" in the book is striking. Except for a cameo appearance in the short Chap. 7 on equivariant differential forms, Fourier analysis, for example, plays no role.

Researchers already active in the field will probably benefit the most from this book, but fun-loving grad students can also profit from it.

Heat Kernels and Dirac Operators.......2000-06-27

The book is devoted to the investigation of Dirac operators , in particular to explaining the relation of the Atiyah-Singer formula for their index with the heat kernel expansion. Dirac operators on Riemannian manifolds were introduced by Lichnerowicz, Atiyah and Singer. Most first order linear differential operators of geometric origin are Dirac operators. The main technique in the book is an explicit construction of the kernel of the heat operator $e\sp{-tD\sp 2}$ associated to the square of the Dirac operator $D$. Note that in the proofs of many theorems due to J. M. Bismut, the original use of probability theory is replaced by the applications of classical asymptotic expansion methods. List of Chapters: Introduction, 1. Background on Differential Geometry, 2. Asymptotic Expansion of Heat Kernel, 3. Clifford Modules and Dirac Operators, 4. Index Density of Dirac Operators, 5. The Exponential Map and the Index Density, 6. The Equivariant Index Theorem, 7. Equivariant Differential Forms, 8. The Kirillov Formula for the Equivariant Index, 9. The Index Bundle, 10.The Family Index Theorem.

Aspects of Boundary Problems in Analysis and Geometry (Operator Theory: Advances and Applications / Advances in Partial Differential Equations)

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Aspects of Boundary Problems in Analysis and Geometry (Operator Theory: Advances and Applications / Advances in Partial Differential Equations)

Manufacturer: Birkhäuser Basel
ProductGroup: Book
Binding: Hardcover

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ASIN: 3764370696

Book Description

Boundary problems constitute an essential field of common mathematical interest. The intention of this volume is to highlight several analytic and geometric aspects of boundary problems with special emphasis on their interplay. It includes surveys on classical topics presented from a modern perspective as well as reports on current research.

The collection splits into two related groups:

- analysis and geometry of geometric operators and their index theory

- elliptic theory of boundary value problems and the Shapiro-Lopatinsky condition

C*-algebras and Elliptic Theory (Trends in Mathematics)

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C*-algebras and Elliptic Theory (Trends in Mathematics)
Dan Burghelea , Richard Melrose , and Victor Nistor
Manufacturer: Birkhäuser Basel
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Clifford Algebra and Spinor-Valued Functions: A Function Theory for the Dirac Operator (Mathematics and Its Applications)

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Advanced encyclopedic treatise on hypercomplex analysis.

Clifford Algebra and Spinor-Valued Functions: A Function Theory for the Dirac Operator (Mathematics and Its Applications)
R. Delanghe , F. Sommen , and V. Soucek
Manufacturer: Springer
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Binding: Hardcover

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Book Description

This volume describes the substantial developments in Clifford analysis which have taken place during the last decade and, in particular, the role of the spin group in the study of null solutions of real and complexified Dirac and Laplace operators. The book has six main chapters. The first two (Chapters 0 and I) present classical results on real and complex Clifford algebras and show how lower-dimensional real Clifford algebras are well-suited for describing basic geometric notions in Euclidean space. Chapters II and III illustrate how Clifford analysis extends and refines the computational tools available in complex analysis in the plane or harmonic analysis in space. In Chapter IV the concept of monogenic differential forms is generalized to the case of spin-manifolds. Chapter V deals with analysis on homogeneous spaces, and shows how Clifford analysis may be connected with the Penrose transform. The volume concludes with some Appendices which present basic results relating to the algebraic and analytic structures discussed. These are made accessible for computational purposes by means of computer algebra programmes written in REDUCE and are contained on an accompanying floppy disk.

Customer Reviews:

Advanced encyclopedic treatise on hypercomplex analysis........2000-07-11

This thick volume constitutes an exhaustive catalogue of the achievements and main interests of the regarded Belgian team working on hypercomplex analysis.

The book is not intended for begineers; you must read their previous monograph "Clifford Analysis" (by Brackx, Delanghe, and Sommen) or any of the other texts at the same level which are available. Also, you must have had a previous acquaintance with spinor geometry, algebraic topology, and differential geometry (Lie algebras), not to mention real, complex, and functional analysis. The style is completely that of a research monograph, though it can be used as a reference in graduate-level courses.

Its contents are: Clifford Algebras over Lower Dimensional Euclidean Spaces; Clifford Algebras and Spinor Spaces; Monogenic Functions; Special Functions and Methods; Monogenic Differential Forms and Residues; Clifford Analysis and the Penrose Transform; 3 appendices. Includes companion diskette with REDUCE software.

Book Description

The Dirac operator has many useful applications in theoretical physics and mathematics. This book provides a clear, concise and self-contained introduction to the global theory of the Dirac operator and to the analysis of spectral asymptotics with local or nonlocal boundary conditions. The theory is introduced at a level suitable for graduate students. Numerous examples are then given to illustrate the peculiar properties of the Dirac operator, and the role of boundary conditions in heat-kernel asymptotics and quantum field theory. Topics covered include the introduction of spin-structures in Riemannian and Lorentzian manifolds; applications of index theory; heat-kernel asymptotics for operators of Laplace type; quark boundary conditions; one-loop quantum cosmology; conformally covariant operators; and the role of the Dirac operator in some recent investigations of four-manifolds. This volume provides graduate students with a rigorous introduction and researchers with a invaluable reference to the Dirac operator and its applications in theoretical physics.

Dirac Operators in Analysis (Research Notes in Mathematics Series)

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Dirac Operators in Analysis (Research Notes in Mathematics Series)
John Ryan , and Daniele C Struppa
Manufacturer: Chapman & Hall/CRC
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Binding: Hardcover

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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.

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