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Spin Geometry
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Download or read book Spin Geometry written by H. Blaine Lawson and published by Princeton University Press. This book was released on 2016-06-02 with total page 442 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a systematic and comprehensive presentation of the concepts of a spin manifold, spinor fields, Dirac operators, and A-genera, which, over the last two decades, have come to play a significant role in many areas of modern mathematics. Since the deeper applications of these ideas require various general forms of the Atiyah-Singer Index Theorem, the theorems and their proofs, together with all prerequisite material, are examined here in detail. The exposition is richly embroidered with examples and applications to a wide spectrum of problems in differential geometry, topology, and mathematical physics. The authors consistently use Clifford algebras and their representations in this exposition. Clifford multiplication and Dirac operator identities are even used in place of the standard tensor calculus. This unique approach unifies all the standard elliptic operators in geometry and brings fresh insights into curvature calculations. The fundamental relationships of Clifford modules to such topics as the theory of Lie groups, K-theory, KR-theory, and Bott Periodicity also receive careful consideration. A special feature of this book is the development of the theory of Cl-linear elliptic operators and the associated index theorem, which connects certain subtle spin-corbordism invariants to classical questions in geometry and has led to some of the most profound relations known between the curvature and topology of manifolds.
Book Synopsis Conformal Groups in Geometry and Spin Structures by : Pierre Anglès
Download or read book Conformal Groups in Geometry and Spin Structures written by Pierre Anglès and published by Springer Science & Business Media. This book was released on 2007-10-16 with total page 307 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a self-contained overview of the role of conformal groups in geometry and mathematical physics. It features a careful development of the material, from the basics of Clifford algebras to more advanced topics. Each chapter covers a specific aspect of conformal groups and conformal spin geometry. All major concepts are introduced and followed by detailed descriptions and definitions, and a comprehensive bibliography and index round out the work. Rich in exercises that are accompanied by full proofs and many hints, the book will be ideal as a course text or self-study volume for senior undergraduates and graduate students.
Book Synopsis Dirac Operators in Riemannian Geometry by : Thomas Friedrich
Download or read book Dirac Operators in Riemannian Geometry written by Thomas Friedrich and published by American Mathematical Soc.. This book was released on 2000 with total page 213 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis The Dirac Spectrum by : Nicolas Ginoux
Download or read book The Dirac Spectrum written by Nicolas Ginoux and published by Springer. This book was released on 2009-05-30 with total page 168 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume surveys the spectral properties of the spin Dirac operator. After a brief introduction to spin geometry, it presents the main known estimates for Dirac eigenvalues on compact manifolds with or without boundaries.
Book Synopsis Introduction to Symplectic Dirac Operators by : Katharina Habermann
Download or read book Introduction to Symplectic Dirac Operators written by Katharina Habermann and published by Springer. This book was released on 2006-10-28 with total page 131 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is the first one that gives a systematic and self-contained introduction to the theory of symplectic Dirac operators and reflects the current state of the subject. At the same time, it is intended to establish the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology, which have become important fields and very active areas of mathematical research.
Book Synopsis The Theory of Spinors by : Élie Cartan
Download or read book The Theory of Spinors written by Élie Cartan and published by Courier Corporation. This book was released on 2012-04-30 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: Describes orthgonal and related Lie groups, using real or complex parameters and indefinite metrics. Develops theory of spinors by giving a purely geometric definition of these mathematical entities.
Book Synopsis Progress in String, Field and Particle Theory by : L. Baulieu
Download or read book Progress in String, Field and Particle Theory written by L. Baulieu and published by Springer Science & Business Media. This book was released on 2003-09-30 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: The NATO Advanced Study Institute and EC Summer School "Progress in String Field and Particle Theory" was held in Cargse from June 25th till July 11th 2002. The main focus of the school was the recent progress in the very ac tive areas of superstring theory, quantum gravity and the theory of elementary particles. It covered topical problems in domains such as duality between gravity and gaugeinteractions, string field theory, tachyon condensation, non-commutative field theory, string cosmology and string phenomenology. The School featured daily introductory lectures and topical seminars. An informal Gong Show session allowed young post-doctoral researchers and senior graduate students to make a concise presentation oftheir current work. The School gave an excellent opportunity to the youngest researchers to establish a close relationship with their seniors and with the lecturers. These proceedings will further serve in fixing the acquired knowledge, and hopefully, become a useful reference for anyone working in this fascinating do main of physics. Some of the contributions provide an elementary introduction to their subject, while other ones are more geared to the specialist. We are deeply indebted to the NATO Division for Scientific Affairs for funding, and for their constant attention for our meetings, and to the European Commission for a High-Level Scientific Conference grant HPCFCT 2001-00298.
Book Synopsis An Introduction to Noncommutative Geometry by : Joseph C. Várilly
Download or read book An Introduction to Noncommutative Geometry written by Joseph C. Várilly and published by European Mathematical Society. This book was released on 2006 with total page 134 pages. Available in PDF, EPUB and Kindle. Book excerpt: Noncommutative geometry, inspired by quantum physics, describes singular spaces by their noncommutative coordinate algebras and metric structures by Dirac-like operators. Such metric geometries are described mathematically by Connes' theory of spectral triples. These lectures, delivered at an EMS Summer School on noncommutative geometry and its applications, provide an overview of spectral triples based on examples. This introduction is aimed at graduate students of both mathematics and theoretical physics. It deals with Dirac operators on spin manifolds, noncommutative tori, Moyal quantization and tangent groupoids, action functionals, and isospectral deformations. The structural framework is the concept of a noncommutative spin geometry; the conditions on spectral triples which determine this concept are developed in detail. The emphasis throughout is on gaining understanding by computing the details of specific examples. The book provides a middle ground between a comprehensive text and a narrowly focused research monograph. It is intended for self-study, enabling the reader to gain access to the essentials of noncommutative geometry. New features since the original course are an expanded bibliography and a survey of more recent examples and applications of spectral triples.
Book Synopsis Geometric Relativity by : Dan A. Lee
Download or read book Geometric Relativity written by Dan A. Lee and published by American Mathematical Soc.. This book was released on 2019-09-25 with total page 361 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many problems in general relativity are essentially geometric in nature, in the sense that they can be understood in terms of Riemannian geometry and partial differential equations. This book is centered around the study of mass in general relativity using the techniques of geometric analysis. Specifically, it provides a comprehensive treatment of the positive mass theorem and closely related results, such as the Penrose inequality, drawing on a variety of tools used in this area of research, including minimal hypersurfaces, conformal geometry, inverse mean curvature flow, conformal flow, spinors and the Dirac operator, marginally outer trapped surfaces, and density theorems. This is the first time these topics have been gathered into a single place and presented with an advanced graduate student audience in mind; several dozen exercises are also included. The main prerequisite for this book is a working understanding of Riemannian geometry and basic knowledge of elliptic linear partial differential equations, with only minimal prior knowledge of physics required. The second part of the book includes a short crash course on general relativity, which provides background for the study of asymptotically flat initial data sets satisfying the dominant energy condition.
Book Synopsis Lectures on Symplectic Geometry by : Ana Cannas da Silva
Download or read book Lectures on Symplectic Geometry written by Ana Cannas da Silva and published by Springer. This book was released on 2004-10-27 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.
Book Synopsis Differential Geometry and Mathematical Physics by : Gerd Rudolph
Download or read book Differential Geometry and Mathematical Physics written by Gerd Rudolph and published by Springer Science & Business Media. This book was released on 2012-11-09 with total page 766 pages. Available in PDF, EPUB and Kindle. Book excerpt: Starting from an undergraduate level, this book systematically develops the basics of • Calculus on manifolds, vector bundles, vector fields and differential forms, • Lie groups and Lie group actions, • Linear symplectic algebra and symplectic geometry, • Hamiltonian systems, symmetries and reduction, integrable systems and Hamilton-Jacobi theory. The topics listed under the first item are relevant for virtually all areas of mathematical physics. The second and third items constitute the link between abstract calculus and the theory of Hamiltonian systems. The last item provides an introduction to various aspects of this theory, including Morse families, the Maslov class and caustics. The book guides the reader from elementary differential geometry to advanced topics in the theory of Hamiltonian systems with the aim of making current research literature accessible. The style is that of a mathematical textbook,with full proofs given in the text or as exercises. The material is illustrated by numerous detailed examples, some of which are taken up several times for demonstrating how the methods evolve and interact.
Book Synopsis Geometry of String Theory Compactifications by : Alessandro Tomasiello
Download or read book Geometry of String Theory Compactifications written by Alessandro Tomasiello and published by Cambridge University Press. This book was released on 2022-01-13 with total page 677 pages. Available in PDF, EPUB and Kindle. Book excerpt: A unified perspective on new and advanced mathematical techniques used in string theory research for graduate students and researchers.
Book Synopsis Geometry of Crystallographic Groups by : Andrzej Szczepański
Download or read book Geometry of Crystallographic Groups written by Andrzej Szczepański and published by World Scientific. This book was released on 2012 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: Crystallographic groups are groups which act in a nice way and via isometries on some n-dimensional Euclidean space. This book gives an example of the torsion free crystallographic group with a trivial center and a trivial outer automorphism group.
Book Synopsis Riemannian Geometry and Geometric Analysis by : Jürgen Jost
Download or read book Riemannian Geometry and Geometric Analysis written by Jürgen Jost and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 460 pages. Available in PDF, EPUB and Kindle. Book excerpt: FROM REVIEWS OF THE FIRST EDITION "a very readable introduction to Riemannian geometry...it is most welcome...The book is made more interesting by the perspectives in various sections, where the author mentions the history and development of the material and provides the reader with references."-MATHEMATICAL REVIEWS
Author :Clay Mathematics Institute. Summer School Publisher :American Mathematical Soc. ISBN 13 :9780821837153 Total Pages :396 pages Book Rating :4.8/5 (371 download)
Book Synopsis Strings and Geometry by : Clay Mathematics Institute. Summer School
Download or read book Strings and Geometry written by Clay Mathematics Institute. Summer School and published by American Mathematical Soc.. This book was released on 2004 with total page 396 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contains selection of expository and research article by lecturers at the school. Highlights current interests of researchers working at the interface between string theory and algebraic supergravity, supersymmetry, D-branes, the McKay correspondence andFourer-Mukai transform.
Book Synopsis On the Hypotheses Which Lie at the Bases of Geometry by : Bernhard Riemann
Download or read book On the Hypotheses Which Lie at the Bases of Geometry written by Bernhard Riemann and published by Birkhäuser. This book was released on 2016-04-19 with total page 181 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents William Clifford’s English translation of Bernhard Riemann’s classic text together with detailed mathematical, historical and philosophical commentary. The basic concepts and ideas, as well as their mathematical background, are provided, putting Riemann’s reasoning into the more general and systematic perspective achieved by later mathematicians and physicists (including Helmholtz, Ricci, Weyl, and Einstein) on the basis of his seminal ideas. Following a historical introduction that positions Riemann’s work in the context of his times, the history of the concept of space in philosophy, physics and mathematics is systematically presented. A subsequent chapter on the reception and influence of the text accompanies the reader from Riemann’s times to contemporary research. Not only mathematicians and historians of the mathematical sciences, but also readers from other disciplines or those with an interest in physics or philosophy will find this work both appealing and insightful.
Book Synopsis A Spinorial Approach to Riemannian and Conformal Geometry by : Jean-Pierre Bourguignon
Download or read book A Spinorial Approach to Riemannian and Conformal Geometry written by Jean-Pierre Bourguignon and published by Erich Schmidt Verlag GmbH & Co. KG. This book was released on 2015 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book gives an elementary and comprehensive introduction to Spin Geometry, with particular emphasis on the Dirac operator, which plays a fundamental role in differential geometry and mathematical physics. After a self-contained presentation of the basic algebraic, geometrical, analytical and topological ingredients, a systematic study of the spectral properties of the Dirac operator on compact spin manifolds is carried out. The classical estimates on eigenvalues and their limiting cases are discussed next, highlighting the subtle interplay of spinors and special geometric structures. Several applications of these ideas are presented, including spinorial proofs of the Positive Mass Theorem or the classification of positive Kahler-Einstein contact manifolds. Representation theory is used to explicitly compute the Dirac spectrum of compact symmetric spaces. The special features of the book include a unified treatment of $\mathrm{Spin^c}$ and conformal spin geometry (with special emphasis on the conformal covariance of the Dirac operator), an overview with proofs of the theory of elliptic differential operators on compact manifolds based on pseudodifferential calculus, a spinorial characterization of special geometries, and a self-contained presentation of the representation-theoretical tools needed in order to apprehend spinors. This book will help advanced graduate students and researchers to get more familiar with this beautiful, though not sufficiently known, domain of mathematics with great relevance to both theoretical physics and geometry.
