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Manifolds And Local Structures A General Theory
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Book Synopsis Manifolds And Local Structures: A General Theory by : Marco Grandis
Download or read book Manifolds And Local Structures: A General Theory written by Marco Grandis and published by World Scientific. This book was released on 2021-02-10 with total page 374 pages. Available in PDF, EPUB and Kindle. Book excerpt: Local structures, like differentiable manifolds, fibre bundles, vector bundles and foliations, can be obtained by gluing together a family of suitable 'elementary spaces', by means of partial homeomorphisms that fix the gluing conditions and form a sort of 'intrinsic atlas', instead of the more usual system of charts living in an external framework.An 'intrinsic manifold' is defined here as such an atlas, in a suitable category of elementary spaces: open euclidean spaces, or trivial bundles, or trivial vector bundles, and so on.This uniform approach allows us to move from one basis to another: for instance, the elementary tangent bundle of an open Euclidean space is automatically extended to the tangent bundle of any differentiable manifold. The same holds for tensor calculus.Technically, the goal of this book is to treat these structures as 'symmetric enriched categories' over a suitable basis, generally an ordered category of partial mappings.This approach to gluing structures is related to Ehresmann's one, based on inductive pseudogroups and inductive categories. A second source was the theory of enriched categories and Lawvere's unusual view of interesting mathematical structures as categories enriched over a suitable basis.
Download or read book Global Calculus written by S. Ramanan and published by American Mathematical Soc.. This book was released on 2005 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: The power that analysis, topology and algebra bring to geometry has revolutionised the way geometers and physicists look at conceptual problems. Some of the key ingredients in this interplay are sheaves, cohomology, Lie groups, connections and differential operators. In Global Calculus, the appropriate formalism for these topics is laid out with numerous examples and applications by one of the experts in differential and algebraic geometry. Ramanan has chosen an uncommon but natural path through the subject. In this almost completely self-contained account, these topics are developed from scratch. The basics of Fourier transforms, Sobolev theory and interior regularity are proved at the same time as symbol calculus, culminating in beautiful results in global analysis, real and complex. Many new perspectives on traditional and modern questions of differential analysis and geometry are the hallmarks of the book. The book is suitable for a first year graduate course on Global Analysis.
Book Synopsis Symmetries And Curvature Structure In General Relativity by : Hall Graham S
Download or read book Symmetries And Curvature Structure In General Relativity written by Hall Graham S and published by World Scientific. This book was released on 2004-04-27 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a text on classical general relativity from a geometrical viewpoint. Introductory chapters are provided on algebra, topology and manifold theory, together with a chapter on the basic ideas of space-time manifolds and Einstein's theory. There is a detailed account of algebraic structures and tensor classification in general relativity and also of the relationships between the metric, connection and curvature structures on space-times. The latter includes chapters on holonomy and sectional curvature. An extensive study is presented of symmetries in general relativity, including isometries, homotheties, conformal symmetries and affine, projective and curvature collineations. Several general properties of such symmetries are studied and a preparatory section on transformation groups and on the properties of Lie algebras of vector fields on manifolds is provided.
Book Synopsis An Introduction to Manifolds by : Loring W. Tu
Download or read book An Introduction to Manifolds written by Loring W. Tu and published by Springer Science & Business Media. This book was released on 2010-10-05 with total page 426 pages. Available in PDF, EPUB and Kindle. Book excerpt: Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'.
Book Synopsis Introduction to Smooth Manifolds by : John M. Lee
Download or read book Introduction to Smooth Manifolds written by John M. Lee and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 646 pages. Available in PDF, EPUB and Kindle. Book excerpt: Author has written several excellent Springer books.; This book is a sequel to Introduction to Topological Manifolds; Careful and illuminating explanations, excellent diagrams and exemplary motivation; Includes short preliminary sections before each section explaining what is ahead and why
Book Synopsis Algebraic L-theory and Topological Manifolds by : Andrew Ranicki
Download or read book Algebraic L-theory and Topological Manifolds written by Andrew Ranicki and published by Cambridge University Press. This book was released on 1992-12-10 with total page 372 pages. Available in PDF, EPUB and Kindle. Book excerpt: Assuming no previous acquaintance with surgery theory and justifying all the algebraic concepts used by their relevance to topology, Dr Ranicki explains the applications of quadratic forms to the classification of topological manifolds, in a unified algebraic framework.
Book Synopsis Mathematical Implications of Einstein-Weyl Causality by : Hans Jürgen Borchers
Download or read book Mathematical Implications of Einstein-Weyl Causality written by Hans Jürgen Borchers and published by Springer. This book was released on 2007-02-22 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt: Here is a systematic approach to such fundamental questions as: What mathematical structures does Einstein-Weyl causality impose on a point-set that has no other previous structure defined on it? The author proposes an axiomatization of the physics inspired notion of Einstein-Weyl causality and investigating the consequences in terms of possible topological spaces. One significant result is that the notion of causality can effectively be extended to discontinuum.
Book Synopsis Algebraic Topology: A Structural Introduction by : Marco Grandis
Download or read book Algebraic Topology: A Structural Introduction written by Marco Grandis and published by World Scientific. This book was released on 2021-12-24 with total page 372 pages. Available in PDF, EPUB and Kindle. Book excerpt: Algebraic Topology is a system and strategy of partial translations, aiming to reduce difficult topological problems to algebraic facts that can be more easily solved. The main subject of this book is singular homology, the simplest of these translations. Studying this theory and its applications, we also investigate its underlying structural layout - the topics of Homological Algebra, Homotopy Theory and Category Theory which occur in its foundation.This book is an introduction to a complex domain, with references to its advanced parts and ramifications. It is written with a moderate amount of prerequisites — basic general topology and little else — and a moderate progression starting from a very elementary beginning. A consistent part of the exposition is organised in the form of exercises, with suitable hints and solutions.It can be used as a textbook for a semester course or self-study, and a guidebook for further study.
Book Synopsis Discrete Causal Theory by : Benjamin F. Dribus
Download or read book Discrete Causal Theory written by Benjamin F. Dribus and published by Springer. This book was released on 2017-04-26 with total page 558 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book evaluates and suggests potentially critical improvements to causal set theory, one of the best-motivated approaches to the outstanding problems of fundamental physics. Spacetime structure is of central importance to physics beyond general relativity and the standard model. The causal metric hypothesis treats causal relations as the basis of this structure. The book develops the consequences of this hypothesis under the assumption of a fundamental scale, with smooth spacetime geometry viewed as emergent. This approach resembles causal set theory, but differs in important ways; for example, the relative viewpoint, emphasizing relations between pairs of events, and relationships between pairs of histories, is central. The book culminates in a dynamical law for quantum spacetime, derived via generalized path summation.
Book Synopsis Introduction to Topological Manifolds by : John M. Lee
Download or read book Introduction to Topological Manifolds written by John M. Lee and published by Springer Science & Business Media. This book was released on 2006-04-06 with total page 395 pages. Available in PDF, EPUB and Kindle. Book excerpt: Manifolds play an important role in topology, geometry, complex analysis, algebra, and classical mechanics. Learning manifolds differs from most other introductory mathematics in that the subject matter is often completely unfamiliar. This introduction guides readers by explaining the roles manifolds play in diverse branches of mathematics and physics. The book begins with the basics of general topology and gently moves to manifolds, the fundamental group, and covering spaces.
Book Synopsis Structure and Regularity of Group Actions on One-Manifolds by : Sang-hyun Kim
Download or read book Structure and Regularity of Group Actions on One-Manifolds written by Sang-hyun Kim and published by Springer Nature. This book was released on 2021-11-19 with total page 323 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the theory of optimal and critical regularities of groups of diffeomorphisms, from the classical work of Denjoy and Herman, up through recent advances. Beginning with an investigation of regularity phenomena for single diffeomorphisms, the book goes on to describes a circle of ideas surrounding Filipkiewicz's Theorem, which recovers the smooth structure of a manifold from its full diffeomorphism group. Topics covered include the simplicity of homeomorphism groups, differentiability of continuous Lie group actions, smooth conjugation of diffeomorphism groups, and the reconstruction of spaces from group actions. Various classical and modern tools are developed for controlling the dynamics of general finitely generated group actions on one-dimensional manifolds, subject to regularity bounds, including material on Thompson's group F, nilpotent groups, right-angled Artin groups, chain groups, finitely generated groups with prescribed critical regularities, and applications to foliation theory and the study of mapping class groups. The book will be of interest to researchers in geometric group theory.
Book Synopsis Differential Geometry of Foliations by : B.L. Reinhart
Download or read book Differential Geometry of Foliations written by B.L. Reinhart and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 205 pages. Available in PDF, EPUB and Kindle. Book excerpt: Whoever you are! How can I but offer you divine leaves . . . ? Walt Whitman The object of study in modern differential geometry is a manifold with a differ ential structure, and usually some additional structure as well. Thus, one is given a topological space M and a family of homeomorphisms, called coordinate sys tems, between open subsets of the space and open subsets of a real vector space V. It is supposed that where two domains overlap, the images are related by a diffeomorphism, called a coordinate transformation, between open subsets of V. M has associated with it a tangent bundle, which is a vector bundle with fiber V and group the general linear group GL(V). The additional structures that occur include Riemannian metrics, connections, complex structures, foliations, and many more. Frequently there is associated to the structure a reduction of the group of the tangent bundle to some subgroup G of GL(V). It is particularly pleasant if one can choose the coordinate systems so that the Jacobian matrices of the coordinate transformations belong to G. A reduction to G is called a G-structure, which is called integrable (or flat) if the condition on the Jacobians is satisfied. The strength of the integrability hypothesis is well-illustrated by the case of the orthogonal group On. An On-structure is given by the choice of a Riemannian metric, and therefore exists on every smooth manifold.
Book Synopsis The Topology of 4-Manifolds by : Robion C. Kirby
Download or read book The Topology of 4-Manifolds written by Robion C. Kirby and published by Springer. This book was released on 2006-11-14 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the classical theorems about simply connected smooth 4-manifolds: intersection forms and homotopy type, oriented and spin bordism, the index theorem, Wall's diffeomorphisms and h-cobordism, and Rohlin's theorem. Most of the proofs are new or are returbishings of post proofs; all are geometric and make us of handlebody theory. There is a new proof of Rohlin's theorem using spin structures. There is an introduction to Casson handles and Freedman's work including a chapter of unpublished proofs on exotic R4's. The reader needs an understanding of smooth manifolds and characteristic classes in low dimensions. The book should be useful to beginning researchers in 4-manifolds.
Book Synopsis Theoretical Foundations of Cosmology by : Michael Heller
Download or read book Theoretical Foundations of Cosmology written by Michael Heller and published by World Scientific. This book was released on 1992-04-15 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book's principal aim is to clarify fundamental concepts, decipher mathematical structures used to model space-time and relativistic worlds, and to disclose their physical meaning. After each chapter, philosophical implications of the presented material are commented upon. Both special and general theories of relativity are presented in the book with the stress on their global aspects. Although global mathematical methods are extensively used throughout the book, the definitions of new concepts, short comments and examples make reading smooth without the need to consult other textbooks or review papers. Contents:Manifold Model of Space-TimeRelativistic Model of Space-TimeMethod of Fibre BundlesThe Frame Bundle and General RelativityThe Cosmological ProblemGeometry of the Standard Cosmological ModelBibliographical EssayAppendix. Evolution of Dynamical Space-Time TheoriesReferencesSymbol IndexAuthor and Subject Index Readership: Cosmologists, astronomers and theoretical physicists. keywords:Cosmology;Universe;Standard Cosmological Model;Space-Time;General Relativity;Special Relativity;Geometry of Space-Time;Differential Geometry;Fibre Bundles;Methodological Questions of Cosmology;Philosophy of Cosmology
Book Synopsis Causality, Measurement Theory and the Differentiable Structure of Space-Time by : R. N. Sen
Download or read book Causality, Measurement Theory and the Differentiable Structure of Space-Time written by R. N. Sen and published by Cambridge University Press. This book was released on 2010-02-11 with total page 413 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introducing graduate students and researchers to mathematical physics, this book discusses two recent developments: the demonstration that causality can be defined on discrete space-times; and Sewell's measurement theory, in which the wave packet is reduced without recourse to the observer's conscious ego, nonlinearities or interaction with the rest of the universe. The definition of causality on a discrete space-time assumes that space-time is made up of geometrical points. Using Sewell's measurement theory, the author concludes that the notion of geometrical points is as meaningful in quantum mechanics as it is in classical mechanics, and that it is impossible to tell whether the differential calculus is a discovery or an invention. Providing a mathematical discourse on the relation between theoretical and experimental physics, the book gives detailed accounts of the mathematically difficult measurement theories of von Neumann and Sewell.
Book Synopsis Tropical and Idempotent Mathematics by : Grigoriĭ Lazarevich Litvinov
Download or read book Tropical and Idempotent Mathematics written by Grigoriĭ Lazarevich Litvinov and published by American Mathematical Soc.. This book was released on 2009 with total page 395 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is a collection of papers from the International Conference on Tropical and Idempotent Mathematics, held in Moscow, Russia in August 2007. This is a relatively new branch of mathematical sciences that has been rapidly developing and gaining popularity over the last decade. Tropical mathematics can be viewed as a result of the Maslov dequantization applied to 'traditional' mathematics over fields. Importantly, applications in econophysics and statistical mechanics lead to an explanation of the nature of financial crises. Another original application provides an analysis of instabilities in electrical power networks. Idempotent analysis, tropical algebra, and tropical geometry are the building blocks of the subject. Contributions to idempotent analysis are focused on the Hamilton-Jacobi semigroup, the max-plus finite element method, and on the representations of eigenfunctions of idempotent linear operators. Tropical algebras, consisting of plurisubharmonic functions and their germs, are examined. The volume also contains important surveys and research papers on tropical linear algebra and tropical convex geometry.
Book Synopsis Relativistic Celestial Mechanics of the Solar System by : Sergei Kopeikin
Download or read book Relativistic Celestial Mechanics of the Solar System written by Sergei Kopeikin and published by John Wiley & Sons. This book was released on 2011-10-25 with total page 897 pages. Available in PDF, EPUB and Kindle. Book excerpt: This authoritative book presents the theoretical development of gravitational physics as it applies to the dynamics of celestial bodies and the analysis of precise astronomical observations. In so doing, it fills the need for a textbook that teaches modern dynamical astronomy with a strong emphasis on the relativistic aspects of the subject produced by the curved geometry of four-dimensional spacetime. The first three chapters review the fundamental principles of celestial mechanics and of special and general relativity. This background material forms the basis for understanding relativistic reference frames, the celestial mechanics of N-body systems, and high-precision astrometry, navigation, and geodesy, which are then treated in the following five chapters. The final chapter provides an overview of the new field of applied relativity, based on recent recommendations from the International Astronomical Union. The book is suitable for teaching advanced undergraduate honors programs and graduate courses, while equally serving as a reference for professional research scientists working in relativity and dynamical astronomy. The authors bring their extensive theoretical and practical experience to the subject. Sergei Kopeikin is a professor at the University of Missouri, while Michael Efroimsky and George Kaplan work at the United States Naval Observatory, one of the world?s premier institutions for expertise in astrometry, celestial mechanics, and timekeeping.
