Download or read book Basic Concepts of Synthetic Differential Geometry written by R. Lavendhomme and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 331 pages. Available in PDF, EPUB and Kindle. Book excerpt: Starting at an introductory level, the book leads rapidly to important and often new results in synthetic differential geometry. From rudimentary analysis the book moves to such important results as: a new proof of De Rham's theorem; the synthetic view of global action, going as far as the Weil characteristic homomorphism; the systematic account of structured Lie objects, such as Riemannian, symplectic, or Poisson Lie objects; the view of global Lie algebras as Lie algebras of a Lie group in the synthetic sense; and lastly the synthetic construction of symplectic structure on the cotangent bundle in general. Thus while the book is limited to a naive point of view developing synthetic differential geometry as a theory in itself, the author nevertheless treats somewhat advanced topics, which are classic in classical differential geometry but new in the synthetic context. Audience: The book is suitable as an introduction to synthetic differential geometry for students as well as more qualified mathematicians.
Download or read book Synthetic Differential Geometry written by Anders Kock and published by Cambridge University Press. This book was released on 2006-06-22 with total page 245 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book, first published in 2006, details how limit processes can be represented algebraically.
Download or read book Synthetic Differential Topology written by Marta Bunge and published by Cambridge University Press. This book was released on 2018-03-29 with total page 234 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book formally introduces synthetic differential topology, a natural extension of the theory of synthetic differential geometry which captures classical concepts of differential geometry and topology by means of the rich categorical structure of a necessarily non-Boolean topos and of the systematic use of logical infinitesimal objects in it. Beginning with an introduction to those parts of topos theory and synthetic differential geometry necessary for the remainder, this clear and comprehensive text covers the general theory of synthetic differential topology and several applications of it to classical mathematics, including the calculus of variations, Mather's theorem, and Morse theory on the classification of singularities. The book represents the state of the art in synthetic differential topology and will be of interest to researchers in topos theory and to mathematicians interested in the categorical foundations of differential geometry and topology.
Download or read book Synthetic Differential Topology written by Marta Bunge and published by Cambridge University Press. This book was released on 2018-03-29 with total page 234 pages. Available in PDF, EPUB and Kindle. Book excerpt: Represents the state of the art in the new field of synthetic differential topology.
Download or read book Synthetic Geometry of Manifolds written by Anders Kock and published by Cambridge University Press. This book was released on 2010 with total page 317 pages. Available in PDF, EPUB and Kindle. Book excerpt: This elegant book is sure to become the standard introduction to synthetic differential geometry. It deals with some classical spaces in differential geometry, namely 'prolongation spaces' or neighborhoods of the diagonal. These spaces enable a natural description of some of the basic constructions in local differential geometry and, in fact, form an inviting gateway to differential geometry, and also to some differential-geometric notions that exist in algebraic geometry. The presentation conveys the real strength of this approach to differential geometry. Concepts are clarified, proofs are streamlined, and the focus on infinitesimal spaces motivates the discussion well. Some of the specific differential-geometric theories dealt with are connection theory (notably affine connections), geometric distributions, differential forms, jet bundles, differentiable groupoids, differential operators, Riemannian metrics, and harmonic maps. Ideal for graduate students and researchers wishing to familiarize themselves with the field.
Download or read book Foundations of Differentiable Manifolds and Lie Groups written by Frank W. Warner and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 283 pages. Available in PDF, EPUB and Kindle. Book excerpt: Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. Coverage includes differentiable manifolds, tensors and differentiable forms, Lie groups and homogenous spaces, and integration on manifolds. The book also provides a proof of the de Rham theorem via sheaf cohomology theory and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem.
Download or read book Models for Smooth Infinitesimal Analysis written by Ieke Moerdijk and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 401 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book is to construct categories of spaces which contain all the C?-manifolds, but in addition infinitesimal spaces and arbitrary function spaces. To this end, the techniques of Grothendieck toposes (and the logic inherent to them) are explained at a leisurely pace and applied. By discussing topics such as integration, cohomology and vector bundles in the new context, the adequacy of these new spaces for analysis and geometry will be illustrated and the connection to the classical approach to C?-manifolds will be explained.
Download or read book A Comprehensive Course in Analysis written by Barry Simon and published by . This book was released on 2015 with total page 749 pages. Available in PDF, EPUB and Kindle. Book excerpt: A Comprehensive Course in Analysis by Poincar Prize winner Barry Simon is a five-volume set that can serve as a graduate-level analysis textbook with a lot of additional bonus information, including hundreds of problems and numerous notes that extend the text and provide important historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis
Download or read book The Continuous the Discrete and the Infinitesimal in Philosophy and Mathematics written by John L. Bell and published by Springer Nature. This book was released on 2019-09-09 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal.
Download or read book New Foundations for Physical Geometry written by Tim Maudlin and published by . This book was released on 2014-02 with total page 374 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tim Maudlin sets out a completely new method for describing the geometrical structure of spaces, and thus a better mathematical tool for describing and understanding space-time. He presents a historical review of the development of geometry and topology, and then his original Theory of Linear Structures.
Download or read book From Calculus to Cohomology written by Ib H. Madsen and published by Cambridge University Press. This book was released on 1997-03-13 with total page 302 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introductory textbook on cohomology and curvature with emphasis on applications.
Download or read book Elementary Categories Elementary Toposes written by Colin McLarty and published by Clarendon Press. This book was released on 1992-06-04 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book covers elementary aspects of category theory and topos theory. It has few mathematical prerequisites, and uses categorical methods throughout rather than beginning with set theoretic foundations. It works with key notions such as cartesian closedness, adjunctions, regular categories, and the internal logic of a topos. Full statements and elementary proofs are given for the central theorems, including the fundamental theorem of toposes, the sheafification theorem, and the construction of Grothendieck toposes over any topos as base. Three chapters discuss applications of toposes in detail, namely to sets, to basic differential geometry, and to recursive analysis. - ;Introduction; PART I: CATEGORIES: Rudimentary structures in a category; Products, equalizers, and their duals; Groups; Sub-objects, pullbacks, and limits; Relations; Cartesian closed categories; Product operators and others; PART II: THE CATEGORY OF CATEGORIES: Functors and categories; Natural transformations; Adjunctions; Slice categories; Mathematical foundations; PART III: TOPOSES: Basics; The internal language; A soundness proof for topos logic; From the internal language to the topos; The fundamental theorem; External semantics; Natural number objects; Categories in a topos; Topologies; PART IV: SOME TOPOSES: Sets; Synthetic differential geometry; The effective topos; Relations in regular categories; Further reading; Bibliography; Index. -
Download or read book K theory written by Michael Atiyah and published by CRC Press. This book was released on 2018-03-05 with total page 181 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes are based on the course of lectures I gave at Harvard in the fall of 1964. They constitute a self-contained account of vector bundles and K-theory assuming only the rudiments of point-set topology and linear algebra. One of the features of the treatment is that no use is made of ordinary homology or cohomology theory. In fact, rational cohomology is defined in terms of K-theory.The theory is taken as far as the solution of the Hopf invariant problem and a start is mode on the J-homomorphism. In addition to the lecture notes proper, two papers of mine published since 1964 have been reproduced at the end. The first, dealing with operations, is a natural supplement to the material in Chapter III. It provides an alternative approach to operations which is less slick but more fundamental than the Grothendieck method of Chapter III, and it relates operations and filtration. Actually, the lectures deal with compact spaces, not cell-complexes, and so the skeleton-filtration does not figure in the notes. The second paper provides a new approach to K-theory and so fills an obvious gap in the lecture notes.
Download or read book Geometry from a Differentiable Viewpoint written by John McCleary and published by Cambridge University Press. This book was released on 2013 with total page 375 pages. Available in PDF, EPUB and Kindle. Book excerpt: A thoroughly revised second edition of a textbook for a first course in differential/modern geometry that introduces methods within a historical context.
Download or read book Lectures On Advanced Mathematical Methods For Physicists written by N Mukunda and published by World Scientific. This book was released on 2010-04-27 with total page 289 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a survey of Topology and Differential Geometry and also, Lie Groups and Algebras, and their Representations. The first topic is indispensable to students of gravitation and related areas of modern physics (including string theory), while the second has applications in gauge theory and particle physics, integrable systems and nuclear physics.Part I provides a simple introduction to basic topology, followed by a survey of homotopy. Calculus of differentiable manifolds is then developed, and a Riemannian metric is introduced along with the key concepts of connections and curvature. The final chapters lay out the basic notions of simplicial homology and de Rham cohomology as well as fibre bundles, particularly tangent and cotangent bundles.Part II starts with a review of group theory, followed by the basics of representation theory. A thorough description of Lie groups and algebras is presented with their structure constants and linear representations. Root systems and their classifications are detailed, and this section of the book concludes with the description of representations of simple Lie algebras, emphasizing spinor representations of orthogonal and pseudo-orthogonal groups.The style of presentation is succinct and precise. Involved mathematical proofs that are not of primary importance to physics student are omitted. The book aims to provide the reader access to a wide variety of sources in the current literature, in addition to being a textbook of advanced mathematical methods for physicists.
Download or read book A History of Geometrical Methods written by Julian Lowell Coolidge and published by Courier Corporation. This book was released on 2013-02-27 with total page 484 pages. Available in PDF, EPUB and Kindle. Book excerpt: Full and authoritative, this history of the techniques for dealing with geometric questions begins with synthetic geometry and its origins in Babylonian and Egyptian mathematics; reviews the contributions of China, Japan, India, and Greece; and discusses the non-Euclidean geometries. Subsequent sections cover algebraic geometry, starting with the precursors and advancing to the great awakening with Descartes; and differential geometry, from the early work of Huygens and Newton to projective and absolute differential geometry. The author's emphasis on proofs and notations, his comparisons between older and newer methods, and his references to over 600 primary and secondary sources make this book an invaluable reference. 1940 edition.
Download or read book The Geometry of Physics written by Theodore Frankel and published by Cambridge University Press. This book was released on 2011-11-03 with total page 749 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a working knowledge of those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms that are essential for a deeper understanding of both classical and modern physics and engineering. Included are discussions of analytical and fluid dynamics, electromagnetism (in flat and curved space), thermodynamics, the Dirac operator and spinors, and gauge fields, including Yang–Mills, the Aharonov–Bohm effect, Berry phase and instanton winding numbers, quarks and quark model for mesons. Before discussing abstract notions of differential geometry, geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space. The book is ideal for graduate and advanced undergraduate students of physics, engineering or mathematics as a course text or for self study. This third edition includes an overview of Cartan's exterior differential forms, which previews many of the geometric concepts developed in the text.
