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Characteristic Classes

Characteristic Classes PDF
Author: John Willard Milnor
Publisher: Princeton University Press
ISBN: 9780691081229
Size: 54.27 MB
Format: PDF, Docs
Category : Mathematics
Languages : en
Pages : 330
View: 3597

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The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds. In this volume, the authors provide a thorough introduction to characteristic classes, with detailed studies of Stiefel-Whitney classes, Chern classes, Pontrjagin classes, and the Euler class. Three appendices cover the basics of cohomology theory and the differential forms approach to characteristic classes, and provide an account of Bernoulli numbers. Based on lecture notes of John Milnor, which first appeared at Princeton University in 1957 and have been widely studied by graduate students of topology ever since, this published version has been completely revised and corrected.

Geometry Of Characteristic Classes

Geometry of Characteristic Classes PDF
Author: Shigeyuki Morita
Publisher: American Mathematical Soc.
ISBN: 0821821393
Size: 29.75 MB
Format: PDF, ePub, Mobi
Category : Mathematics
Languages : en
Pages : 185
View: 6071

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This is an inexpensive paper volume that will appeal to upper level students. Professor Morita is a world-class authority on this topic.

Loop Spaces Characteristic Classes And Geometric Quantization

Loop Spaces  Characteristic Classes and Geometric Quantization PDF
Author: Jean-Luc Brylinski
Publisher: Springer Science & Business Media
ISBN: 0817647317
Size: 77.77 MB
Format: PDF, Mobi
Category : Mathematics
Languages : en
Pages : 302
View: 2955

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This book examines the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Applications presented in the book involve anomaly line bundles on loop spaces and anomaly functionals, central extensions of loop groups, Kähler geometry of the space of knots, and Cheeger--Chern--Simons secondary characteristics classes. It also covers the Dirac monopole and Dirac’s quantization of the electrical charge.

Symplectic Geometry And Secondary Characteristic Classes

Symplectic Geometry and Secondary Characteristic Classes PDF
Author: Izu Vaisman
Publisher: Springer Science & Business Media
ISBN: 1475719604
Size: 16.43 MB
Format: PDF, Docs
Category : Mathematics
Languages : en
Pages : 216
View: 4142

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The present work grew out of a study of the Maslov class (e. g. (37]), which is a fundamental invariant in asymptotic analysis of partial differential equations of quantum physics. One of the many in terpretations of this class was given by F. Kamber and Ph. Tondeur (43], and it indicates that the Maslov class is a secondary characteristic class of a complex trivial vector bundle endowed with a real reduction of its structure group. (In the basic paper of V. I. Arnold about the Maslov class (2], it is also pointed out without details that the Maslov class is characteristic in the category of vector bundles mentioned pre viously. ) Accordingly, we wanted to study the whole range of secondary characteristic classes involved in this interpretation, and we gave a short description of the results in (83]. It turned out that a complete exposition of this theory was rather lengthy, and, moreover, I felt that many potential readers would have to use a lot of scattered references in order to find the necessary information from either symplectic geometry or the theory of the secondary characteristic classes. On the otherhand, both these subjects are of a much larger interest in differential geome try and topology, and in the applications to physical theories.

Characteristic Classes Am 76 Volume 76

Characteristic Classes   AM 76   Volume 76 PDF
Author: John Milnor
Publisher: Princeton University Press
ISBN: 140088182X
Size: 30.91 MB
Format: PDF
Category : Mathematics
Languages : en
Pages : 340
View: 1592

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The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds. In this volume, the authors provide a thorough introduction to characteristic classes, with detailed studies of Stiefel-Whitney classes, Chern classes, Pontrjagin classes, and the Euler class. Three appendices cover the basics of cohomology theory and the differential forms approach to characteristic classes, and provide an account of Bernoulli numbers. Based on lecture notes of John Milnor, which first appeared at Princeton University in 1957 and have been widely studied by graduate students of topology ever since, this published version has been completely revised and corrected.

Nilpotent Orbits Primitive Ideals And Characteristic Classes

Nilpotent Orbits  Primitive Ideals  and Characteristic Classes PDF
Author: Walter Borho
Publisher: Springer Science & Business Media
ISBN: 1461245583
Size: 23.58 MB
Format: PDF, Kindle
Category : Mathematics
Languages : en
Pages : 134
View: 1926

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1. The Subject Matter. Consider a complex semisimple Lie group G with Lie algebra g and Weyl group W. In this book, we present a geometric perspective on the following circle of ideas: polynomials The "vertices" of this graph are some of the most important objects in representation theory. Each has a theory in its own right, and each has had its own independent historical development. - A nilpotent orbit is an orbit of the adjoint action of G on g which contains the zero element of g in its closure. (For the special linear group 2 G = SL(n,C), whose Lie algebra 9 is all n x n matrices with trace zero, an adjoint orbit consists of all matrices with a given Jordan canonical form; such an orbit is nilpotent if the Jordan form has only zeros on the diagonal. In this case, the nilpotent orbits are classified by partitions of n, given by the sizes of the Jordan blocks.) The closures of the nilpotent orbits are singular in general, and understanding their singularities is an important problem. - The classification of irreducible Weyl group representations is quite old.

Differential Geometry

Differential Geometry PDF
Author: Loring W. Tu
Publisher: Springer
ISBN: 3319550845
Size: 12.81 MB
Format: PDF
Category : Mathematics
Languages : en
Pages : 347
View: 1114

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This text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text. Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to make the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior power are included. Differential geometry, as its name implies, is the study of geometry using differential calculus. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. It is not too far-fetched to argue that differential geometry should be in every mathematician's arsenal.

From Calculus To Cohomology

From Calculus to Cohomology PDF
Author: Ib H. Madsen
Publisher: Cambridge University Press
ISBN: 9780521589567
Size: 41.76 MB
Format: PDF, Mobi
Category : Mathematics
Languages : en
Pages : 286
View: 7591

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An introductory textbook on cohomology and curvature with emphasis on applications.

Geometry Topology And Physics Second Edition

Geometry  Topology and Physics  Second Edition PDF
Author: Mikio Nakahara
Publisher: CRC Press
ISBN: 9780750306065
Size: 35.18 MB
Format: PDF, ePub, Docs
Category : Mathematics
Languages : en
Pages : 596
View: 4516

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Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields. The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view. Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics.

A Concise Course In Algebraic Topology

A Concise Course in Algebraic Topology PDF
Author: J. P. May
Publisher: University of Chicago Press
ISBN: 9780226511832
Size: 73.26 MB
Format: PDF, Kindle
Category : Mathematics
Languages : en
Pages : 243
View: 316

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Algebraic topology is a basic part of modern mathematics, and some knowledge of this area is indispensable for any advanced work relating to geometry, including topology itself, differential geometry, algebraic geometry, and Lie groups. This book provides a detailed treatment of algebraic topology both for teachers of the subject and for advanced graduate students in mathematics either specializing in this area or continuing on to other fields. J. Peter May's approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. The final four chapters provide sketches of substantial areas of algebraic topology that are normally omitted from introductory texts, and the book concludes with a list of suggested readings for those interested in delving further into the field.