Characteristic Classes

Characteristic Classes PDF
Author: John Willard Milnor
Publisher: Princeton University Press
Category : Mathematics
Languages : en
Pages : 330
View: 4744

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Book Description:
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.
Category : Mathematics
Languages : en
Pages : 185
View: 6722

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


Symplectic Geometry And Secondary Characteristic Classes

Symplectic Geometry and Secondary Characteristic Classes PDF
Author: Izu Vaisman
Publisher: Springer Science & Business Media
Category : Mathematics
Languages : en
Pages : 216
View: 3310

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Book Description:
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.


Loop Spaces Characteristic Classes And Geometric Quantization

Loop Spaces  Characteristic Classes and Geometric Quantization PDF
Author: Jean-Luc Brylinski
Publisher: Springer Science & Business Media
Category : Mathematics
Languages : en
Pages : 302
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Book Description:
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.


Nilpotent Orbits Primitive Ideals And Characteristic Classes

Nilpotent Orbits  Primitive Ideals  and Characteristic Classes PDF
Author: Walter Borho
Publisher: Springer Science & Business Media
Category : Mathematics
Languages : en
Pages : 134
View: 4958

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Book Description:
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.


Algorithms To Compute Characteristic Classes

Algorithms to Compute Characteristic Classes PDF
Author: Martin Helmer
Publisher:
Category :
Languages : en
Pages : 370
View: 2280

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Book Description:
In this thesis we develop several new algorithms to compute characteristics classes in a variety of settings. In addition to algorithms for the computation of the Euler characteristic, a classical topological invariant, we also give algorithms to compute the Segre class and Chern-Schwartz-MacPherson (CSM) class. These invariants can in turn be used to compute other common invariants such as the Chern-Fulton class (or the Chern class in smooth cases). We begin with subschemes of a projective space over an algebraically closed field of characteristic zero. In this setting we give effective algorithms to compute the CSM class, Segre class and the Euler characteristic. The algorithms can be implemented using either symbolic or numerical methods. The algorithms are based on a new method for calculating the projective degrees of a rational map defined by a homogeneous ideal. Running time bounds are given for these algorithms and the algorithms are found to perform favourably compared to other applicable algorithms. Relations between our algorithms and other existing algorithms are explored. In the special case of a complete intersection subcheme we develop a second algorithm to compute CSM classes and Euler characteristics in a more direct and efficient manner. Each of these algorithms are generalized to subschemes of a product of projective spaces. Running time bounds for the generalized algorithms to compute the CSM class, Segre class and the Euler characteristic are given. Our Segre class algorithm is tested in comparison to another applicable algorithm and is found to perform favourably. To the best of our knowledge there are no other algorithms in the literature which compute the CSM class and Euler characteristic in the multi-projective setting. For complete simplical toric varieties defined by a fan we give a strictly combinatorial algorithm to compute the CSM class and Euler characteristic and a second combinatorial algorithm with reduced running time to compute only the Euler characteristic. We also prove several Bezout type bounds in multi-projective space. An application of these bounds to obtain a sharper degree bound on a certain system with a natural bi-projective structure is demonstrated.


From Calculus To Cohomology

From Calculus to Cohomology PDF
Author: Ib H. Madsen
Publisher: Cambridge University Press
Category : Mathematics
Languages : en
Pages : 286
View: 3312

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Book Description:
An introductory textbook on cohomology and curvature with emphasis on applications.