• Mathematics

K-theory


Author: Michael Atiyah
Publisher: CRC Press
ISBN: 0429973179
Category: Mathematics
Page: 240
View: 9396
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.

    • Mathematics

The $K$-book

An Introduction to Algebraic $K$-theory
Author: Charles A. Weibel
Publisher: American Mathematical Soc.
ISBN: 0821891324
Category: Mathematics
Page: 618
View: 1202
Informally, $K$-theory is a tool for probing the structure of a mathematical object such as a ring or a topological space in terms of suitably parameterized vector spaces and producing important intrinsic invariants which are useful in the study of algebr

    • Mathematics

Algebraic K-Theory


Author: Vasudevan Srinivas
Publisher: Springer Science & Business Media
ISBN: 0817647392
Category: Mathematics
Page: 341
View: 8841
Algebraic K-Theory has become an increasingly active area of research. With its connections to algebra, algebraic geometry, topology, and number theory, it has implications for a wide variety of researchers and students in mathematics. This book is based on lectures given by the author at the Tata Institute in Bombay and elsewhere. This new edition includes an appendix on algebraic geometry that contains required definitions and results needed to understand the core of the book.

    • Mathematics

Algebraic K-Theory and Its Applications


Author: Jonathan Rosenberg
Publisher: Springer Science & Business Media
ISBN: 1461243149
Category: Mathematics
Page: 394
View: 8004
Algebraic K-Theory is crucial in many areas of modern mathematics, especially algebraic topology, number theory, algebraic geometry, and operator theory. This text is designed to help graduate students in other areas learn the basics of K-Theory and get a feel for its many applications. Topics include algebraic topology, homological algebra, algebraic number theory, and an introduction to cyclic homology and its interrelationship with K-Theory.

    • Mathematics

K-Theory

An Introduction
Author: Max Karoubi
Publisher: Springer Science & Business Media
ISBN: 3540798900
Category: Mathematics
Page: 316
View: 1384
From the Preface: K-theory was introduced by A. Grothendieck in his formulation of the Riemann- Roch theorem. For each projective algebraic variety, Grothendieck constructed a group from the category of coherent algebraic sheaves, and showed that it had many nice properties. Atiyah and Hirzebruch considered a topological analog defined for any compact space X, a group K{X) constructed from the category of vector bundles on X. It is this ''topological K-theory" that this book will study. Topological K-theory has become an important tool in topology. Using K- theory, Adams and Atiyah were able to give a simple proof that the only spheres which can be provided with H-space structures are S1, S3 and S7. Moreover, it is possible to derive a substantial part of stable homotopy theory from K-theory. The purpose of this book is to provide advanced students and mathematicians in other fields with the fundamental material in this subject. In addition, several applications of the type described above are included. In general we have tried to make this book self-contained, beginning with elementary concepts wherever possible; however, we assume that the reader is familiar with the basic definitions of homotopy theory: homotopy classes of maps and homotopy groups.Thus this book might be regarded as a fairly self-contained introduction to a "generalized cohomology theory".

    • Mathematics

An Algebraic Introduction to K-Theory


Author: Bruce A. Magurn
Publisher: Cambridge University Press
ISBN: 9780521800785
Category: Mathematics
Page: 676
View: 4291
An introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra.

    • Mathematics

Modern Classical Homotopy Theory


Author: Jeffrey Strom
Publisher: American Mathematical Soc.
ISBN: 0821852868
Category: Mathematics
Page: 835
View: 8599
The core of classical homotopy theory is a body of ideas and theorems that emerged in the 1950s and was later largely codified in the notion of a model category. This core includes the notions of fibration and cofibration; CW complexes; long fiber and cofiber sequences; loop spaces and suspensions; and so on. Brown's representability theorems show that homology and cohomology are also contained in classical homotopy theory. This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. The exposition is guided by the principle that it is generally preferable to prove topological results using topology (rather than algebra). The language and basic theory of homotopy limits and colimits make it possible to penetrate deep into the subject with just the rudiments of algebra. The text does reach advanced territory, including the Steenrod algebra, Bott periodicity, localization, the Exponent Theorem of Cohen, Moore, and Neisendorfer, and Miller's Theorem on the Sullivan Conjecture. Thus the reader is given the tools needed to understand and participate in research at (part of) the current frontier of homotopy theory. Proofs are not provided outright. Rather, they are presented in the form of directed problem sets. To the expert, these read as terse proofs; to novices they are challenges that draw them in and help them to thoroughly understand the arguments.

    • Business & Economics

Classics in Game Theory


Author: Harold William Kuhn
Publisher: Princeton University Press
ISBN: 0691011923
Category: Business & Economics
Page: 362
View: 4259
A subfield of mathematics and economics, the theory of games simulates situations in which individuals compete and cooperate with each other to hypothesize a conclusion. The contributions collected here are "classics" from the groundbreaking era of research launched in the late 1940s. These 18 essays constitute the core of game theory as it exists today. An invaluable tool for researchers and students of the sciences.

    • Mathematics

K-theory


Author: Michael Atiyah
Publisher: CRC Press
ISBN: 0429973179
Category: Mathematics
Page: 240
View: 8664
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.

    • Science

Quantum Many-particle Systems


Author: John W. Negele
Publisher: CRC Press
ISBN: 0429966474
Category: Science
Page: 476
View: 7053
This book explains the fundamental concepts and theoretical techniques used to understand the properties of quantum systems having large numbers of degrees of freedom. A number of complimentary approaches are developed, including perturbation theory; nonperturbative approximations based on functional integrals; general arguments based on order parameters, symmetry, and Fermi liquid theory; and stochastic methods.

    • Science

Quantum Electrodynamics


Author: Richard P. Feynman
Publisher: CRC Press
ISBN: 0429972873
Category: Science
Page: 212
View: 306
This classic work presents the main results and calculational procedures of quantum electrodynamics in a simple and straightforward way. Designed for the student of experimental physics who does not intend to take more advanced graduate courses in theoretical physics, the material consists of notes on the third of a three-semester course given at the California Institute of Technology.

    • Literary Criticism

Chronoschisms

Time, Narrative, and Postmodernism
Author: Ursula K. Heise
Publisher: Cambridge University Press
ISBN: 9780521555449
Category: Literary Criticism
Page: 286
View: 1234
An analysis of the way postmodern novels respond to changes in the experience of time.

    • History

Theory for Classics

A Student's Guide
Author: Louise Hitchcock
Publisher: Taylor & Francis
ISBN: 0415454972
Category: History
Page: 213
View: 1488
This student's guide is a clear and concise handbook to the key connections between Classical Studies and critical theory in the twentieth century. Louise Hitchcock looks at the way Classics has been engaged across a number of disciplines. Beginning with four foundational figures - Freud, Marx, Nietzshe and Saussure - Hitchcock goes on to provide guided introductions of the major theoretical thinkers of the past century, from Adorno to Williams. Each entry offers biographical, theoretical and bibliographical information along with a discussion of each figure's relevance to Classical Studies and suggestions for future research. Theory for Classics, adapted from Theory for Religious Studies, by William E. Deal and Timothy K. Beal, is a brisk, thoughtful, provocative, and engaging title, which will be an essential first volume for anyone interested in the intersection between theory and classical studies today.

    • Literary Collections

Theory of Literature and Other Critical Writings


Author: Sōseki Natsume,Michael K. Bourdaghs,Atsuko Ueda
Publisher: Columbia University Press
ISBN: 9780231146562
Category: Literary Collections
Page: 287
View: 1399
"The Theory of Literature foreshadows the ideas and concepts that would later form the critical foundations of formalism, structuralism, reader-response theory, cognitive science, and postcolonialism. It remains an unprecedented work of literary theory, unmistakably modern yet also clearly (and self-consciously) non-Western. In a later series of lectures and essays, Soseki continued to develop his ideas. This material, some of it never before translated into English, is also included in the volume. The editors offer a critical introduction that contextualizes Soseki's theoretical project historically and explores its contemporary legacy."--BOOK JACKET.

    • Mathematics

Simplicial Homotopy Theory


Author: Paul G. Goerss,John F. Jardine
Publisher: Birkhäuser
ISBN: 3034887078
Category: Mathematics
Page: 510
View: 5394

    • Mathematics

Complex Topological K-Theory


Author: Efton Park
Publisher: Cambridge University Press
ISBN: 1139469746
Category: Mathematics
Page: N.A
View: 9744
Topological K-theory is a key tool in topology, differential geometry and index theory, yet this is the first contemporary introduction for graduate students new to the subject. No background in algebraic topology is assumed; the reader need only have taken the standard first courses in real analysis, abstract algebra, and point-set topology. The book begins with a detailed discussion of vector bundles and related algebraic notions, followed by the definition of K-theory and proofs of the most important theorems in the subject, such as the Bott periodicity theorem and the Thom isomorphism theorem. The multiplicative structure of K-theory and the Adams operations are also discussed and the final chapter details the construction and computation of characteristic classes. With every important aspect of the topic covered, and exercises at the end of each chapter, this is the definitive book for a first course in topological K-theory.

    • Science

Einstein's Theory of Relativity


Author: Max Born
Publisher: Courier Corporation
ISBN: 0486142124
Category: Science
Page: 400
View: 8384
Semi-technical account includes a review of classical physics (origin of space and time measurements, Ptolemaic and Copernican astronomy, laws of motion, inertia, more) and of Einstein's theories of relativity.

    • Mathematics

Characteristic Classes. (AM-76)


Author: John Milnor,James D. Stasheff
Publisher: Princeton University Press
ISBN: 140088182X
Category: Mathematics
Page: 340
View: 7626
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.

    • Science

Advanced MR Neuroimaging

From Theory to Clinical Practice
Author: Ioannis Tsougos
Publisher: CRC Press
ISBN: 135121652X
Category: Science
Page: 221
View: 9927
Over the last decade, some of the greatest achievements in the field of neuroimaging have been related to remarkable advances in magnetic resonance techniques, including diffusion, perfusion, magnetic resonance spectroscopy, and functional MRI. Such techniques have provided valuable insights into tissue microstructure, microvasculature, metabolism and brain connectivity. Previously available mostly in research environments, these techniques are now becoming part of everyday clinical practice in a plethora of clinical MR systems. Nevertheless, despite growing interest and wider acceptance, there remains a lack of a comprehensive body of knowledge on the subject, exploring the intrinsic complexity and physical difficulty of the techniques. This book focuses on the basic principles and theories of diffusion, perfusion, magnetic resonance spectroscopy, and functional MRI. It also explores their clinical applications and places emphasis on the associated artifacts and pitfalls with a comprehensive and didactic approach. This book aims to bridge the gap between research applications and clinical practice. It will serve as an educational manual for neuroimaging researchers and radiologists, neurologists, neurosurgeons, and physicists with an interest in advanced MR techniques. It will also be a useful reference text for experienced clinical scientists who wish to optimize their multi-parametric imaging approach.