The first edition was published in 1988, and much has happened in the field since then. This second edition is adds updates at the end of each chapter, a new appendix reprinting a relevant article from Mathematical Reviews, an enlarged bibliography, and an index. Mathematicians Moore (U. of California-Berkeley) and Schochet (Wayne State U.) based this work on their lecture notes to serve as an introduction to the field and as a reference for students and researchers. In it they develop a variety of aspects of analysis and geometry on foliated spaces applicable to many contexts, dealing with the topics of locally traceable operators, tangential cohomology, transverse measures, characteristics classes, operator algebras, pseudodifferential operators, and the index theorem. Annotation ©2006 Book News, Inc., Portland, OR (booknews.com)
Foliated spaces look locally like products, but their global structure is generally not a product, and tangential differential operators are correspondingly more complex. In the 1980s, Alain Connes founded what is now known as noncommutative geometry. One of the first results was his generalization of the Atiyah-Singer index theorem to compute the analytic index associated with a tangential (pseudo)-differential operator and an invariant transverse measure on a foliated manifold, in terms of topological data on the manifold and the operator. This book presents a complete proof of this beautiful result, generalized to foliated spaces (not just manifolds).
This book presents a complete proof of Connes' Index Theorem generalized to foliated spaces, including coverage of new developments and applications.
Arvustused
Praise for the first edition 'The quest for the proof leads through functional analysis, C^* and von Neumann algebras, topological groupoids, characteristic classes and K-theory along a foliation, and the theory of pseudodifferential operators. It is a long but very rewarding journey and Moore and Schochet have performed a valuable service in putting all this material in one place in an easily readable form The book contains a wealth of information. It is not for those who wish an overview...However, for those wishing a comprehensive proof this book is indispensable.' AMS Bulletin 'This book presents a complete proof of this beautiful result, generalized to foliated spaces (not just manifolds). It includes the necessary background from analysis geometry and topology. This second edition has improved exposition, an updated bibliography, an index, and additional material covering developments and applications since the first edition came out.' L'enseignement mathematique
Muu info
This book presents a complete proof of Connes' Index Theorem generalized to foliated spaces, including coverage of new developments and applications.
| Preface to the Second Edition |
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xi | |
| Preface to the First Edition |
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xiii | |
| Introduction |
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1 | (12) |
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Locally Traceable Operators |
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13 | (18) |
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31 | (24) |
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55 | (20) |
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75 | (34) |
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109 | (20) |
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129 | (38) |
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Pseudodifferential Operators |
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167 | (42) |
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Pseudodifferential Operators |
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169 | (21) |
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Differential Operators and Finite Propagation |
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190 | (7) |
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Dirac Operators and the McKean--Singer Formula |
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197 | (5) |
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Superoperators and the Asymptotic Expansion |
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202 | (7) |
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209 | (16) |
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Appendix A. The ∂-Operator |
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225 | (24) |
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Average Euler Characteristic |
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226 | (3) |
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The ∂-Index Theorem and Riemann-Roch |
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229 | (2) |
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Foliations by Surfaces (Complex Lines or k = 1) |
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231 | (5) |
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236 | (6) |
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Examples of Complex Foliations of Three-Manifolds |
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242 | (7) |
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Appendix B. L2 Harmonic Forms on Noncompact Manifolds |
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249 | (6) |
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Appendix C. Positive Scalar Curvature Along the Leaves |
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255 | (4) |
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259 | (8) |
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| References |
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267 | (13) |
| Notation |
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280 | (5) |
| Index |
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285 | |
Calvin C. Moore received his Ph.D. from Harvard in 1960 under George Mackey in topological groups and their representations. His research interests have extended over time to include ergodic theory, operator algebras, and applications of these to number theory, algebra, and geometry. He spent from 196061 as Postdoc at the University of Chicago and has been on UC Berkeley Mathematics faculty since 1961. He was co-founder (with S. S. Chern and I. M. Singer) of the Mathematical Sciences Research Institute, and has held various administrative posts within the University of California. He is a Fellow of the American Association for the Advancement of Sciences and the American Academy of Arts and Sciences. Claude L. Schochet received his Ph.D. at the University of Chicago under J. P. May, in algebraic topology. His research interests have extended to include operator algebras, foliated spaces, K-theory and non-commutative topology. He taught at Aarhus University (Denmark), Hebrew University (Jerusalem), Indiana University, and has been at WSU since 1976. Since then, he has spent his year long sabbatical leaves at StonyBrook, UCLA, MSRI, U. Maryland, Technion (Haifa, Israel) and has made shorter visits to many other institutions, including Hautes Etudes Sci., University of Copenhagen, and University of California, Berkeley. He has co-authored an AMS Memoir, edited volumes and published many articles. He is a member of the American Mathematical Society, London Mathematical Society, European Mathematical Society, and Israel Mathematics Union.
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