Index Theory Beyond The Fredholm Case
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Index Theory Beyond The Fredholm Case
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Author : Alan Carey
language : en
Publisher: Springer Nature
Release Date : 2022-11-30
Index Theory Beyond The Fredholm Case written by Alan Carey and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-11-30 with Mathematics categories.
This book is about extending index theory to some examples where non-Fredholm operators arise. It focuses on one aspect of the problem of what replaces the notion of spectral flow and the Fredholm index when the operators in question have zero in their essential spectrum. Most work in this topic stems from the so-called Witten index that is discussed at length here. The new direction described in these notes is the introduction of `spectral flow beyond the Fredholm case'. Creating a coherent picture of numerous investigations and scattered notions of the past 50 years, this work carefully introduces spectral flow, the Witten index and the spectral shift function and describes their relationship. After the introduction, Chapter 2 carefully reviews Double Operator Integrals, Chapter 3 describes the class of so-called p-relative trace class perturbations, followed by the construction of Krein's spectral shift function in Chapter 4. Chapter 5 reviews the analytic approach to spectral flow, culminating in Chapter 6 in the main abstract result of the book, namely the so-called principal trace formula. Chapter 7 completes the work with illustrations of the main results using explicit computations on two examples: the Dirac operator in Rd, and a differential operator on an interval. Throughout, attention is paid to the history of the subject and earlier references are provided accordingly. The book is aimed at experts in index theory as well as newcomers to the field.
Spectral Flow
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Author : Nora Doll
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2023-06-19
Spectral Flow written by Nora Doll and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-06-19 with Mathematics categories.
This is the first treatment entirely dedicated to an analytic study of spectral flow for paths of selfadjoint Fredholm operators, possibly unbounded or understood in a semifinite sense. The importance of spectral flow for homotopy and index theory is discussed in detail. Applications concern eta-invariants, the Bott-Maslov and Conley-Zehnder indices, Sturm-Liouville oscillation theory, the spectral localizer and bifurcation theory.
Index Theory In Von Neumann Algebras
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Author : Catherine Louise Olsen
language : en
Publisher: American Mathematical Soc.
Release Date : 1984
Index Theory In Von Neumann Algebras written by Catherine Louise Olsen and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984 with Mathematics categories.
The object of this paper is to define a natural analytic index function on an arbitrary von Neumann algebra relative to an arbitrary ideal. This index map enables us to develop a complete Fredholm and semi-Fredholm theory in this setting which is parallel to classical Fredholm and semi-Fredholm theory.
Geometric And Topological Methods For Quantum Field Theory Proceedings Of The Summer School
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Author : Alexander Cardona
language : en
Publisher: World Scientific
Release Date : 2003-03-21
Geometric And Topological Methods For Quantum Field Theory Proceedings Of The Summer School written by Alexander Cardona and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-03-21 with Mathematics categories.
This volume offers an introduction to recent developments in several active topics of research at the interface between geometry, topology and quantum field theory. These include Hopf algebras underlying renormalization schemes in quantum field theory, noncommutative geometry with applications to index theory on one hand and the study of aperiodic solids on the other, geometry and topology of low dimensional manifolds with applications to topological field theory, Chern-Simons supergravity and the anti de Sitter/conformal field theory correspondence. It comprises seven lectures organized around three main topics, noncommutative geometry, topological field theory, followed by supergravity and string theory, complemented by some short communications by young participants of the school.
Index Theory Of Elliptic Operators Foliations And Operator Algebras
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Author : Jerome Kaminker
language : en
Publisher: American Mathematical Soc.
Release Date : 1988
Index Theory Of Elliptic Operators Foliations And Operator Algebras written by Jerome Kaminker and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1988 with Mathematics categories.
Combining analysis, geometry, and topology, this volume provides an introduction to current ideas involving the application of $K$-theory of operator algebras to index theory and geometry. In particular, the articles follow two main themes: the use of operator algebras to reflect properties of geometric objects and the application of index theory in settings where the relevant elliptic operators are invertible modulo a $C^*$-algebra other than that of the compact operators. The papers in this collection are the proceedings of the special sessions held at two AMS meetings: the Annual meeting in New Orleans in January 1986, and the Central Section meeting in April 1986. Jonathan Rosenberg's exposition supplies the best available introduction to Kasparov's $KK$-theory and its applications to representation theory and geometry. A striking application of these ideas is found in Thierry Fack's paper, which provides a complete and detailed proof of the Novikov Conjecture for fundamental groups of manifolds of non-positive curvature. Some of the papers involve Connes' foliation algebra and its $K$-theory, while others examine $C^*$-algebras associated to groups and group actions on spaces.
The Localization Problem In Index Theory Of Elliptic Operators
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Author : Vladimir Nazaikinskii
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-11-26
The Localization Problem In Index Theory Of Elliptic Operators written by Vladimir Nazaikinskii and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-26 with Mathematics categories.
The book deals with the localization approach to the index problem for elliptic operators. Localization ideas have been widely used for solving various specific index problems for a long time, but the fact that there is actually a fundamental localization principle underlying all these solutions has mostly passed unnoticed. The ignorance of this general principle has often necessitated using various artificial tricks and hindered the solution of new important problems in index theory. So far, the localization principle has been only scarcely covered in journal papers and not covered at all in monographs. The suggested book is intended to fill the gap. So far, it is the first and only monograph dealing with the topic. Both the general localization principle and its applications to specific problems, existing and new, are covered. The book will be of interest to working mathematicians as well as graduate and postgraduate university students specializing in differential equations and related topics.
Higher Index Theory
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Author : Rufus Willett
language : en
Publisher: Cambridge University Press
Release Date : 2020-07-02
Higher Index Theory written by Rufus Willett and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-07-02 with Mathematics categories.
A friendly introduction to higher index theory, a rapidly-developing subject at the intersection of geometry, topology and operator algebras. A well-balanced combination of introductory material (with exercises), cutting-edge developments and references to the wider literature make this book a valuable guide for graduate students and experts alike.
Spectral Theory And Mathematical Physics
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Author : Marius Mantoiu
language : en
Publisher: Birkhäuser
Release Date : 2016-06-30
Spectral Theory And Mathematical Physics written by Marius Mantoiu and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-06-30 with Mathematics categories.
The present volume contains the Proceedings of the International Conference on Spectral Theory and Mathematical Physics held in Santiago de Chile in November 2014. Main topics are: Ergodic Quantum Hamiltonians, Magnetic Schrödinger Operators, Quantum Field Theory, Quantum Integrable Systems, Scattering Theory, Semiclassical and Microlocal Analysis, Spectral Shift Function and Quantum Resonances. The book presents survey articles as well as original research papers on these topics. It will be of interest to researchers and graduate students in Mathematics and Mathematical Physics.
Surveys In Noncommutative Geometry
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Author : Nigel Higson
language : en
Publisher: American Mathematical Soc.
Release Date : 2006
Surveys In Noncommutative Geometry written by Nigel Higson and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Mathematics categories.
In June 2000, the Clay Mathematics Institute organized an Instructional Symposium on Noncommutative Geometry in conjunction with the AMS-IMS-SIAM Joint Summer Research Conference. These events were held at Mount Holyoke College in Massachusetts from June 18 to 29, 2000. The Instructional Symposium consisted of several series of expository lectures which were intended to introduce key topics in noncommutative geometry to mathematicians unfamiliar with the subject. Those expository lectures have been edited and are reproduced in this volume. The lectures of Rosenberg and Weinberger discuss various applications of noncommutative geometry to problems in ordinary geometry and topology. The lectures of Lagarias and Tretkoff discuss the Riemann hypothesis and the possible application of the methods of noncommutative geometry in number theory. Higson gives an account of the residue index theorem of Connes and Moscovici. Noncommutative geometry is to an unusual extent the creation of a single mathematician, Alain Connes. The present volume gives an extended introduction to several aspects of Connes' work in this fascinating area.
New Trends In Approximation Theory
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Author : Javad Mashreghi
language : en
Publisher: Springer
Release Date : 2018-03-28
New Trends In Approximation Theory written by Javad Mashreghi and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-03-28 with Mathematics categories.
The international conference entitled "New Trends in Approximation Theory" was held at the Fields Institute, in Toronto, from July 25 until July 29, 2016. The conference was fondly dedicated to the memory of our unique friend and colleague, André Boivin, who gave tireless service in Canada until his very last moment of his life in October 2014. The impact of his warm personality and his fine work on Complex Approximation Theory was reflected by the mathematical excellence and the wide research range of the 37 participants. In total there were 27 talks, delivered by well-established mathematicians and young researchers. In particular, 19 invited lectures were delivered by leading experts of the field, from 8 different countries. The wide variety of presentations composed a mosaic of aspects of approximation theory, highlighting interesting connections with important contemporary areas of Analysis. Primary topics discussed include application of approximation theory (isoperimetric inequalities, construction of entire order-isomorphisms, dynamical sampling); approximation by harmonic and holomorphic functions (especially uniform and tangential approximation), polynomial and rational approximation; zeros of approximants and zero-free approximation; tools used in approximation theory; approximation on complex manifolds, in product domains, and in function spaces; and boundary behaviour and universality properties of Taylor and Dirichlet series.