From Differential Geometry to Non-commutative Geometry and Topology
This book aims to provide a friendly introduction to non-commutative geometry. It studies index theory from a classical differential geometry perspective up to the point where classical differential geometry methods become insufficient. It then presents n
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From Differential Geometry to Non-commutative Geometry and Topology
From Differential Geometry to Non-commutative Geometry and Topology
Neculai S. Teleman
From Differential Geometry to Non-commutative Geometry and Topology
Neculai S. Teleman Dipartimento di Scienze Matematiche Universit`a Politecnica delle Marche Ancona, Italy
ISBN 978-3-030-28432-9 ISBN 978-3-030-28433-6 (eBook) https://doi.org/10.1007/978-3-030-28433-6 Mathematics Subject Classification (2010): 53-XX © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
This book is a tribute to the memory of Professor Enzo Martinelli, with deep esteem and gratitude. Neculai S. Teleman
Foreword
Neculai S. Teleman takes the reader on a fascinating expedition exploring the lands between the smooth and the continuous domains. For a very long time a continuous function was assumed to be differentiable perhaps with the exception of a finite or numerable but discrete set of points. The Weierstrass example (1872) of a continuous but nowhere differentiable function came as a great shock to mathematicians of the nineteenth century. A similar attitude was prevalent in the middle of the previous century when a topological manifold was considered to carry the unique smooth structure inducing the initial topology of the manifold. Then in 1956 John Milnor showed that on the 7-sphere there are several different smooth exotic structures. But in 1963 Michael F. Atiyah and Isadore M. Singer announced their index theorem which asserted that the index of the Laplace operator (associated with the smooth Riemannian metric) is equal to the topological index of the smooth manifold. Therefore, the index obtained from the
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