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<!DOCTYPE html><html lang="en"><head><meta charSet="utf-8"/><meta name="viewport" content="initial-scale=1.0, width=device-width"/><title>Basic Noncommutative Geometry | EMS Press</title><meta name="description" content="Basic Noncommutative Geometry, Second edition, by Masoud Khalkhali. Published by EMS Press"/><meta name="author" content="Masoud Khalkhali"/><meta property="og:url" content="https://ems.press/books/elm/225"/><meta property="og:type" content="website"/><meta property="og:title" content="Basic Noncommutative Geometry | EMS Press"/><meta property="og:description" content="Basic Noncommutative Geometry, Second edition, by Masoud Khalkhali. Published by EMS Press"/><meta property="og:image" content="https://content.ems.press/assets/public/images/books/cover-225.png"/><link rel="canonical" href="https://ems.press/books/elm/225"/><meta name="citation_title" content="Basic Noncommutative Geometry"/><meta name="citation_author" content="Masoud Khalkhali"/><meta name="citation_publication_date" content="2013/12/13"/><meta name="citation_issn" content="2523-5176"/><meta name="citation_issn" content="2523-5184"/><meta name="citation_isbn" content="978-3-03719-128-6"/><meta name="citation_isbn" content="978-3-03719-628-1"/><meta name="citation_doi" content="10.4171/128"/><meta name="citation_pdf_url" content="https://ems.press/content/book-files/23588"/><script type="application/ld+json">{"@context":"https://schema.org","@type":["Product","Book"],"name":"Basic Noncommutative Geometry","description":"This text provides an introduction to noncommutative geometry and some of its applications. It can be used either as a textbook for a graduate course or for self-study. It will be useful for graduate students and researchers in mathematics and theoretical physics and all those who are interested in gaining an understanding of the subject. One feature of this book is the wealth of examples and exercises that help the reader to navigate through the subject. While background material is provided in the text and in several appendices, some familiarity with basic notions of functional analysis, algebraic topology, differential geometry and homological algebra at a first year graduate level is helpful.\n\nDeveloped by Alain Connes since the late 1970s, noncommutative geometry has found many applications to long-standing conjectures in topology and geometry and has recently made headways in theoretical physics and number theory. The book starts with a detailed description of some of the most pertinent algebra-geometry correspondences by casting geometric notions in algebraic terms, then proceeds in the second chapter to the idea of a noncommutative space and how it is constructed. The last two chapters deal with homological tools: cyclic cohomology and Connes–Chern characters in K-theory and K-homology, culminating in one commutative diagram expressing the equality of topological and analytic index in a noncommutative setting. Applications to integrality of noncommutative topological invariants are given as well.\n\nTwo new sections have been added to this second edition: one concerns the Gauss–Bonnet theorem and the definition and computation of the scalar curvature of the curved noncommutative two torus, and the second is a brief introduction to Hopf cyclic cohomology. The bibliography has been extended and some new examples are presented.","image":"https://content.ems.press/assets/public/images/books/cover-225.png","isbn":"978-3-03719-128-6","offers":[{"@type":"Offer","price":"49.00","priceCurrency":"EUR","availability":"InStock"},{"@type":"Offer","price":"53.00","priceCurrency":"USD","availability":"InStock"},{"@type":"Offer","price":"49.00","priceCurrency":"EUR","availability":"InStock"},{"@type":"Offer","price":"53.00","priceCurrency":"USD","availability":"InStock"}]}</script><link rel="preload" as="image" imageSrcSet="/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fbooks%2Fcover-225.png&w=256&q=90 256w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fbooks%2Fcover-225.png&w=384&q=90 384w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fbooks%2Fcover-225.png&w=640&q=90 640w, 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12.707 7.414"><path d="M15.586,12l-5.293,5.293L5,12" transform="translate(-3.939 -10.939)" fill="none" stroke="black" stroke-linecap="round" stroke-width="1.5"></path></svg></button><ul class="jsx-494347439 nav-list jsx-24792961 secondary-navigation hidden"><li class="jsx-494347439"><a class="jsx-2412542125 secondary jsx-2462825770" href="/info/authors">Authors</a></li><li class="jsx-494347439"><a class="jsx-2412542125 secondary jsx-2462825770" href="/info/libraries">Libraries</a></li><li class="jsx-494347439"><a class="jsx-2412542125 secondary jsx-2462825770" href="/info/booksellers">Booksellers</a></li></ul></li></ul></div></div></div></nav></header><main class="jsx-2bcaf0ec39f1a89c"><noscript></noscript><article class="jsx-310690786 jsx-2857661604 detail-page"><div class="jsx-310690786 jsx-2857661604 container with-actions with-sections"><div class="jsx-310690786 jsx-2857661604 title"><header class="jsx-2448132927 title undefined"><span class="jsx-2448132927 roofline"><div class="jsx-1179927685 breadcrumbs"><span class="jsx-1179927685 step"><a class="link jsx-2462825770" href="/books">Books</a></span><span class="jsx-1179927685 step"><a class="link jsx-2462825770" href="/books/elm">elm</a></span><span class="jsx-1179927685 label">Monograph</span></div></span><h1 class="jsx-2448132927 primary-title">Basic Noncommutative Geometry</h1><p class="jsx-2448132927 primary-subtitle">Second edition</p></header><div class="jsx-7eb328d3ab251c5 person-group"><div role="group" class="jsx-2097570476 person-group undefined"><ul class="jsx-2097570476 list"><li class="jsx-2097570476 list-item"><section class="jsx-2177874377 person"><header class="jsx-2177874377 title-wrapper"><h3 class="jsx-2177874377 title">Masoud Khalkhali</h3><span class="jsx-2177874377 organisation">The University of Western Ontario, London, Canada</span></header></section></li></ul></div></div></div><nav class="jsx-2370296075 tab-navigation tab-navigation "><ul class="jsx-2370296075 list"><li class="jsx-2370296075"><a class="link active jsx-2462825770" href="/books/elm/225"><svg class="stripes" xmlns="http://www.w3.org/2000/svg" width="70" viewBox="0 0 70 27"><g><rect x="0" y="0" width="100%" height="9" fill="#FDB913"></rect><rect x="0" y="18" width="100%" height="9" fill="#FDB913"></rect></g></svg>Overview</a></li><li class="jsx-2370296075"><a class="link jsx-2462825770" href="/books/elm/225/contents">Contents</a></li><li class="jsx-2370296075"><a class="link jsx-2462825770" href="/books/elm/225/reviews">Reviews</a></li></ul></nav><div class="jsx-310690786 jsx-2857661604 cover"><img alt="Basic Noncommutative Geometry cover" fetchpriority="high" width="1500" height="2117" decoding="async" data-nimg="1" style="color:transparent;width:100%;height:auto" sizes="33vw" srcSet="/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fbooks%2Fcover-225.png&w=256&q=90 256w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fbooks%2Fcover-225.png&w=384&q=90 384w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fbooks%2Fcover-225.png&w=640&q=90 640w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fbooks%2Fcover-225.png&w=750&q=90 750w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fbooks%2Fcover-225.png&w=828&q=90 828w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fbooks%2Fcover-225.png&w=1080&q=90 1080w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fbooks%2Fcover-225.png&w=1200&q=90 1200w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fbooks%2Fcover-225.png&w=1920&q=90 1920w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fbooks%2Fcover-225.png&w=2048&q=90 2048w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fbooks%2Fcover-225.png&w=3840&q=90 3840w" src="/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fbooks%2Fcover-225.png&w=3840&q=90"/><div class="jsx-310690786 jsx-2857661604 actions"><div class="jsx-3326522818 book-actions"><div class="jsx-3326522818 buttons"><a class="button buy-button jsx-1368592022 jsx-2462825770" href="/books/elm/225/buy">Buy from <strong class="jsx-3326522818 price">€49.00</strong></a><a class="button jsx-1554637200 download-button jsx-2661684887 jsx-2462825770" href="/content/book-files/23588?nt=1">Download PDF</a></div><div class="jsx-3326522818 access-info"><p class="jsx-1012291113">A subscription is required to access this book.</p></div></div></div></div><main style="grid-area:main;margin-top:1.8rem"><p>This text provides an introduction to noncommutative geometry and some of its applications. It can be used either as a textbook for a graduate course or for self-study. It will be useful for graduate students and researchers in mathematics and theoretical physics and all those who are interested in gaining an understanding of the subject. One feature of this book is the wealth of examples and exercises that help the reader to navigate through the subject. While background material is provided in the text and in several appendices, some familiarity with basic notions of functional analysis, algebraic topology, differential geometry and homological algebra at a first year graduate level is helpful.</p> <p>Developed by Alain Connes since the late 1970s, noncommutative geometry has found many applications to long-standing conjectures in topology and geometry and has recently made headways in theoretical physics and number theory. The book starts with a detailed description of some of the most pertinent algebra-geometry correspondences by casting geometric notions in algebraic terms, then proceeds in the second chapter to the idea of a noncommutative space and how it is constructed. The last two chapters deal with homological tools: cyclic cohomology and Connes–Chern characters in K-theory and K-homology, culminating in one commutative diagram expressing the equality of topological and analytic index in a noncommutative setting. Applications to integrality of noncommutative topological invariants are given as well.</p> <p>Two new sections have been added to this second edition: one concerns the Gauss–Bonnet theorem and the definition and computation of the scalar curvature of the curved noncommutative two torus, and the second is a brief introduction to Hopf cyclic cohomology. The bibliography has been extended and some new examples are presented.</p></main><aside class="jsx-1279956873 jsx-1604905398 metadata-box"><h2 class="jsx-1279956873 jsx-1604905398">Dates</h2><dl class="jsx-1279956873 jsx-1604905398"><dt class="jsx-1279956873 jsx-1604905398">Published</dt><dd class="jsx-1279956873 jsx-1604905398">13 December 2013</dd></dl><h2 class="jsx-1279956873 jsx-1604905398">Identifiers</h2><dl class="jsx-1279956873 jsx-1604905398"><dt class="jsx-1279956873 jsx-1604905398">DOI</dt><dd class="jsx-1279956873 jsx-1604905398"><a href="https://doi.org/10.4171/128" class="jsx-1279956873 jsx-1604905398">10.4171/128</a></dd><dt class="jsx-1279956873 jsx-1604905398">ISBN print</dt><dd class="jsx-1279956873 jsx-1604905398">978-3-03719-128-6</dd><dt class="jsx-1279956873 jsx-1604905398">ISBN digital</dt><dd class="jsx-1279956873 jsx-1604905398">978-3-03719-628-1</dd></dl><h2 class="jsx-1279956873 jsx-1604905398">Keywords</h2><ul class="jsx-1279956873 jsx-1604905398 keywords"><li class="jsx-1279956873 jsx-1604905398">Noncommutative space</li><li class="jsx-1279956873 jsx-1604905398">noncommutative quotient</li><li class="jsx-1279956873 jsx-1604905398">groupoid</li><li class="jsx-1279956873 jsx-1604905398">cyclic cohomology</li><li class="jsx-1279956873 jsx-1604905398">Connes–Chern character</li><li class="jsx-1279956873 jsx-1604905398">index formula</li></ul><h2 class="jsx-1279956873 jsx-1604905398">Mathematics Subject Classification</h2><ul class="jsx-1279956873 jsx-1604905398 keywords"><li class="jsx-1279956873 jsx-1604905398">58-02</li><li class="jsx-1279956873 jsx-1604905398">58B34</li></ul><h2 class="jsx-1279956873 jsx-1604905398">Print</h2><p class="jsx-1279956873 jsx-1604905398">Softcover, 257 pages, 17cm x 24cm</p><p class="jsx-1279956873 jsx-1604905398 copyright">© EMS Press</p></aside></div></article></main><footer class="jsx-812847572 footer"><div class="jsx-812847572 inner layout-container"><nav title="Footer Navigation" 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imprint of the European Mathematical Society - EMS - Publishing House GmbH,<br class="jsx-812847572"/> a subsidiary of the European Mathematical Society.</p></div></footer></div></div><script id="__NEXT_DATA__" type="application/json">{"props":{"pageProps":{"error":false,"data":{"baseUrl":"https://ems.press","emsSocietyUrl":"https://euromathsoc.org/","requiredConsentVersion":2,"sentryDSN":"https://c18d3379cdda4e76b7b57ad1c555d51c@o1351767.ingest.sentry.io/6634446","sentryEnabled":true,"sentryEnvironment":"production","googleAnalyticsMeasurementID":"G-FR28GMJ82P","isMultiVolumeChild":false,"book":{"type":"books","id":"225","titleTex":"Basic Noncommutative Geometry","subtitleTex":"Second edition","doi":"10.4171/128","isbn":"978-3-03719-128-6","eIsbn":"978-3-03719-628-1","bookType":"monograph","copyright":"© EMS Press","publishedAt":"2013-12-13T00:00:00.000Z","binding":"softcover","pages":257,"printWidthCm":17,"printHeightCm":24,"keywords":["Noncommutative space","noncommutative 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