Analysis and Mathematical Physics

243 Pages·2017·4.772 MB·English

by  Shaun Bullett, Tom Fearn, Frank Smith

Analysis and
Mathematical Physics

Q0029_9781786340986_TP.indd 1 29/11/16 5:39 PM
LTCC Advanced Mathematics Series
Series Editors: Shaun Bullett (Queen Mary University of London, UK)
Tom Fearn (University College London, UK)
Frank Smith (University College London, UK)
Published
Vol. 1 Advanced Techniques in Applied Mathematics
edited by Shaun Bullett, Tom Fearn & Frank Smith
Vol. 2 Fluid and Solid Mechanics
edited by Shaun Bullett, Tom Fearn & Frank Smith
Vol. 3 Algebra, Logic and Combinatorics
edited by Shaun Bullett, Tom Fearn & Frank Smith
Vol. 5 Dynamical and Complex Systems
edited by Shaun Bullett, Tom Fearn & Frank Smith
Vol. 6 Analysis and Mathematical Physics
edited by Shaun Bullett, Tom Fearn & Frank Smith
Forthcoming
Vol. 4 Geometry in Advanced Pure Mathematics
edited by Shaun Bullett, Tom Fearn & Frank Smith

Vishnu Mohan - Analysis and Mathematical Physics.indd 1 28-11-16 8:56:25 AM
LTCC
Advanced Mathematics Series — Volume 6
Analysis and
Mathematical Physics
Editors
Shaun Bullett
Queen Mary University of London, UK
Tom Fearn
University College London, UK
Frank Smith
University College London, UK

World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
Q0029_9781786340986_TP.indd 2 29/11/16 5:39 PM
Published by
World Scientific Publishing Europe Ltd.
57 Shelton Street, Covent Garden, London WC2H 9HE
Head office: 5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
Library of Congress Cataloging-in-Publication Data
Names: Bullett, Shaun, 1947– editor. | Fearn, T., 1949– editor. |
Smith, F. T. (Frank T.), 1948– editor.
Title: Analysis and mathematical physics / edited by Shaun Bullett (Queen Mary University of
London, UK), Tom Fearn (University College London, UK),
Frank Smith (University College London, UK).
Other titles: Analysis and mathematical physics (Hackensack, N.J.)
Description: [Hackensack, N.J.] : World Scientific, 2017. | Series: LTCC advanced
mathematics series ; volume 6 | Includes bibliographical references.
Identifiers: LCCN 2016036916 | ISBN 9781786340986 (hc : alk. paper)
Subjects: LCSH: Mathematical analysis. | Mathematical physics. | Geometry, Differential.
Classification: LCC QC20.7.A5 A534 2017 | DDC 530.15--dc23
LC record available at https://lccn.loc.gov/2016036916
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2017 by World Scientific Publishing Europe Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy
is not required from the publisher.
Desk Editors: V. Vishnu Mohan/Mary Simpson
Typeset by Stallion Press
Email: [email protected]
Printed in Singapore
Vishnu Mohan - Analysis and Mathematical Physics.indd 2 28-11-16 8:56:25 AM
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Chapter 2 was contributed by Professor Yuri Safarov. Yuri was highly
regarded as a teacher, researcher and colleague. We dedicate this volume
as a tribute to his memory.

v
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Preface
The London Taught Course Centre(LTCC) for PhD students in the Math-
ematical Sciences has the objective of introducing research students to a
broad range of topics. For some students, some of these topics might be of
obvious relevance to their PhD projects, but the relevance of most will be
muchlessobviousorapparentlynon-existent.Howeverallofus involvedin
mathematical research have experienced that extraordinary moment when
thepennydropsandsometinygemofinformationfromoutsideonesimme-
diateresearchfieldturnsouttobethekeytounravellingaseeminglyinsol-
uble problem, or to opening up a new vista of mathematical structure. By
offering our students advanced introductions to a range of different areas
of mathematics, we hope to open their eyes to new possibilities that they
might not otherwise encounter.
Each volume in this series consists of chapters on a group of related
themes, based on modules taught at the LTCC by their authors. These
modules were already short (five two-hour lectures) and in most cases the
lecture notes here are even shorter, covering perhaps three-quarters of the

content of the original LTCC course. This brevity was quite deliberate on
the part of the editors — we asked the authors to confine themselves to
around35pagesineachchapter,inordertoallowasmanytopicsaspossible
to be included in each volume, while keeping the volumes digestible. The
chaptersare“advancedintroductions”,andreaderswhowishtolearnmore
are encouragedto continue elsewhere. There has been no attempt to make
the coverage of topics comprehensive. That would be impossible in any
case — any book or series of books which included all that a PhD student
in mathematics might need to know would be so large as to be totally
unreadable.Insteadwhatwepresentinthisseriesisacross-sectionofsome
vii
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viii Preface
ofthetopics,bothclassicalandnew,thathaveappearedinLTCCmodules
in the nine years since it was founded.
Thepresentvolumecoverstopicsinanalysisandmathematicalphysics.
The main readers are likely to be graduate students and more experienced
researchersinthe mathematicalsciences,lookingforintroductionstoareas
with which they are unfamiliar. The mathematics presented is intended to
be accessibleto firstyearPhDstudents,whatevertheir specialisedareasof
research. Whatever your mathematical background, we encourage you to
dive in, and we hope that you will enjoy the experience of widening your
mathematical knowledge by reading these concise introductory accounts
written by experts at the forefront of current research.
Shaun Bullett, Tom Fearn, Frank Smith
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Contents
Preface vii
Chapter 1. Differential Geometry and Mathematical Physics 1
Andrew Hone and Steffen Krusch
Chapter 2. Distributions, Fourier Transforms and Microlocal
Analysis 41
Yuri Safarov
Chapter 3. C*-algebras 73
Cho-Ho Chu
Chapter 4. Special Functions 109
Rod Halburd
Chapter 5. Non-commutative Differential Geometry 139
Shahn Majid

Chapter 6. Mathematical Problems of General Relativity 177
Juan A. Valiente Kroon
Chapter 7. Value Distribution of Meromorphic Functions 209
Rod Halburd
ix
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Chapter 1
Differential Geometry
and Mathematical Physics
Andrew Hone and Steffen Krusch
School of Mathematics, Statistics & Actuarial Science
University of Kent, Canterbury CT2 7NF, UK
The chapter will illustrate how concepts in differential geometry arise
naturally in different areas of mathematical physics. We will describe
manifolds, fibre bundles, (co)tangent bundles, metrics and symplectic
structures,and theirapplications to Lagrangian mechanics, field theory
and Hamiltonian systems, including various examples related to inte-
grable systems and topological solitons.
1. Manifolds
Manifolds are a central concept in mathematics and have natural applica-
tions to problems in physics. Here we provide an example-led introduction
to manifolds and introduce important additional structures.
Asimple,yetnon-trivial,exampleofamanifoldisthe2-sphereS2,given
by the set of points
(cid:1) (cid:3)
(cid:2)3
S2 = (x ,x ,x ): x2 =1 .
1 2 3 i
i=1
Often, we label the points on the sphere by polar coordinates:
x = cosφsinθ, x = sinφsinθ, x = cosθ,
1 2 3
where 0≤φ<2π, 0≤θ ≤π.
However, there is the following problem: we cannot label S2 with a single
coordinate system such that nearby points have nearby coordinates, and
every point has unique coordinates.
1