BOOKS

Topological Groups: Yesterday, Today, Tomorrow

Edited by Sidney A. Morris

MDPI (Basel, Switzerland)

vii+217pp, 2016, ISBN 978-03842-268-6

Photo

In 1900, David Hilbert asked whether each locally euclidean topological group admits a Lie group structure. This was the fifth of his famous 23 questions which foreshadowed much of the mathematical creativity of the twentieth century. It required half a century of effort by several generations of eminent mathematicians until it was settled in the affirmative. These efforts resulted over time in the Peter-Weyl Theorem, the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups, and finally the solution of Hilbert 5 and the structure theory of locally compact groups, through the combined work of Andrew Gleason, Kenkichi Iwasawa, Deane Montgomery, and Leon Zippin. For a presentation of Hilbert 5 see the 2014 book "Hilbert's Fifth Problem and Related Topics" by the winner of a 2006 Fields Medal and 2014 Breakthrough Prize in Mathematics, Terence Tao. It is not possible to describe briefly the richness of the topological group theory and the many directions taken since Hilbert 5. The 900 page reference book in 2013 "The Structure of Compact Groups" by Karl H. Hofmann and Sidney A. Morris, deals with one aspect of compact group theory. There are several books on profinite groups including those written by John S. Wilson (1998) and by Luis Ribes and Pavel Zalesskii (2012). The 2007 book "The Lie Theory of Connected Pro-Lie Groups" by Karl Hofmann and Sidney A. Morris, demonstrates how powerful Lie Theory is in exposing the structure of infinite-dimensional Lie groups. The study of free topological groups initiated by A.A. Markov, M.I. Graev and S. Kakutani, has resulted in a wealth of interesting results, in particular those of A.V. Arkhangel?ski? and many of his former students who developed this topic and its relations with topology. The book "Topological Groups and Related Structures" by Alexander Arkhangel?skii and Mikhail Tkachenko has a diverse content including much material on free topological groups. Compactness conditions in topological groups, especially pseudocompactness as exemplified in the many papers of W.W. Comfort, has been another direction which has proved very fruitful to the present day.



The Lie Theory of Connected Pro-Lie Groups

A Structure Theory for Pro-Lie Algebras,
Pro-Lie Groups, and Connected Locally
Compact Groups

European Mathematical Society Publishing House

xv+678pp, May 2007, ISBN 978-3-03719-032-6

Photo

For further publication information see the Publisher's website
Lie groups were introduced in 1870 by the Norwegian mathematician Sophus Lie. A century later Jean Dieudonné quipped that Lie groups had moved to the center of mathematics and that one cannot undertake anything without them. If a complete topological group G can be approximated by Lie groups in the sense that every identity neighborhood U of G contains a normal subgroup N such that G/N is a Lie group, then it is called a pro-Lie group. Every locally compact connected topological group and every compact group is a pro-Lie group. While the class of locally compact groups is not closed under the formation of arbitrary products, the class of pro-Lie groups is.

For half a century, locally compact pro-Lie groups have drifted through the literature, yet this is the first book which systematically treats the Lie and structure theory of pro-Lie groups irrespective of local compactness. This study fits very well into that current trend which addresses infinite dimensional Lie groups. The results of this text are based on a theory of pro-Lie algebras which parallels the structure theory of finite dimensional real Lie algebras to an astonishing degree even though it has to overcome greater technical obstacles.

This book exposes a Lie theory of connected pro-Lie groups (and hence of connected locally compact groups) and illuminates the manifold ways in which their structure theory reduces to that of compact groups on the one hand and of finite dimensional Lie groups on the other. It is a continuation of the authors' fundamental monograph on the structure of compact groups (1998, 2006), and is an invaluable tool for researchers in topological groups, Lie theory, harmonic analysis and representation theory. It is written to be accessible to advanced graduate students wishing to study this fascinating and important area of current research, which has so many fruitful interactions with other fields of mathematics.

Recent Enhancements
Review in Jahresbericht der Deutschen Mathematiker 111(2009)
The Structure of Compact Groups
A Primer for the Student -
A Handbook for the Expert
3rd Edition, 2013

Photo

The subject matter of compact groups is frequently cited in fields like algebra, topology, functional analysis, and theoretical physics. This book serves the dual purpose of providing a textbook on it for upper level graduate courses or seminars, and of serving as a source book for research specialists who need to apply the structure and representation theory of compact groups.

After a gentle introduction to compact groups and their representation theory, the book presents self-contained courses on linear Lie groups, on compact Lie groups, and on locally compact abelian groups. Separate appended chapters contain the material for courses on abelian groups and on category theory. However, the thrust of the book points in the direction of the structure theory of not necessarily finite dimensional, nor necessarily commutative, compact groups, unfettered by weight restrictions or dimensional bounds. In the process it utilizes infinite dimensional Lie algebras and the exponential function of arbitrary compact groups.

The first edition of 1998 and the second edition of 2006 were well received by reviewers and have been frequently quoted in the areas of instruction and research. For the present new edition the text has been cleaned of typographical flaws and the content has been conceptually sharpened in some places and polished and improved in others. New material has been added to various sections taking into account the progress of research on compact groups both by the authors and other writers. Motivation was provided, among other things, by questions about the structure of compact groups put to the authors by readers through the years following the earlier editions. Accordingly, the authors wished to clarify some aspects of the book which they felt needed improvement. The list of references has increased as the authors included recent publications pertinent to the content of the book.

For further publication information see Publisher's website

Errata and Addenda to 3rd Edition

Errata and Addenda to the previous edition
Pontryagin Duality and the Structure of
Locally Compact Abelian Groups

Review
Abstract Algebra and
Famous Impossibilities

Review



Sidney A. Morris

Pontryagin Duality and the Structure of
Locally Compact Abelian Groups

Cambridge University Press
1977, 136 pp.

ISBN 0 521 21543 9


The book was translated
into Russian and published by
Mir Publishers (Moscow), 1980.

pdf file of book

Review of book
A review of this book which appeared in Acta Scientarium Mathematicarum
"One of the central results in the theory of locally compact abelian groups is the Pontryagin-van Kampen duality theorem which implies that a locally compact abelian group is completely determined by its dual and thus yields a powerful method to study the structure of such groups. Utilizing this fact, the author gives an approach to the structure theory of locally compact abelian groups which proceeds simultaneously with the derivation of the duality theorem. This approach is made possible by a new and simple proof of the duality theorem, which beyond some basic facts from group theory and topology, presupposes only the Peter-Weyl theorem.

First, a concise general introduction to the theory of topological groups, some basic facts concerning subgroups and quotient groups of Rn and concerning uniform spaces are given. Then dual groups are introduced. The duality theorem is proved first for compact and discrete abelian groups and then extended to all locally compact abelian groups. The structure theory of locally compact abelian groups including the Principal Structure Theorem is derived simultaneously. Then some consequences of the duality theorem and applications in diophantine approximations are discussed. The structure theory is further developed by considering its relations to the structure theory of general locally compact groups. At last some important results are given concerning the structure of non-abelian locally compact groups. Each chapter contains a number of stimulating and illustrating exercises, which help to develop the reader's technique.
The author's skill and exceptional knowledge of the subject enabled him to achieve his purpose completely. The lecture note is very clearly and elegantly written and can be recommended as a text for first year graduate courses both by its content and by the educational value of its presentation.
J. Szenthe (Budapest)"



Arthur Jones, Sidney A. Morris,
and Kenneth R. Pearson


Abstract Algebra and Famous Impossibilities

Springer-Verlag Publishers
New York, Berlin etc.

187 pp. 27 figs., Softcover

1st ed. 1991.
ISBN 0-387-97661-2

Corr. 2nd printing 1993.
ISBN 3-540-97661-2

Review of book

The famous problems of squaring the circle, doubling the cube, and trisecting the angle have captured the imagination of both professional and amateur mathematician for over two thousand years. These problems, however, have not yielded to purely geometrical methods. It was only the development of abstract algebra in the nineteenth century which enabled mathematicians to arrive at the surprising conclusion that these constructions are not possible. This text develops the abstract algebra and geometry necessary to prove that these constructions are impossible. It is written at a level suitable for students who have already taken a course in linear algebra and contains thorough discussion of the new concepts along with many illustrative examples. A large number of carefully graded examples are included.

A review of this book which appeared in the Australian Mathematical Society Gazette
This book gives complete proofs that it is not possible, using only straight edge and compass, to trisect an arbitrary angle, double a cube or square a circle. It is intended to be used as a text at about second year level.

The proofs that these geometric constructions are impossible make a good framework for a second year course. The impossibility assertions provide a clear objective which can be stated and understood with only a rudimentary knowledge of geometry. Along the way to this objective students learn some quite sophisticated algebra and calculus.

The authors have taught this topic over a number of years and the benefit of that experience is evident in the text. It is written clearly and great care is taken to explain what each chapter, section, theorem and step is about and how they all fit together to form the proofs of the impossibilities. The material is, necessarily, abstract and involves many proofs but the authors present it in a way which ought to make it accessible to most students. There are exercises after each section for the students to practise what they have just read about. For those students (or instructors) who want to know more, there is at the end of each chapter an inviting list of ‘Additional Reading.

The first chapter is called "Algebraic Preliminaries" and it summarises the basics of fields, rings and vector spaces. Students are assumed to be already familiar with fields and vector spaces. Straight edge and compass constructions are then defined and the proofs of the impossibilities completed. Knowledge of basic integration theory, but not complex analysis, is assumed for the proof that pi is transcendental.

....

This linking, and solving, of problems in geometry, calculus, and the theory of equations, even though they are at first sight quite unrelated, illustrates very well the power of abstract thinking and the unity of mathematics. That relatively modern techniques were required to solve ancient problems shows the historical scope of mathematics. We should be showing students such things and ‘Abstract Algebra and Famous Impossibilities’ does that very clearly.

George Willis (Newcastle)





Updated November 19, 2013
Direct comments and questions to: Sid Morris