available from August 2021
FORMAT: Hardback ISBN: 9781316519134
FORMAT: Paperback ISBN: 9781009001625
DIMENSIONS: 244 x 170
Often it is more instructive to know 'what can go wrong' and to understand 'why a result fails' than to plod through yet another piece of theory. In this text, the authors gather more than 300 counterexamples - some of them both surprising and amusing - showing the limitations, hidden traps and pitfalls of measure and integration. Many examples are put into context, explaining relevant parts of the theory, and pointing out further reading. The text starts with a self-contained, non-technical overview on the fundamentals of measure and integration. A companion to the successful undergraduate textbook Measures, Integrals and Martingales, it is accessible to advanced undergraduate students, requiring only modest prerequisites. More specialized concepts are summarized at the beginning of each chapter, allowing for self-study as well as supplementary reading for any course covering measures and integrals. For researchers, it provides ample examples and warnings as to the limitations of general measure theory. This book forms a sister volume to Rene Schilling's other book Measures, Integrals and Martingales (www.cambridge.org/9781316620243).
More than 300 examples and counterexamples illustrating the (im)possibilities of measure and integration theory
Concise non-technical overview of the main points of measure and integration
Companion volume to the popular textbook Measures, Integrals and Martingales, now in its second edition
Preface
User's guide
List of topics and phenomena
1. A panorama of Lebesgue integration
2. A refresher of topology and ordinal numbers
3. Riemann is not enough
4. Families of sets
5. Set functions and measures
6. Range and support of a measure
7. Measurable and non-measurable sets
8. Measurable maps and functions
9. Inner and outer measure
10. Integrable functions
11. Modes of convergence
12. Convergence theorems
13. Continuity and a.e. continuity
14. Integration and differentiation
15. Measurability on product spaces
16. Product measures
17. Radon?Nikodym and related results
18. Function spaces
19. Convergence of measures
References
Index.
Not yet published - available from September 2021
FORMAT: Hardback
ISBN: 9781108838924
DIMENSIONS: 228 x 152 mm
Galois Theory, the theory of polynomial equations and their solutions, is one of the most fascinating and beautiful subjects of pure mathematics. Using group theory and field theory, it provides a complete answer to the problem of the solubility of polynomial equations by radicals: that is, determining when and how a polynomial equation can be solved by repeatedly extracting roots using elementary algebraic operations. This textbook contains a fully detailed account of Galois Theory and the algebra that it needs and is suitable both for those following a course of lectures and the independent reader (who is assumed to have no previous knowledge of Galois Theory). The second edition has been significantly revised and re-ordered; the first part develops the basic algebra that is needed, and the second a comprehensive account of Galois Theory. There are applications to ruler-and- compass constructions, and to the solution of classical mathematical problems of ancient times. There are new exercises throughout, and carefully-selected examples will help the reader develop a clear understanding of the mathematical theory.
The revised second edition, with more examples and additional background material
Contains a wealth of new exercises to challenge the reader
Gives a direct and straightforward account of the mathematical theory
'Garling's book presents Galois theory in a style which is at once readable and compact. The necessary prerequisites are developed in the early chapters only to the extent that they are needed later. The proofs of the lemmas and main theorems are presented in as concrete a manner as possible, without unnecessary abstraction. Yet they seem remarkably short, without the difficulties being glossed over. In fact the approach throughout the book is down-to-earth and concrete c I can heartily recommend this book as an undergraduate text.' Bulletin of the London Mathematical Society
Part I. The Algebraic Background:
1. Groups
2. Integral domains
3. Vector spaces and determinants
Part II. The Theory of Fields, and Galois Theory:
4. Field extensions
5. Ruler and compass constructions
6. Splitting fields
7. Normal extensions
8. Separability
9. The fundamental theorem of Galois theory
10. The discriminant
11. Cyclotomic polynomials and cyclic extensions
12. Solution by radicals
13. Regular polygons
14. Polynomials of low degree
15. Finite fields
16. Quintic polynomials
17. Further theory
18. The algebraic closure of a field
19. Transcendental elements and algebraic independence
20. Generic and symmetric polynomials
Appendix: the axiom of choice
Index.
Not yet published - available from September 2021
FORMAT: Paperback
ISBN: 9781009009195
DIMENSIONS: 254 x 178 mm
Beginning graduate students in mathematical sciences and related areas in physical and computer sciences and engineering are expected to be familiar with a daunting breadth of mathematics, but few have such a background. This bestselling book helps students fill in the gaps in their knowledge. Prize-winning mathematician Thomas A. Garrity explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The explanations are accompanied by numerous examples, exercises and suggestions for further reading that allow the reader to test and develop their understanding of these core topics. Featuring four new chapters and many other improvements, this second edition of All the Math You Missed is an essential resource for advanced undergraduates and beginning graduate students who need to learn some serious mathematics quickly.
New edition of a bestseller (over 20,000 copies worldwide), with four new chapters and numerous other improvements
Provides an overview of all the key topics in undergraduate mathematics, explaining the basic concepts with numerous examples and exercises
Each chapter includes an annotated bibliography offering a guide to further study and more rigorous foundations
Topics include: linear algebra, vector calculus, differential geometry, number theory, real and complex analysis, topology, differential equations, probability theory, abstract algebra, category theory and more
'Reading Garrity is like talking with your favorite uncle - he tells you the essential stories, in a clear and colorful way, and you get just what you need to explore further. The topics are well chosen (and there are more in this new edition). His points of view enrich the reader - not only do you learn what to know, but how to know it. I wish I had had this book when I started graduate school.' John McCleary, Vassar College
On the structure of mathematics
Brief summaries of topics
1. Linear Algebra
2. à and  real analysis
3. Calculus for vector-valued functions
4. Point set topology
5. Classical Stokes' theorems
6. Diff erential forms and Stokes' theorem
7. Curvature for curves and surfaces
8. Geometry
9. Countability and the Axiom of Choice
10. Elementary number theory
11. Algebra
12. Algebraic number theory
13. Complex analysis
14. Analytic number theory
15. Lebesgue integration
16. Fourier analysis
17. Diff erential equations
18. Combinatorics and probability theory
19. Algorithms
20. Category theory
Appendix A. Equivalence relations
References
Index.
Zurich Lectures in Advanced Mathematics, 26
ISBN print 978-3-98547-000-6,
DOI 10.4171/ZLAM/26
March 2021, 86 pages, softcover, 17 x 24 cm.
These lecture notes cover the classification of hyperbolic structures and measured foliations on surfaces in a minimalist way. While the inspiration is obviously taken from the excellent books Primer on mapping class groups and Travaux de Thurston sur les surfaces, the author aimed at including a little bit more of hyperbolic trigonometry, including a proof of Basmajian's identity on the orthogeodesic spectrum, while keeping the rest short.
Keywords: mapping class group, Dehn twist, pseudo-Anosov diffeomorphism, hyperbolic surface, Basmajian identity, measured foliation, Teichmu?ller theory, Thurston classification
EMS Series of Lectures in Mathematics, 32
ISBN print 978-3-98547-004-4
DOI 10.4171/ELM/32
April 2021, 250 pages, softcover, 17 x 24 cm.
The theory of von Neumann algebras, originating with the work of F. J. Murray and J. von Neumann in the late 1930s, has grown into a rich discipline with connections to different branches of mathematics and physics. Following the breakthrough of Tomita?Takesaki theory, many great advances were made throughout the 1970s by H. Araki, A. Connes, U. Haagerup, M. Takesaki and others.
These lecture notes aim to present a fast-track study of some important topics in classical parts of von Neumann algebra theory that were developed in the 1970s. Starting with Tomita?Takesaki theory, this book covers topics such as the standard form, Connesf cocycle derivatives, operator-valued weights, type III structure theory and non-commutative integration theory.
The self-contained presentation of the material makes this book useful not only to graduate students and researchers who want to know the fundamentals of von Neumann algebras, but also to interested undergraduates who have a basic knowledge of functional analysis and measure theory.
von Neumann algebra, Tomita?Takesaki theory, modular operator, standard form, Connesf cocycle derivative, operator-valued weight, relative modular operator, crossed product, KMS condition, Takesakifs duality theorem,
The Ladies' Diary was an annual almanac published in England from 1704 to 1840. It was designed to provide useful information to women; the subtitle reveals the purpose, Containing New Improvements in Arts and Sciences, and Many Entertaining Particulars: Designed for the Use and Diversion of the Fair Sex. It contained meteorological and astronomical information, recipes, health and medical advice, scientific information, and mathematical puzzles and problems. Readers were encouraged to, and did, send solutions and original problems and puzzles of their own for publication in the next year's issue.
Frank Swetz, one of the founding Editors of Convergence, the MAA's online journal of the history of mathematics, wondered about the historical and sociological conditions that supported The Ladies' Diary. In this volume he unearths the story of the Diary's creation and of the community of people surrounding it. We learn who the editors were and something about the contributors and readers. Swetz explores the sociological and cultural circumstances that made this unique almanac full of mathematics popular for over a century. The book includes scores of puzzles from the Diary, many in the form of riddles, rebuses, and poems.
Graduate students and researchers interested in the history of mathematics.
Spectrum, Volume: 101; 2021; 169 pp; Softcover
MSC: Primary 01; 97;
Print ISBN: 978-1-4704-6266-6
Product Code: SPEC/101
Expected publication date May 1, 2021