Topology Books
Princeton University Press Visual Differential Geometry and Forms
Book SynopsisTrade Review"Finalist for the PROSE Award in Mathematics, Association of American Publishers""Needham proposes to provide a truly geometric 'visual' explication of differential geometry, and he succeeds brilliantly. I know nothing like it in the literature."---Frank Morgan, EMS Magazine"[The] book offers a truly unique and original take on differential geometry, and it amply deserves inclusion within the pantheon of textbook deities."---Eric Poisson, Notices of the AMS"This is a valuable and beautifully created guide to what can at first seem a confusing area of mathematical physics. There are other contenders that try to teach this subject, but this is the best that I have come across so far and I will continue to enjoy learning from it (and almost certainly teaching from it) over the coming years, I am sure."---Jonathan Shock, Mathemafrica"[Proactively] rethinks the way this important area of mathematics should be considered and taught." * MathSciNet *"The book is a remarkable and highly original approach to the basic stem of differential geometry. And that mathematical trunk has roots and branches in so many other unexpected yet related subjects, each of which can be equally well approached from the same geometrical point of view."---Adhemar Bultheel, MAA Reviews"[Visual Differential Geometry and Forms] its peers. It is fun to read and provides a unique and intuitive approach to differential geometry. The author’s passion for the subject is evident throughout the book. Although Needham’s approach is unorthodox, it is rewarding, and complements the exposition found in standard textbooks."---Sean M. Eli & Krešmir Josić, SIAM Review
£35.70
Cambridge University Press Homotopy Theory of Enriched Mackey Functors
Book SynopsisThis work develops techniques and basic results concerning the homotopy theory of enriched diagrams and enriched Mackey functors. Presentation of a category of interest as a diagram category has become a standard and powerful technique in a range of applications. Diagrams that carry enriched structures provide deeper and more robust applications. With an eye to such applications, this work provides further development of both the categorical algebra of enriched diagrams, and the homotopy theoretic applications in K-theory spectra. The title refers to certain enriched presheaves, known as Mackey functors, whose homotopy theory classifies that of equivariant spectra. More generally, certain stable model categories are classified as modules - in the form of enriched presheaves - over categories of generating objects. This text contains complete definitions, detailed proofs, and all the background material needed to understand the topic. It will be indispensable for graduate students and researchers alike.
£67.50
Cambridge University Press Topological Phases of Matter
Book SynopsisTopological Phases of Matter are an exceptionally dynamic field of research: several of the most exciting recent experimental discoveries and conceptual advances in modern physics have originated in this field. These have generated new, topological, notions of order, interactions and excitations. This text provides an accessible, unified and comprehensive introduction to the phenomena surrounding topological matter, with detailed expositions of the underlying theoretical tools and conceptual framework, alongside accounts of the central experimental breakthroughs. Among the systems covered are topological insulators, magnets, semimetals, and superconductors. The emergence of new particles with remarkable properties such as fractional charge and statistics is discussed alongside possible applications such as fault-tolerant topological quantum computing. Suitable as a textbook for graduate or advanced undergraduate students, or as a reference for more experienced researchers, the book assTrade Review'… a timely and valuable introduction to the most important theoretical concepts in the topological study of matter … brief treatment of a vast, rapidly evolving subject that currently dominates condensed matter physics … This book is appropriate for physics collections within all university libraries.' M. C. Ogilvie, Choice ConnectTable of ContentsPreface; Acknowledgements; 1. Introduction; 2. Basic concepts of topology and condensed matter; 3. Integer topological phases; 4. Geometry and topology of wavefunctions in crystals; 5. Hydrogen atoms for fractionalisation; 6. Gauge and topological field theories; 7. Topology in gapless matter; 8. Disorder and defects in topological phases; 9. Topological quantum computation via non-Abelian statistics; 10. Topology out of equilibrium; 11. Symmetry, topology, and information; Appendix; References; Index.
£60.99
CRC Press The Shape of Space
Book SynopsisThe Shape of Space, Third Edition maintains the standard of excellence set by the previous editions. This lighthearted textbook covers the basic geometry and topology of two- and three-dimensional spacesâstretching studentsâ minds as they learn to visualize new possibilities for the shape of our universe.Written by a master expositor, leading researcher in the field, and MacArthur Fellow, its informal exposition and engaging exercises appeal to an exceptionally broad audience, from liberal arts students to math undergraduate and graduate students looking for a clear intuitive understanding to supplement more formal texts, and even to laypeople seeking an entertaining self-study book to expand their understanding of space.Features of the Third Edition: Full-color figures throughout Picture proofs have replaced algebraic proofs Simpler handles-and-crosscaps approach to surfaces Updated discussiTable of ContentsPart I Surfaces and Three-Manifolds Flatland Gluing Vocabulary Orientability Classification of Surfaces Products Flat Manifolds Orientability vs. Two-Sidedness Part II Geometries on Surfaces The Sphere The Hyperbolic Plane Geometries on Surfaces Gauss-Bonnet Formula and Euler Number Part III Geometries on Three-Manifolds Four-Dimensional Space The Hypersphere Hyperbolic Space Geometries on Three-Manifolds I Bundles Geometries on Three-Manifolds II Part IV The Universe The Universe The History of Space Appendix A: Answers Appendix B: Bibliography Appendix C: Conway’s ZIP Proof
£47.49
Princeton University Press Eulers Gem The Polyhedron Formula and the Birth
Book SynopsisLeonhard Euler's polyhedron formula describes the structure of many objects - from soccer balls and gemstones to Buckminster Fuller's buildings and giant all-carbon molecules. Yet Euler's formula is so simple it can be explained to a child. This title tells the story of this indispensable mathematical idea.Trade ReviewWinner of the 2010 Euler Book Prize, Mathematical Association of America One of Choice's Outstanding Academic Titles for 2009 "The author has achieved a remarkable feat, introducing a naive reader to a rich history without compromising the insights and without leaving out a delicious detail. Furthermore, he describes the development of topology from a suggestion by Gottfried Leibniz to its algebraic formulation by Emmy Noether, relating all to Euler's formula. This book will be valuable to every library with patrons looking for an awe-inspiring experience."--Choice "This is an excellent book about a great man and a timeless formula."--Charles Ashbacher, Journal of Recreational Mathematics "I liked Richeson's style of writing. He is enthusiastic and humorous. It was a pleasure reading this book, and I recommend it to everyone who is not afraid of mathematical arguments and has ever wondered what this field of 'rubbersheet geometry' is about. You will not be disappointed."--Jeanine Daems, Mathematical Intelligencer "The book is a pleasure to read for professional mathematicians, students of mathematicians or anyone with a general interest in mathematics."--European Mathematical Society Newsletter "I found much more to like than to criticize in Euler's Gem. At its best, the book succeeds at showing the reader a lot of attractive mathematics with a well-chosen level of technical detail. I recommend it both to professional mathematicians and to their seatmates."--Jeremy L. Martin, Notices of the AMS "I highly recommend this book for teachers interested in geometry or topology, particularly for university faculty. The examples, proofs, and historical anecdotes are interesting, informative, and useful for encouraging classroom discussions. Advanced students will also glimpse the broad horizons of mathematics by reading (and working through) the book."--Dustin L. Jones, Mathematics Teacher "The book should interest non-mathematicians as well as mathematicians. It is written in a lively way, mathematical properties are explained well and several biographical details are included."--Krzysztof Ciesielski, Mathematical ReviewsTable of ContentsPreface ix Introduction 1 Chapter 1: Leonhard Euler and His Three "Great" Friends 10 Chapter 2: What Is a Polyhedron? 27 Chapter 3: The Five Perfect Bodies 31 Chapter 4: The Pythagorean Brotherhood and Plato's Atomic Theory 36 Chapter 5: Euclid and His Elements 44 Chapter 6: Kepler's Polyhedral Universe 51 Chapter 7: Euler's Gem 63 Chapter 8: Platonic Solids, Golf Balls, Fullerenes, and Geodesic Domes 75 Chapter 9: Scooped by Descartes? 81 Chapter 10: Legendre Gets It Right 87 Chapter 11: A Stroll through Konigsberg 100 Chapter 12: Cauchy's Flattened Polyhedra 112 Chapter 13: Planar Graphs, Geoboards, and Brussels Sprouts 119 Chapter 14: It's a Colorful World 130 Chapter 15: New Problems and New Proofs 145 Chapter 16: Rubber Sheets, Hollow Doughnuts, and Crazy Bottles 156 Chapter 17: Are They the Same, or Are They Different? 173 Chapter 18: A Knotty Problem 186 Chapter 19: Combing the Hair on a Coconut 202 Chapter 20: When Topology Controls Geometry 219 Chapter 21: The Topology of Curvy Surfaces 231 Chapter 22: Navigating in n Dimensions 241 Chapter 23: Henri Poincare and the Ascendance of Topology 253 Epilogue The Million-Dollar Question 265 Acknowledgements 271 Appendix A Build Your Own Polyhedra and Surfaces 273 Appendix B Recommended Readings 283 Notes 287 References 295 Illustration Credits 309 Index 311
£16.19
Dover Publications Inc. A Combinatorial Introduction to Topology Dover
Book SynopsisExcellent text covers vector fields, plane homology and the Jordan Curve Theorem, surfaces, homology of complexes, more. Problems and exercises. Some knowledge of differential equations and multivariate calculus required.Bibliography. 1979 edition.
£11.24
Princeton University Press Higher Topos Theory
Book SynopsisHigher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. This title presents the foundations of this theory.Trade Review"This book is a remarkable achievement, and the reviewer thinks it marks the beginning of a major change in algebraic topology."--Mark Hovey, Mathematical ReviewsTable of ContentsPreface vii Chapter 1. An Overview of Higher Category Theory 1 1.1 Foundations for Higher Category Theory 1 1.2 The Language of Higher Category Theory 26 Chapter 2. Fibrations of Simplicial Sets 53 2.1 Left Fibrations 55 2.2 Simplicial Categories and 1-Categories 72 2.3 Inner Fibrations 95 2.4 Cartesian Fibrations 114 Chapter 3. The 1-Category of 1-Categories 145 3.1 Marked Simplicial Sets 147 3.2 Straightening and Unstraightening 169 3.3 Applications 204 Chapter 4. Limits and Colimits 223 4.1 Co_nality 223 4.2 Techniques for Computing Colimits 240 4.3 Kan Extensions 261 4.4 Examples of Colimits 292 Chapter 5. Presentable and Accessible 1-Categories 311 5.1 1-Categories of Presheaves 312 5.2 Adjoint Functors 331 5.3 1-Categories of Inductive Limits 377 5.4 Accessible 1-Categories 414 5.5 Presentable 1-Categories 455 Chapter 6. 1-Topoi 526 6.1 1-Topoi: De_nitions and Characterizations 527 6.2 Constructions of 1-Topoi 569 6.3 The 1-Category of 1-Topoi 593 6.4 n-Topoi 632 6.5 Homotopy Theory in an 1-Topos 651 Chapter 7. Higher Topos Theory in Topology 682 7.1 Paracompact Spaces 683 7.2 Dimension Theory 711 7.3 The Proper Base Change Theorem 742 Appendix. Appendix 781 A.1 Category Theory 781 A.2 Model Categories 803 A.3 Simplicial Categories 844 Bibliography 909 General Index 915 Index of Notation 923
£74.80
Dover Publications Inc. Introduction to Topology Third Edition Dover
Book SynopsisConcise undergraduate introduction to fundamentals of topology â clearly and engagingly written, and filled with stimulating, imaginative exercises. Topics include set theory, metric and topological spaces, connectedness, and compactness. 1975 edition.
£13.77
MP-AMM American Mathematical New Horizons in Geometry
Book SynopsisRepresents the fruits of 15 years of work in geometry by prize-winning authors Tom Apostol and Mamikon Mnatsakanian. Using new and intuitively rich methods, they give beautifully illustrated proofs of results, the majority of which are new, and frequently develop extensions of familiar theorems.Trade ReviewIn a remarkable display of mathematical versatility and imagination, the authors present us with a wealth of geometrical gems. These beautiful and often surprising results deal with a multitude of geometric forms, their interrelationships, and in many cases, their connection with patterns underlying the laws of nature."" - Don Chakerian""New Horizons in Geometry is a compendium of joint work produced by the authors during the period 1998-2012, most of it published in the American Mathematical Monthly, Math Horizons, Mathematics Magazine, and The Mathematical Gazette. The published papers have been edited, augmented and rearranged into 15 chapters dealing with several parts of classical geometry. The authors provide fresh and powerful insights into geometry that requires only a modest background in mathematics. Using new and intuitively rich methods, they give beautifully illustrated proofs of results and extensions of familiar theorems. Lengths, areas and volumes of curves, surfaces and solids are explored from a visually captivating perspective. Powerful geometric methods are used to solve standard calculus problems. Constructions and mechanical interpretations in the spirit of Archimedes involving centroids and moments are carried to new heights and to higher dimensional spaces. The hundreds of full color illustrations are visually enticing and provide great motivation to read further and savor the wonderful results. This book is a must have for any geometer."" - Dirk Keppen, Zentrallblatt MATH""Readers of New Horizons in Geometry are in for a great ride in the spirit of Archimedes through a beautiful geometrical landscape that will give you considerable pleasure and a heightened appreciation for a wonderful subject."" - Don Albers, former Director of MAA Publications
£67.50
Springer-Verlag New York Inc. Fixed Point Theory for LipschitzianType Mappings
Book SynopsisIn recent years, the fixed point theory of Lipschitzian-type mappings has rapidly grown into an important field of study in both pure and applied mathematics.Trade ReviewFrom the reviews: “The present book explains many of the basic techniques and … the classical results of fixed point theory, and normal structure properties. … Exercises are included in each chapter. As such, it is a self-contained book that can be used in a course for graduate students.” (Srinivasa Swaminathan, Zentralblatt MATH, Vol. 1176, 2010) “This book provides a presentation of fixed point theory for Lipschitzian type mappings in metric and Banach spaces. … An exercise section is included at the end of each chapter, containing interesting and well chosen material in order to cover topics complementing the main body of the text. … It is worthwhile to point out that a beginner in this area is certainly well served with this text … . A book including all … topics together for sure should be welcomed for graduate students.” (Enrique Llorens-Fuster, Mathematical Reviews, Issue 2010 e)Table of ContentsFundamentals.- Convexity, Smoothness, and Duality Mappings.- Geometric Coefficients of Banach Spaces.- Existence Theorems in Metric Spaces.- Existence Theorems in Banach Spaces.- Approximation of Fixed Points.- Strong Convergence Theorems.- Applications of Fixed Point Theorems.
£58.12
World Scientific Publishing Co Pte Ltd Shadows Of The Circle: Conic Sections, Optimal
Book SynopsisThe aim of this book is to throw light on various facets of geometry through development of four geometrical themes.The first theme is about the ellipse, the shape of the shadow cast by a circle. The next, a natural continuation of the first, is a study of all three types of conic sections, the ellipse, the parabola and the hyperbola.The third theme is about certain properties of geometrical figures related to the problem of finding the largest area that can be enclosed by a curve of given length. This problem is called the isoperimetric problem. In itself, this topic contains motivation for major parts of the curriculum in mathematics at college level and sets the stage for more advanced mathematical subjects such as functions of several variables and the calculus of variations.The emergence of non-Euclidean geometries in the beginning of the nineteenth century represents one of the dramatic episodes in the history of mathematics. In the last theme the non-Euclidean geometry in the Poincaré disc model of the hyperbolic plane is developed.Table of ContentsAn ellipse in the shadow; with conic sections in the light; optimal plane figures; the Poincare disc model of non-Euclidean geometry; exercises.
£16.15
Springer Fachmedien Wiesbaden Topologie: Eine anschauliche Einführung in die
Book SynopsisDas Ziel des Buches ist eine umfassende Einführung sowohl in die geometrische wie die algebraische Topologie. Dabei werden lediglich gute Kenntnisse aus dem 1. Studienjahr in der Mathematik vorausgesetzt, die über die Analysis und lineare Algebra kaum hinausgehen; alle weiteren Hilfsmittel, wie die Grundbegriffe der mengentheoretischen Topologie, die Theorie der topologischen Gruppen und die algebraischen Grundlagen werden ebenfalls ausführlich dargestellt. Im Vordergrund stehen jedoch nicht die hieraus hervorgehenden technischen Apparate, sondern die geometrischen Fragestellungen, die erst den Anlass zu ihrer Entwicklung gaben.Table of ContentsEinführung - Allgemeine Topologie - Homotopie - Lie-Gruppen und homogene Räume - Homologie
£26.59
American Mathematical Society The Knot Book An Elementary Introduction to the
Book SynopsisKnots are familiar objects. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. This work offers an introduction to this theory, starting with our understanding of knots. It presents the applications of knot theory to modern chemistry, biology and physics.Table of ContentsIntroduction Tabulating knots Invariants of knots Surfaces and knots Types of knots Polynomials Biology, chemistry, and physics Knots, links, and graphs Topology Higher dimensional knotting Knot jokes and pastimes Appendix Suggested readings and references Index Corrections to the 2004 AMS printing.
£38.90
Cambridge University Press New Spaces in Mathematics Volume 1
Book SynopsisAfter the development of manifolds and algebraic varieties in the previous century, mathematicians and physicists have continued to advance concepts of space. This book and its companion explore various new notions of space, including both formal and conceptual points of view, as presented by leading experts at the New Spaces in Mathematics and Physics workshop held at the Institut Henri Poincaré in 2015. The chapters in this volume cover a broad range of topics in mathematics, including diffeologies, synthetic differential geometry, microlocal analysis, topos theory, infinity-groupoids, homotopy type theory, category-theoretic methods in geometry, stacks, derived geometry, and noncommutative geometry. It is addressed primarily to mathematicians and mathematical physicists, but also to historians and philosophers of these disciplines.Trade Review'The essays are self-contained, providing motivation to read selectively. Examples in each chapter illustrate the usefulness of these new notions of space … Recommended.' M. Clay, Choice MagazineTable of ContentsIntroduction Mathieu Anel and Gabriel Catren; Part I. Differential geometry: 1. An Introduction to diffeology Patrick Iglesias-Zemmour; 2. New methods for old spaces: synthetic differential geometry Anders Kock; 3. Microlocal analysis and beyond Pierre Schapira; Part II. Topology and algebraic topology: 4. Topo-logie Mathieu Anel and André Joyal; 5. Spaces as infinity-groupoids Timothy Porter; 6. Homotopy type theory: the logic of space Michael Shulman; Part III. Algebraic geometry: 7. Sheaves and functors of points Michel Vaquié; 8. Stacks Nicole Mestrano and Carlos Simpson; 9. The geometry of ambiguity: an introduction to the ideas of derived geometry Mathieu Anel; 10. Geometry in dg categories Maxim Kontsevich.
£57.94
American Mathematical Society A Panoply of Polygons
Book SynopsisPresents and organises hundreds of beautiful, surprising and intriguing results about polygons with more than four sides. This panoply consists of eight chapters, one dedicated to polygonal basics, the next ones dedicated to pentagons, hexagons, heptagons, octagons and many-sided polygons.Table of Contents Polygon basics Pentagons Hexagons Heptagons Octagons Many-sided polygons Miscellaneous classes of polygons Polygonal numbers Solutions to the challenges Credits and permissions Index
£51.30
Princeton University Press Eulers Gem
Book SynopsisTrade Review"Everything in the book is very well illustrated with insightful graphics that, together with the text, make results almost like being obvious."---Adhemar Bultheel, European Mathematical Society
£16.19
Penguin Books Ltd The Poincaré Conjecture
Book SynopsisDonal O'Shea is professor of mathematics and dean of faculty at Mount Holyoke College. He has written scholarly books and monographs, and his research articles have appeared in numerous journals and collections. He lives in South Hadley, Massachusetts.Trade ReviewConveys topology's mind-bending contortions with great flair * New Scientist *One can't read The Poincaré Conjecture without an overwhelming awe at the infinite depths and richness of a mathematical realm not made by us * Martin Gardner, author of The Annotated Alice *Reveals the human story behind the challenge of the conjecture, and gives us a glimpse of the weird world inhabited by mathematicians * BBC Focus *Beautifully written * American Scientist *Intriguing * The Times *A truly marvellous book * Martin Gardner *One can't read The Poincaré Conjecture without an overwhelming awe at the infinite depths and richness of a mathematical realm not made by us * Martin Gardner, author of The Annotated Alice *
£11.69
Clarendon Press Projective Geometries Over Finite Fields Oxford Mathematical Monographs
Book SynopsisThis is a complete reworking of the out-of-print first volume of a three-volume treatise on finite projective spaces. There are numerous articles in journals, but this is the only extended work in the area. It also includes a comprehensive bibliography of more than 3000 items.Trade ReviewThe first edition of this work appeared in 1979, and was immediately recognized as an outstanding contribution to the field of finite geometry; the present volume is a complete revision of the earlier work...The book can serve as the text for a basic course...or as a detailed reference work for all topics in spaces of dimension two...The work is an indispensable aid to all workers in finite geometry, from beginning students to advanced researchers. * Short Book Reviews *...A complete reworking [of the first edition]. As before, the volume is concisely but clearly written, and contains a wealth of interesting material...the new trilogy, comprising the 1998, 1986 and 1991 volumes, looks set to be the standard reference work on projective spaces over finite fields for many years to come. * Bulletin of the London Mathematical Society *'Snap it up !' Bulletin London mathematical SocietyTable of Contents1. Finite fields ; 2. Projective spaces and algebraic varieties ; 3. Subspaces ; 4. Partitions ; 5. Canonical forms for varieties and polarities ; 6. The line ; 7. First properties of the plane ; 8. Ovals ; 9. Arithmetic of arcs of degree two ; 10. Arcs in ovals ; 11. Cubic curves ; 12. Arcs of higher degree ; 13. Blocking sets ; 14. Small planes ; Appendix ; Notation ; References
£227.25
Oxford University Press Groups and Geometry
Book SynopsisAvailable for the first time in published form, Groups and Geometry presents the Oxford Mathematical Institute notes for undergraduates and first year postgraduates. The content is guided by the Oxford syllabus but includes much more material than is included on the syllabus. This book is about the measurement of symmetry: covering groups and geometry with the symbiotic relationship between the two more than justifying the union. A number of exercises are included in this sylish text to help the reader gain a full understanding of this branch of mathematics.Trade Review'develops a comprehensive group-theoretic approach to affine, projective and inversive geometry ... It ends with a fascinating chapter on the group theory behind the Rubik cube.' Ian Stewart, New Scientist'Both parts contain a number of exercises that will be invaluable to any reader wishing to gain a fuller understanding of this area of mathematics.' Extrait de L'Enseignement Mathematique, T. 40 1994The book can be recommended warmly for any interested reader. "Monatshefte fur Mathematik No.3 1996.delightful book ... The group theory is directed towards group actions, but all the basic material is there. * Mathematika, 41 (1994) *Table of Contents1. A survey of some group theory ; 2. A menagerie of groups ; 3. Actions of groups ; 4. A garden of G-spaces ; 5. Transitivity and orbits ; 6. The classification of transitive G-spaces ; 7. G-morphisms ; 8. Group actions in group theory ; 9. Actions count ; 10. Geometry: an introduction ; 11. The axiomatisation of geometry ; 12. Affine geometry ; 13. Projective geometry ; 14. Euclidean geometry ; 15. Finite groups of isometries ; 16. Complex numbers and quaternions ; 17. Inversive geometry ; 18. Topological considerations ; 19. The groups theory of Rubik's magic cube ; Index
£70.20
Oxford University Press Quasiconformal Maps and Teichmüller Theory
Book SynopsisBased on a series of graduate lectures given by Vladimir Markovic at the University of Warwick in spring 2003, this book is accessible to those with a grounding in complex analysis looking for an introduction to the theory of quasiconformal maps and Teichmüller theory. Assuming some familiarity with Riemann surfaces and hyperbolic geometry, topics covered include the Grötzch argument, analytical properties of quasiconformal maps, the Beltrami differential equation, holomorphic motions and Teichmüller spaces. Where proofs are omitted, references to where they may be found are always given, and the text is clearly illustrated throughout with diagrams, examples, and exercises for the reader.Table of ContentsPreface ; 1. The Grotzch argument ; 2. Geometric definition of quasiconformal maps ; 3. Analytic properties of quasiconformal maps ; 4. Quasi-isometries and quasisymmetric maps ; 5. The Beltrami differential equation ; 6. Holomorphic motions and applications ; 7. Teichmuller spaces ; 8. Extremal quasiconformal mappings ; 9. Unique extremality ; 10. Isomorphisms of Teichmuller space ; 11. Local rigidity of Teichmuller spaces ; References ; Index
£117.50
Oxford University Press Undergraduate Topology A Working Textbook
Book SynopsisThis textbook offers an accessible, modern introduction at undergraduate level to an area known variously as general topology, point-set topology or analytic topology with a particular focus on helping students to build theory for themselves. It is the result of several years of the authors'' combined university teaching experience stimulated by sustained interest in advanced mathematical thinking and learning, alongside established research careers in analytic topology. Point-set topology is a discipline that needs relatively little background knowledge, but sufficient determination to grasp ideas precisely and to argue with straight and careful logic. Research and long experience in undergraduate mathematics education suggests that an optimal way to learn such a subject is to teach it to yourself, pro-actively, by guided reading of brief skeleton notes and by doing your own spadework to fill in the details and to flesh out the examples. This text will facilitate such an approach for Trade ReviewThe text offers a step-by-step learning with motivation, help and guidance; lucidity takes priority over elegant prose ... in only 144 pages, the excellent book covers most of the topics that a well-read graduate student is expected to know in basic general topology. * Peter Shiu, Mathematical Gazette *General Topology is a vast fundamental domain of mathematics. Its basic concepts, and central ideas of convergence and continuity, are connected by a multitude of links to practically every domain of mathematics. This makes writing a reasonably balanced undergraduate textbook on General Topology a formidable task. The authors have resoundingly accomplished this task in the creation of a unique textbook, less than 150 pages long, which offers the core basics of General Topology in a manner that allows instructor and student alike to take an active and insightful approach to their teaching and learning. Even more, students can select their own measure of initiative and creativity as they proceed. * A. V. Arhangel'skii, Ohio University *This approachable and engaging book should help students gain confidence and practice in building mathematical theory by themselves. * Stephen Watson, York University Toronto *This text marks a timely addition to the literature; it provides a rich resource and opportunity for the contemporary student to learn deeply, to learn as a âworkingâ mathematician, to learn by doing mathematics . . . That learning is cleverly and subtly supported throughout by the text's novel and engaging presentation. * Jan van Mill, University of Amsterdam *Table of ContentsPreface ; 1. Introduction ; 2. Topological Spaces ; 3. Continuity and Convergence ; 4. Invariants ; 5. Base and Product ; 6. Separation Axioms ; Essential Exercises ; Solutions to Selected Exercises ; Further Reading
£39.96
Oxford University Press 4Manifolds
Book SynopsisThis book presents the topology of smooth 4-manifolds in an intuitive self-contained way, developed over a number of years by Professor Akbulut. The text is aimed at graduate students and focuses on the teaching and learning of the subject, giving a direct approach to constructions and theorems which are supplemented by exercises to help the reader work through the details not covered in the proofs. The book contains a hundred colour illustrations to demonstrate the ideas rather than providing long-winded and potentially unclear explanations. Key results have been selected that relate to the material discussed and the author has provided examples of how to analyse them with the techniques developed in earlier chapters.Trade ReviewThis book is about the art of proving theorems about 4-manifolds by mental visualization and direct interaction with their depictions. Its existence is justifed by the need for an updated treatment...and the need to set out a unifed presentation of the tools and knowledge which have proved most useful over the relatively unique research history of its author. * Jonathan D. Williams, MathSciNet *Table of Contents1: 4-manifold handlebodies 2: Building low dimensional manifolds 3: Gluing 4 manifolds along their boundaries 4: Bundles 5: 3-manifolds 6: Operations 7: Lefschetz Fibrations 8: Symplectic Manifolds 9: Exotic 4-manifolds 10: Cork decomposition 11: Covering spaces 12: Complex surfaces 13: Seiberg-Witten invariants 14: Some applications
£74.10
Oxford University Press Topology A Geometric Approach
Book SynopsisThis new-in-paperback introduction to topology emphasizes a geometric approach with a focus on surfaces. A primary feature is a large collection of exercises and projects, which fosters a teaching style that encourages the student to be an active class participant. A wide range of material at different levels supports flexible use of the book for a variety of students. Part I is appropriate for a one-semester or two-quarter course, and Part II (which is problem based) allows the book to be used for a year-long course which supports a variety of syllabuses.The over 750 exercises range from simple checks of omitted details in arguments, to reinforce the material and increase student involvement, to the development of substantial theorems that have been broken into many steps. The style encourages an active student role. Solutions to selected exercises are included as an appendix, with solutions to all exercises available to the instructor on a companion website.Trade ReviewIdeas are introduced in a geometric context and development of material always maintains its excitement. Long live mathematics! * Peter Ruane, MAA Reviews *Table of ContentsPART I: A GEOMETRIC INTRODUCTION TO TOPOLOGY; PART II: COVERING SPACES, CW COMPLEXES AND HOMOLOGY
£71.40
The University of Chicago Press Inversions Popular Lectures in Mathematics
Book SynopsisIn this book, I. Ya. Bakel'man introduces inversion transformations in the Euclidean plane and discusses the interrelationships among more general mathematical concepts. The author begins by defining and giving examples of the concept of a transformation in the Euclidean plane, and then explains the point of infinity and the stereographic projection of the sphere onto the plane. With this preparation, the student is capable of applying the theory of inversions to classical construction problems in the plane. The author also discusses the theory of pencils of circles, and he uses the acquired techniques in a proof of Ptolemy's theorem. In the final chapter, the idea of a group is introduced with applications of group theory to geometry. The author demonstrates the group-theoretic basis for the distinction between Euclidean and Lobachevskian geometry.
£22.80
The University of Chicago Press Geometry of Nonpositively Curved Manifolds
Book SynopsisStarting from the foundations, this text presents an almost entirely self-contained treatment of differentiable spaces of nonpositive curvature, focusing on the symmetric spaces in which every geodesic lies in a flat Euclidean space of dimension at least two.
£117.80
The University of Chicago Press Geometry Rigidity and Group Actions
Book SynopsisExplores the role of group actions and rigidity in several areas of mathematics, including ergodic theory, dynamics, geometry, topology, and the algebraic properties of representation varieties.
£72.20
The University of Chicago Press Topics in Geometric Group Theory
Book SynopsisThis work seeks to offer a concise introduction to geometric group theory - a method for studying infinite groups via their intrinsic geometry. Basic combinatorial and geometric group theory is presented, along with research on the growth of groups, and exercises and problems.
£34.20
The University of Chicago Press Lie Algebras and Locally Compact Groups Chicago
Book SynopsisThis volume presents lecture notes based on the author's courses on Lie algebras and the solution of Hilbert's fifth problem. In chapter 1, Lie Algebras, the structure theory of semi-simple Lie algebras in characteristic zero is presented, following the ideas of Killing and Cartan. Chapter 2, The Structure of Locally Compact Groups, deals with the solution of Hilbert's fifth problem given by Gleason, Montgomery, and Zipplin in 1952.
£38.00
The University of Chicago Press Groups of Circle Diffeomorphisms
Book SynopsisGroup actions are an efficient way of describing symmetries in objects by defining the essential elements of a given object as a set. The symmetries of the object are then defined as the symmetry group of this set. This title explores group actions on the simplest closed manifold, the circle.
£42.75
The University of Chicago Press The Topological Classification of Stratified
Book SynopsisThis text provides topologists with a new way of looking at the classification theory of singular spaces. Divided into three parts, the book begins with an overview of high-dimensional manifold theory. It then offers the parallel theory for stratified spaces. Applications are also included.Table of ContentsPart 1 Manifold theory: algebraic K-theory and topology; surgery theory; spacification and functoriality; applications. Part 2 General theory: definitions and examples; classification of stratified spaces; transverse stratified classification; PT category; controlled topology; proof of main theorems in topology. Part 3 Applications and illustrations: manifolds and embedding theory revisited; supernormal spaces and varieties; group actions; rigidity conjectures.
£28.50
The University of Chicago Press Geometrical Vectors
Book SynopsisA guide for any student of vector analysis, this text separates those relationships which are topologically invariant from those which are not. Based on the essentially geometric nature of the subject, this approach builds consistently on students' prior knowledge and geometrical intuition.
£26.60
MIT Press Ltd Topology A Categorical Approach
Book SynopsisA graduate-level textbook that presents basic topology from the perspective of category theory.This graduate-level textbook on topology takes a unique approach: it reintroduces basic, point-set topology from a more modern, categorical perspective. Many graduate students are familiar with the ideas of point-set topology and they are ready to learn something new about them. Teaching the subject using category theory—a contemporary branch of mathematics that provides a way to represent abstract concepts—both deepens students' understanding of elementary topology and lays a solid foundation for future work in advanced topics. After presenting the basics of both category theory and topology, the book covers the universal properties of familiar constructions and three main topological properties—connectedness, Hausdorff, and compactness. It presents a fine-grained approach to convergence of sequences and filters; explores categorical limits and colimits, wi
£31.35
University of Texas Press Notes on the Witt Classification of Hermitian
Book SynopsisTable of Contents Introduction I. Relative Quadratic Extensions 1. Extension of primes 2. Hilbert symbols 3. The group Gen(E/F) 4. The group Iso(E/F) 5. The unramified case 6. Examples II. The Witt Ring H(E) 1. General definitions 2. Anisotropic representatives 3. Invariants for H(E) 4. Algebraic number fields III. Torsion Forms 1. Torsion OE-modules 2. The quotient E/K 3. Torsion innerproducts 4. Localizers 5. The inverse different IV. The Group Hu(K) 1. Basic definitions 2. The group Iso(E/F) again 3. The Knebusch exact sequence 4. Localization 5. Computing Hu(K) 6. The ring H(OE) 7. The Cokernel of δ V. The Witt Ring W(OF) 1. Symbols 2. The boundary operator 3. The ring W(OF) References Symbol List
£17.99
Taylor & Francis Ltd Math and Art
Book SynopsisMath and Art: An Introduction to Visual Mathematics explores the potential of mathematics to generate visually appealing objects and reveals some of the beauty of mathematics. It includes numerous illustrations, computer-generated graphics, photographs, and art reproductions to demonstrate how mathematics can inspire or generate art.Focusing on accessible, visually interesting, and mathematically relevant topics, the text unifies mathematics subjects through their visual and conceptual beauty. Sequentially organized according to mathematical maturity level, each chapter covers a cross section of mathematics, from fundamental Euclidean geometry, tilings, and fractals to hyperbolic geometry, platonic solids, and topology. For art students, the book stresses an understanding of the mathematical background of relatively complicated yet intriguing visual objects. For science students, it presents various elegant mathematical theories and notions.Features Provides an accessible introduction to mathematics in art Supports the narrative with a self-contained mathematical theory, with complete proofs of the main results (including the classification theorem for similarities) Presents hundreds of figures, illustrations, computer-generated graphics, designs, photographs, and art reproductions, mainly presented in full color Includes 21 projects and approximately 280 exercises, about half of which are fully solved Covers Euclidean geometry, golden section, Fibonacci numbers, symmetries, tilings, similarities, fractals, cellular automata, inversion, hyperbolic geometry, perspective drawing, Platonic and Archimedean solids, and topology New to the Second Edition New exercises, projects and artworks Revised, reorganized and expanded chapters More use of color throughout Trade Review"A beautiful book that brings out a wide range of mathematics, ancient to modern, with rich and often unexpected connections to the visual arts."– Catherine A. Gorini, Maharishi International University"Kalajdzievski takes us on a fascinating journey through the most visual subjects in mathematics. This book has the rare quality of not only organizing topics in a sequence that reveals how geometric concepts build upon one another, but also presenting each topic in a compact and self-contained manner for readers who prefer to browse for different entry points into the text. Although verbal explanations and mathematical formulae abound here, it is the colorful diagrams and photographs that capture the attention and enchant the eye. "– James Mai, Professor of Art, Illinois State University"The book presents mathematical and geometrical topics which can be expressed as the artistic pieces and serve to inspiring the artists to explore visual beauty and power of mathematics. In comparison with the first edition (of 2008), this book is noticeably extended to 280 exercises (from 190 originally) with solutions given to a half of them, 740 figures and artworks (from 556 previously), and 21 projects suggested for students.[. . . ] The book contains various illustrations and computer-generated graphics, photographs and art reproductions almost in each page, revealing an astonishing interaction of mathematics and artistic findings in human civilization and culture. [. . . ] The book can be useful to instructors and students, and interesting to any readers wishing to extend their knowledge and understanding of the esthetics and science of the visual math and mathematical art."– Technometrics"There are many books about mathematics and art; this one distinguishes itself as an “unorthodox geometry textbook,” with exercises and fun art projects. The book is based on 20 years of offering a course to more than 10,000 students. It stops short of covering some of the mathematics (groups are mentioned but not defined), though one theorem (classification of similarities) is proved in an appendix. Topics are Euclidean geometry, transformations of the plane, similarities and fractals, hyperbolic geometry, perspective, three-dimensional objects, and topology. The book averages two figures per page, with many utterly beautiful in color. You might be surprised at the sophisticated mathematical content of some crop circles (no doubt made by aliens!), and amazed by some of the illustrations of artworks."– Mathematics Magazine, MAAPraise for the First Edition"This delightful book grew out of set of teaching notes for an interdisciplinary course called Math in Art that was co-taught by a mathematician and an artist or architect. … The mathematical ideas are presented visually in a way that seems quite natural, and it engages the reader through explorations with lots of hands-on exercises. The mathematical presentation is solid, and the choice of topics puts the focus on the visual presentation of mathematical concepts. The illustrations are beautiful! … This text is very readable. The mathematics is accessible to those with little mathematical background, and yet the presentation is still engaging for those with more background."—MAA Reviews, March 2009"All in all, this work offers an excellent account of art inspired by mathematics and art generated by mathematics, and it should interest readers in both fields. Summing Up: Highly Recommended."– R.M. Davis, emeritus, Albion College, in Choice: Current Review for Academic Libraries, February 2009, Vol. 46, No. 6Table of ContentsChapter 1. Euclidean Geometry. 1.0. Introduction. 1.1. The Five Axioms of Euclidean Geometry. 1.2. Ruler and Compass Constructions. 1.3. The Golden Ratio. 1.4. Fibonacci Numbers. Chapter 2. Plane Transformations. 2.1. Plane Symmetries. 2.2.* Plane Symmetries, Vectors, and Matrices (Optional). 2.3. Groups of Symmetries Of Planar Objects. 2.4. Frieze Patterns. 2.5. Wallpaper Designs and Tilings of the Plane. 2.6. Tilings and Art. Chapter 3. Similarities, Fractals, and Cellular Automata. 3.1. Similarities and some other Planar Transformations. 3.2.* Complex Numbers (Optional). 3.3. Fractals: Definition and Some Examples. 3.4. Julia Sets. 3.5. Cellular Automata. Chapter 4. Hyperbolic Geometry. 4.1. Non-Euclidean Geometries: Background and Some History. 4.2. Inversion. 4.3. Hyperbolic Geometry. 4.4. Some Basic Constructions in the Poincaré Model. 4.5. Tilings of the Hyperbolic Plane. Chapter 5. Perspective. 5.1. Perspective: A brief overview of the Evolution of the rules of perspective. 5.2. Perspective Drawing and Constructions of Some Two-Dimensional (Planar) Objects. 5.3. Perspective Images of Three-Dimensional Objects. 5.4.* Mathematics of Perspective Drawing: A Brief Overview (Optional). Chapter 6. Some Three-Dimensional Objects. 6.1. Regular and Other Polyhedra. 6.2. Sphere, Cylinder, Cone, and Conic Sections. 6.3. Geometry, Tilings, Fractals, and Cellular Automata in Three Dimensions. Chapter 7. Topology. 7.1. Homotopy of Spaces: An Informal Introduction. 7.2. Two-Manifolds and The Euler Characteristic. 7.3. Non-Orientable Two-Manifolds and Three-Manifolds. Appendix: Classification Theorem for Similarities. Solutions.
£56.04
Taylor & Francis Ltd Introduction to Lorentz Geometry
Book SynopsisLorentz Geometry is a very important intersection between Mathematics and Physics, being the mathematical language of General Relativity. Learning this type of geometry is the first step in properly understanding questions regarding the structure of the universe, such as: What is the shape of the universe? What is a spacetime? What is the relation between gravity and curvature? Why exactly is time treated in a different manner than other spatial dimensions? Introduction to Lorentz Geometry: Curves and Surfaces intends to provide the reader with the minimum mathematical background needed to pursue these very interesting questions, by presenting the classical theory of curves and surfaces in both Euclidean and Lorentzian ambient spaces simultaneously. Features: Over 300 exercises Suitable for senior undergraduates and graduates studying Mathematics and Physics Written in an accessible style wTable of Contents1. Welcome to Lorentz-Minkowski Space. 1.1. Pseudo–Euclidean Spaces. 1.2. Subspaces of Rᵑᵥ. 1.3. Contextualization in Special Relativity. 1.4. Isometries in Rᵑᵥ. 1.5. Investigating O1(2, R) And O1(3, R). 1.6 Cross Product in Rᵑᵥ. 2. Local Theory of Curves. 2.1. Parametrized Curves in Rᵑᵥ. 2.2. Curves in the Plane. 2.3. Curves in Space. 3. Surfaces in Space. 3.1. Basic Topology of Surfaces. 3.2. Casual type of Surfaces, First Fundamental Form. 3.3. Second Fundamental Form and Curvatures. 3.4. The Diagonalization Problem. 3.5. Curves in Surface. 3.6. Geodesics, Variational Methods and Energy. 3.7. The Fundamental Theorem of Surfaces. 4. Abstract Surfaces and Further Topics. 4.1. Pseudo-Riemannian Metrics. 4.2. Riemann’s Classification Theorem. 4.3. Split-Complex Numbers and Critical Surfaces. 4.4 Digression: Completeness and Causality
£35.14
Springer New York Foundations of Hyperbolic Manifolds
Book SynopsisThis heavily class-tested book is an exposition of the theoretical foundations of hyperbolic manifolds. The first part is concerned with hyperbolic geometry and discrete groups. The second part is devoted to the theory of hyperbolic manifolds. The third part integrates the first two parts in a development of the theory of hyperbolic orbifolds.Trade ReviewFrom the reviews of the second edition: "Designed to be useful as both textbook and a reference, this book renders a real service to the mathematical community by putting together the tools and prerequisites needed to enter the territory of Thurston’s formidable theory of hyperbolic 3-mainfolds … . Every chapter is followed by historical notes, with attributions to the relevant literature, both of the originators of the idea present in the chapter and of modern presentation thereof. The bibliography contains 463 entries." (Victor V. Pambuccian, Zentralblatt MATH, Vol. 1106 (8), 2007)Table of ContentsEuclidean Geometry.- Spherical Geometry.- Hyperbolic Geometry.- Inversive Geometry.- Isometries of Hyperbolic Space.- Geometry of Discrete Groups.- Classical Discrete Groups.- Geometric Manifolds.- Geometric Surfaces.- Hyperbolic 3-Manifolds.- Hyperbolic n-Manifolds.- Geometrically Finite n-Manifolds.- Geometric Orbifolds.
£47.49
Springer-Verlag New York Inc. Braid Groups
Book SynopsisIn this well-written presentation, motivated by numerous examples and problems, the authors introduce the basic theory of braid groups, highlighting several definitions that show their equivalence;Trade ReviewFrom the reviews:"Details on … braid groups are carefully provided by Kassel and Turaev’s text Braid Groups. … Braid Groups is very well written. The proofs are detailed, clear, and complete. ... The text is to be praised for its level of detail. … For people … who want to understand current research in braid group related areas, Braid Groups is an excellent, in fact indispensable, text." (Scott Taylor, The Mathematical Association of America, October, 2008)"This is a very useful, carefully written book that will bring the reader up to date with some of the recent important advances in the study of the braid groups and their generalizations. It continues the tradition of these high quality graduate texts in mathematics. The book could easily be used as a text for a year course on braid groups for graduate students, one advantage being that the chapters are largely independent of each other." (Stephen P. Humphries, Mathematical Reviews, Issue 2009 e)“This book is a comprehensive introduction to the theory of braid groups. Assuming only a basic knowledge of topology and algebra, it is intended mainly for graduate and postdoctoral students.” (Hirokazu Nishimura, Zentralblatt MATH, Vol. 1208, 2011)“The book of Kassel and Turaev is a textbook … for graduate students and researchers. As such, it covers the basic material on braids, knots, and links … at a level which requires minimal background, yet moves rapidly to non-trivial topics. … It is a carefully planned and well-written book; the authors are true experts, and it fills a gap. … it will have many readers.” (Joan S. Birman, Bulletin of the American Mathematical Society, Vol. 48 (1), January, 2011)Table of ContentsBraids and Braid Groups.- Braids, Knots, and Links.- Homological Representations of the Braid Groups.- Symmetric Groups and Iwahori#x2013;Hecke Algebras.- Representations of the Iwahori#x2013;Hecke Algebras.- Garside Monoids and Braid Monoids.- An Order on the Braid Groups.- Presentations of SL(Z) and PSL(Z).- Fibrations and Homotopy Sequences.- The Birman#x2013;Murakami#x2013;Wenzl Algebras.- Left Self-Distributive Sets.
£71.99
Springer A Topological Picturebook
Book SynopsisAims to encourage mathematicians to illustrate their work and to help artists understand the ideas expressed by such drawings. This book explains the graphic design of illustrations from Thurston's world of low-dimensional geometry and topology. It presents the principles of linear and aerial perspective from the viewpoint of projective geometry.Trade ReviewFrom the reviews: "I was very pleasantly surprised when I opened the book. … this is really a much richer book. Indeed, the approach it offers to drawing can have a significant impact on how we teach and think about mathematics. … If you are good at visualization and illustration, this book can help you become better yet. … this will give you concrete and specific suggestions for developing your skills. If you just appreciate skillful drawing and illustration, this book deserves a look." (William J. Satzer, MathDL, December, 2006) "This book is a drawing manual for mathematicians. It is written by a great expert in the subject … . The author explained his techniques of drawing and of shading pictures, he explains when to use perspective and when not to use it, and so on. The numerous illustrations that are contained in this book as examples for drawing of mathematical objects are delightful … . This book is unique in the mathematical literature, and the present second printing is most welcome." (Athanase Papadopoulos, Zentralblatt MATH, Vol. 1105 (7), 2007)Table of Contents1 Descriptive Topology.- 2 Methods and Media.- 3 Pictures in Perspective.- 4 The Impossible Tribar.- 5 Shadows from Higher Dimension.- 6 Sphere Eversions.- 7 Group Pictures.- 8 The Figure Eight Knot.- Postscript.
£52.24
Springer Measure Topology and Fractal Geometry
Book SynopsisFractal Examples.- Metric Topology.- Topological Dimension.- Self-Similarity.- Measure Theory.- Fractal Dimension.- Additional Topics.Trade ReviewFrom the reviews of the second edition: "As a non-specialist, I found this book very helpful. It gave me a better understanding of the nature of fractals, and of the technical issues involved in the theory. I think it will be valuable as a textbook for undergraduate students in mathematics, and also for researchers wanting to learn fractal geometry from scratch. The material is well-organized and the proofs are clear; the abundance of examples is an asset for a book on measure theory and topology." (Fabio Mainardi, MathDL, February, 2008) "This is the second edition of a well-known textbook in the field … . The book may serve as a textbook for a one-semester (advanced) undergraduate course in mathematics. … the book is also suitable for readers interested in theoretical fractal geometry coming from other disciplines (e.g. physics, computer sciences) with a basic knowledge of mathematics. The presentation of the material is appealing … and the style is clear and motivating. … the book will remain as a standard reference in the field." (José-Manuel Rey, Zentralblatt MATH, Vol. 1152, 2009)Table of ContentsFractal Examples.- Metric Topology.- Topological Dimension.- Self-Similarity.- Measure Theory.- Fractal Dimension.- Additional Topics.
£36.99
Springer New York Stable Mappings and Their Singularities 14 Graduate Texts in Mathematics
Book SynopsisThe study of stable singularities is based on the now classical theories of Hassler Whitney, who determined the generic singularities (or lack of them) of Rn ~ Rm (m ~ 2n - 1) and R2 ~ R2, and Marston Morse, for mappings who studied these singularities for Rn ~ R.Table of ContentsI: Preliminaries on Manifolds.- §1. Manifolds.- §2. Differentiable Mappings and Submanifolds.- §3. Tangent Spaces.- §4. Partitions of Unity.- §5. Vector Bundles.- §6. Integration of Vector Fields.- II: Transversality.- §1. Sard’s Theorem.- §2. Jet Bundles.- §3. The Whitney C? Topology.- §4. Transversality.- §5. The Whitney Embedding Theorem.- §6. Morse Theory.- §7. The Tubular Neighborhood Theorem.- III: Stable Mappings.- §1. Stable and Infinitesimally Stable Mappings.- §2. Examples.- §3. Immersions with Normal Crossings.- §4. Submersions with Folds.- IV: The Malgrange Preparation Theorem.- §1. The Weierstrass Preparation Theorem.- §2. The Malgrange Preparation Theorem.- §3. The Generalized Malgrange Preparation Theorem.- V: Various Equivalent Notions of Stability.- §1. Another Formulation of Infinitesimal Stability.- §2. Stability Under Deformations.- §3. A Characterization of Trivial Deformations.- §4. Infinitesimal Stability => Stability.- §5. Local Transverse Stability.- §6. Transverse Stability.- §7. Summary.- VI: Classification of Singularities, Part I: The Thom-Boardman Invariants.- §1. The Sr Classification.- §2. The Whitney Theory for Generic Mappings between 2-Manifolds.- §3. The Intrinsic Derivative.- §4. The Sr,s Singularities.- §5. The Thom-Boardman Stratification.- §6. Stable Maps Are Not Dense.- VII: Classification of Singularities, Part II: The Local Ring of a Singularity.- §1. Introduction.- §2. Finite Mappings.- §3. Contact Classes and Morin Singularities.- §4. Canonical Forms for Morin Singularities.- §5. Umbilics.- §6. Stable Mappings in Low Dimensions.- §A. Lie Groups.- Symbol Index.
£71.24
Springer New York Differential Topology
Book SynopsisPresents a comprehensive treatment of most of the basic material in differential topology, as far as is accessible without the methods of algebraic topology.Trade ReviewM.W. Hirsch Differential Topology "A very valuable book. In little over 200 pages, it presents a well-organized and surprisingly comprehensive treatment of most of the basic material in differential topology, as far as is accessible without the methods of algebraic topology. Newly introduced concepts are usually well motivated, and often the historical development of an idea is described. There is an abundance of exercises, which supply many beautiful examples and much interesting additional information, and help the reader to become thoroughly familiar with the material of the main text. "—MATHEMATICAL REVIEWSTable of Contents1 : Manifolds and Maps.- 0. Submanifolds of ?n+k.- 1. Differential Structures.- 2. Differentiable Maps and the Tangent Bundle.- 3. Embeddings and Immersions.- 4. Manifolds with Boundary.- 5. A Convention.- 2 : Function Spaces.- 1. The Weak and Strong Topologies on Cr(M, N).- 2. Approximations.- 3. Approximations on ?-Manifolds and Manifold Pairs.- 4. Jets and the Baire Property.- 5. Analytic Approximations.- 3 : Transversality.- 1. The Morse-Sard Theorem.- 2. Transversality.- 4 : Vector Bundles and Tubular Neighborhoods.- 1. Vector Bundles.- 2. Constructions with Vector Bundles.- 3. The Classification of Vector Bundles.- 4. Oriented Vector Bundles.- 5. Tubular Neighborhoods.- 6. Collars and Tubular Neighborhoods of Neat Submanifolds.- 7. Analytic Differential Structures.- 5 : Degrees, Intersection Numbers, and the Euler Characteristic.- 1. Degrees of Maps.- 2. Intersection Numbers and the Euler Characteristic.- 3. Historical Remarks.- 6 : Morse Theory.- 1. Morse Functions.- 2. Differential Equations and Regular Level Surfaces.- 3. Passing Critical Levels and Attaching Cells.- 4. CW-Complexes.- 7 : Cobordism.- 1. Cobordism and Transversality.- 2. The Thorn Homomorphism.- 8 : Isotopy.- 1. Extending Isotopies.- 2. Gluing Manifolds Together.- 3. Isotopies of Disks.- 9 : Surfaces.- 1. Models of Surfaces.- 2. Characterization of the Disk.- 3. The Classification of Compact Surfaces.
£56.99
Springer New York Linear Topological Spaces 36 Graduate Texts in Mathematics
£27.99
Springer Counterexamples in Topology
Book SynopsisI Basic Definitions.- 1. General Introduction.- 2. Separation Axioms.- 3. Compactness.- 4. Connectedness.- 5. Metric Spaces.- II Counterexamples.- III Metrization Theory.- Conjectures and Counterexamples.- IV Appendices.- Special Reference Charts.- Separation Axiom Chart.- Compactness Chart.- Paracompactness Chart.- Connectedness Chart.- Disconnectedness Chart.- Metrizability Chart.- General Reference Chart.- Problems.- Notes.Table of ContentsI Basic Definitions.- 1. General Introduction.- Limit Points.- Closures and Interiors.- Countability Properties.- Functions.- Filters.- 2. Separation Axioms.- Regular and Normal Spaces.- Completely Hausdorff Spaces.- Completely Regular Spaces.- Functions, Products, and Subspaces.- Additional Separation Properties.- 3. Compactness.- Global Compactness Properties.- Localized Compactness Properties.- Countability Axioms and Separability.- Paracompactness.- Compactness Properties and Ti Axioms.- Invariance Properties.- 4. Connectedness.- Functions and Products.- Disconnectedness.- Biconnectedness and Continua.- 5. Metric Spaces.- Complete Metric Spaces.- Metrizability.- Uniformities.- Metric Uniformities.- II Counterexamples.- 1. Finite Discrete Topology.- 2. Countable Discrete Topology.- 3. Uncountable Discrete Topology.- 4. Indiscrete Topology.- 5. Partition Topology.- 6. Odd-Even Topology.- 7. Deleted Integer Topology.- 8. Finite Particular Point Topology.- 9. Countable Particular Point Topology.- 10. Uncountable Particular Point Topology.- 11. Sierpinski Space.- 12. Closed Extension Topology.- 13. Finite Excluded Point Topology.- 14. Countable Excluded Point Topology.- 15. Uncountable Excluded Point Topology.- 16. Open Extension Topology.- 17. Either-Or Topology.- 18. Finite Complement Topology on a Countable Space.- 19. Finite Complement Topology on an Uncountable Space.- 20. Countable Complement Topology.- 21. Double Pointed Countable Complement Topology.- 22. Compact Complement Topology.- 23. Countable Fort Space.- 24. Uncountable Fort Space.- 25. Fortissimo Space.- 26. Arens-Fort Space.- 27. Modified Fort Space.- 28. Euclidean Topology.- 29. The Cantor Set.- 30. The Rational Numbers.- 31. The Irrational Numbers.- 32. Special Subsets of the Real Line.- 33. Special Subsets of the Plane.- 34. One Point Compactification Topology.- 35. One Point Compactification of the Rationals.- 36. Hilbert Space.- 37. Fréchet Space.- 38. Hilbert Cube.- 39. Order Topology.- 40. Open Ordinal Space [0,?) (? < ?).- 41. Closed Ordinal Space [0,?] (? < ?).- 42. Open Ordinal Space [0,?).- 43. Closed Ordinal Space [0,?].- 44. Uncountable Discrete Ordinal Space.- 45. The Long Line.- 46. The Extended Long Line.- 47. An Altered Long Line.- 48. Lexicographic Ordering on the Unit Square.- 49. Right Order Topology.- 50. Right Order Topology on R.- 51. Right Half-Open Interval Topology.- 52. Nested Interval Topology.- 53. Overlapping Interval Topology.- 54. Interlocking Interval Topology.- 55. Hjalmar Ekdal Topology.- 56. Prime Ideal Topology.- 57. Divisor Topology.- 58. Evenly Spaced Integer Topology.- 59. The p-adic Topology on Z.- 60. Relatively Prime Integer Topology.- 61. Prime Integer Topology.- 62. Double Pointed Reals.- 63. Countable Complement Extension Topology.- 64. Smirnov’s Deleted Sequence Topology.- 65. Rational Sequence Topology.- 66. Indiscrete Rational Extension of R.- 67. Indiscrete Irrational Extension of R.- 68. Pointed Rational Extension of R.- 69. Pointed Irrational Extension of R.- 70. Discrete Rational Extension of R.- 71. Discrete Irrational Extension of R.- 72. Rational Extension in the Plane.- 73. Telophase Topology.- 74. Double Origin Topology.- 75. Irrational Slope Topology.- 76. Deleted Diameter Topology.- 77. Deleted Radius Topology.- 78. Half-Disc Topology.- 79. Irregular Lattice Topology.- 80. Arens Square.- 81. Simplified Arens Square.- 82. Niemytzki’s Tangent Disc Topology.- 83. Metrizable Tangent Disc Topology.- 84. Sorgenfrey’s Half-Open Square Topology.- 85. Michael’s Product Topology.- 86. Tychonoff Plank.- 87. Deleted Tychonoff Plank.- 88. Alexandroff Plank.- 89. Dieudonne Plank.- 90. Tychonoff Corkscrew.- 91. Deleted Tychonoff Corkscrew.- 92. Hewitt’s Condensed Corkscrew.- 93. Thomas’ Plank.- 94. Thomas’ Corkscrew.- 95. Weak Parallel Line Topology.- 96. Strong Parallel Line Topology.- 97. Concentric Circles.- 98. Appert Space.- 99. Maximal Compact Topology.- 100. Minimal Hausdorff Topology.- 101. Alexandroff Square.- 102. ZZ.- 103. Uncountable Products of Z+.- 104. Baire Product Metric on Rw.- 105. II.- 106. [0,?) × II.- 107. Helly Space.- 108. C[0,1].- 109. Box Product Topology on Rw.- 110. Stone-?ech Compactification.- 111. Stone-?ech Compactification of the Integers.- 112. Novak Space.- 113. Strong Ultrafilter Topology.- 114. Single Ultrafilter Topology.- 115. Nested Rectangles.- 116. Topologist’s Sine Curve.- 117. Closed Topologist’s Sine Curve.- 118. Extended Topologist’s Sine Curve.- 119. The Infinite Broom.- 120. The Closed Infinite Broom.- 121. The Integer Broom.- 122. Nested Angles.- 123. The Infinite Cage.- 124. Bernstein’s Connected Sets.- 125. Gustin’s Sequence Space.- 126. Roy’s Lattice Space.- 127. Roy’s Lattice Subspace.- 128. Cantor’s Leaky Tent.- 129. Cantor’s Teepee.- 130. A Pseudo-Arc.- 131. Miller’s Biconnected Set.- 132. Wheel without Its Hub.- 133. Tangora’s Connected Space.- 134. Bounded Metrics.- 135. Sierpinski’s Metric Space.- 136. Duncan’s Space.- 137. Cauchy Completion.- 138. Hausdorff’s Metric Topology.- 139. The Post Office Metric.- 140. The Radial Metric.- 141. Radial Interval Topology.- 142. Bing’s Discrete Extension Space.- 143. Michael’s Closed Subspace.- III Metrization Theory.- Conjectures and Counterexamples.- IV Appendices.- Special Reference Charts.- Separation Axiom Chart.- Compactness Chart.- Paracompactness Chart.- Connectedness Chart.- Disconnectedness Chart.- Metrizability Chart.- General Reference Chart.- Problems.- Notes.
£85.49
Springer New York Foundations of Differentiable Manifolds and Lie Groups 94 Graduate Texts in Mathematics
Book SynopsisFoundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. Coverage includes differentiable manifolds, tensors and differentiable forms, Lie groups and homogenous spaces, and integration on manifolds.Table of Contents1 Manifolds.- 2 Tensors and Differential Forms.- 3 Lie Groups.- 4 Integration on Manifolds.- 5 Sheaves, Cohomology, and the de Rham Theorem.- 6 The Hodge Theorem.- Supplement to the Bibliography.- Index of Notation.
£47.49
Springer New York Homology Theory
Book Synopsis1 Singular Homology Theory.- 2 Attaching Spaces with Maps.- 3 The Eilenberg-Steenrod Axioms.- 4 Covering Spaces.- 5 Products.- 6 Manifolds and Poincaré Duality.- 7 Fixed-Point Theory.- Appendix I.- Appendix II.- References.- Books and Historical Articles Since 1973.- Books and Notes.- Survey and Expository Articles.Table of Contents1 Singular Homology Theory.- 2 Attaching Spaces with Maps.- 3 The Eilenberg-Steenrod Axioms.- 4 Covering Spaces.- 5 Products.- 6 Manifolds and Poincaré Duality.- 7 Fixed-Point Theory.- Appendix I.- Appendix II.- References.- Books and Historical Articles Since 1973.- Books and Notes.- Survey and Expository Articles.
£71.24
Springer New York Algebraic Topology
Book SynopsisRather than choosing one point of view of modem topology (homotopy theory, simplicial complexes, singular theory, axiomatic homology, differ ential topology, etc.), we concentrate our attention on concrete prob lems in low dimensions, introducing only as much algebraic machin ery as necessary for the problems we meet.Table of ContentsI Calculus in the Plane.- 1 Path Integrals.- 1a. Differential Forms and Path Integrals.- 1b. When Are Path Integrals Independent of Path?.- 1c. A Criterion for Exactness.- 2 Angles and Deformations.- 2a. Angle Functions and Winding Numbers.- 2b. Reparametrizing and Deforming Paths.- 2c. Vector Fields and Fluid Flow.- II Winding Numbers.- 3 The Winding Number.- 3a. Definition of the Winding Number.- 3b. Homotopy and Reparametrization.- 3c. Varying the Point.- 3d. Degrees and Local Degrees.- 4 Applications of Winding Numbers.- 4a. The Fundamental Theorem of Algebra.- 4b. Fixed Points and Retractions.- 4c. Antipodes.- 4d. Sandwiches.- III Cohomology and Homology, I.- 5 De Rham Cohomology and the Jordan Curve Theorem.- 5a. Definitions of the De Rham Groups.- 5b. The Coboundary Map.- 5c. The Jordan Curve Theorem.- 5d. Applications and Variations.- 6 Homology.- 6a. Chains, Cycles, and H0U.- 6b. Boundaries, H1U, and Winding Numbers.- 6c. Chains on Grids.- 6d. Maps and Homology.- 6e. The First Homology Group for General Spaces.- IV Vector Fields.- 7 Indices of Vector Fields.- 7a. Vector Fields in the Plane.- 7b. Changing Coordinates.- 7c. Vector Fields on a Sphere.- 8 Vector Fields on Surfaces.- 8a. Vector Fields on a Torus and Other Surfaces.- 8b. The Euler Characteristic.- V Cohomology and Homology, II.- 9 Holes and Integrals.- 9a. Multiply Connected Regions.- 9b. Integration over Continuous Paths and Chains.- 9c. Periods of Integrals.- 9d. Complex Integration.- 10 Mayer—Vietoris.- 10a. The Boundary Map.- 10b. Mayer—Vietoris for Homology.- 10c. Variations and Applications.- 10d. Mayer—Vietoris for Cohomology.- VI Covering Spaces and Fundamental Groups, I.- 11 Covering Spaces.- 11a. Definitions.- 11b. Lifting Paths and Homotopies.- 11c. G-Coverings.- 11d. Covering Transformations.- 12 The Fundamental Group.- 12a. Definitions and Basic Properties.- 12b. Homotopy.- 12c. Fundamental Group and Homology.- VII Covering Spaces and Fundamental Groups, II.- 13 The Fundamental Group and Covering Spaces.- 13a. Fundamental Group and Coverings.- 13b. Automorphisms of Coverings.- 13c. The Universal Covering.- 13d. Coverings and Subgroups of the Fundamental Group.- 14 The Van Kampen Theorem.- 14a. G-Coverings from the Universal Covering.- 14b. Patching Coverings Together.- 14c. The Van Kampen Theorem.- 14d. Applications: Graphs and Free Groups.- VIII Cohomology and Homology, III.- 15 Cohomology.- 15a. Patching Coverings and ?ech Cohomology.- 15b. ?ech Cohomology and Homology.- 15c. De Rham Cohomology and Homology.- 15d. Proof of Mayer—Vietoris for De Rham Cohomology.- 16 Variations.- 16a. The Orientation Covering.- 16b. Coverings from 1-Forms.- 16c. Another Cohomology Group.- 16d. G-Sets and Coverings.- 16e. Coverings and Group Homomorphisms.- 16f. G-Coverings and Cocycles.- IX Topology of Surfaces.- 17 The Topology of Surfaces.- 17a. Triangulation and Polygons with Sides Identified.- 17b. Classification of Compact Oriented Surfaces.- 17c. The Fundamental Group of a Surface.- 18 Cohomology on Surfaces.- 18a. 1-Forms and Homology.- 18b. Integrals of 2-Forms.- 18c. Wedges and the Intersection Pairing.- 18d. De Rham Theory on Surfaces.- X Riemann Surfaces.- 19 Riemann Surfaces.- 19a. Riemann Surfaces and Analytic Mappings.- 19b. Branched Coverings.- 19c. The Riemann—Hurwitz Formula.- 20 Riemann Surfaces and Algebraic Curves.- 20a. The Riemann Surface of an Algebraic Curve.- 20b. Meromorphic Functions on a Riemann Surface.- 20c. Holomorphic and Meromorphic 1-Forms.- 20d. Riemann’s Bilinear Relations and the Jacobian.- 20e. Elliptic and Hyperelliptic Curves.- 21 The Riemann—Roch Theorem.- 21a. Spaces of Functions and 1-Forms.- 21b. Adeles.- 21c. Riemann—Roch.- 21d. The Abel—Jacobi Theorem.- XI Higher Dimensions.- 22 Toward Higher Dimensions.- 22a. Holes and Forms in 3-Space.- 22b. Knots.- 22c. Higher Homotopy Groups.- 22d. Higher De Rham Cohomology.- 22e. Cohomology with Compact Supports.- 23 Higher Homology.- 23a. Homology Groups.- 23b. Mayer—Vietoris for Homology.- 23c. Spheres and Degree.- 23d. Generalized Jordan Curve Theorem.- 24 Duality.- 24a. Two Lemmas from Homological Algebra.- 24b. Homology and De Rham Cohomology.- 24c. Cohomology and Cohomology with Compact Supports.- 24d. Simplicial Complexes.- Appendices.- Appendix A Point Set Topology.- A1. Some Basic Notions in Topology.- A2. Connected Components.- A3. Patching.- A4. Lebesgue Lemma.- Appendix B Analysis.- B1. Results from Plane Calculus.- B2. Partition of Unity.- Appendix C Algebra.- C1. Linear Algebra.- C2. Groups; Free Abelian Groups.- C3. Polynomials; Gauss’s Lemma.- Appendix D On Surfaces.- D1. Vector Fields on Plane Domains.- D2. Charts and Vector Fields.- D3. Differential Forms on a Surface.- Appendix E Proof of Borsuk’s Theorem.- Hints and Answers.- References.- Index of Symbols.
£33.99
Springer-Verlag New York Inc. Classical Descriptive Set Theory
Book SynopsisDescriptive set theory has been one of the main areas of research in set theory for almost a century. It includes a wide variety of examples, more than 400 exercises, and applications, in order to illustrate the general concepts and results of the theory.Table of ContentsI Polish Spaces.- 1. Topological and Metric Spaces.- 1.A Topological Spaces.- 1.B Metric Spaces.- 2. Trees.- 2.A Basic Concepts.- 2.B Trees and Closed Sets.- 2.C Trees on Produtcs.- 2.D Leftmost Branches.- 2.E Well-founded Trees and Rank.- 2.F The Well-founded Part of a Tree.- 2.G The Kleene-Brouwer Ordering.- 3. Polish Spaces.- 3.A Definitions and Examples.- 3.B Extensions of Continuous Functions and Homeomorphisms.- 3.C Polish Subspaces of Polish Spaces.- 4. Compact Metrizable Spaces.- 4.A Basic Facts.- 4.B Examples.- 4.C A Universality Property of the Hilbert Cube.- 4.D Continuous Images of the Cantor Space.- 4.E The Space of Continuous Functions on a Compact Space.- 4.F The Hyperspace of Compact Sets.- 5. Locally Compact Spaces.- 6. Perfect Polish Spaces.- 6.A Embedding the Cantor Space in Perfect Polish Spaces.- 6.B The Cantor-Bendixson Theorem.- 6.C Cantor-Bendixson Derivatives and Ranks.- 7.Zero-dimensional Spaces.- 7.A Basic Facts.- 7.B A Topological Characterization of the Cantor Space.- 7.C A Topological Characterization of the Baire Space.- 7.D Zero-dimensional Spaces aa Subspaces of the Baire Space.- 7.F Polish Spaces as Continuous Images of the Baire Space.- 7.F Closed Subsets Homcomorphic to the Baire Space.- 8. Baire Category.- 8.A Meager Sets.- 8.B Baire Spaces.- 8.C Choquet Games and Spaces.- 8.D Strong Choquet Games and Spaces.- 8.E A Characterization of Polish Spaces.- 8.F Sets with the Baire Property.- 8.G Localization.- 8.H The Banach-Mazur Game.- 8.I Baire Measurable Functions.- 8.J Category Quantifiers.- 8.K The Kuratowski-Ulam Theorem.- 8.L Some Applications.- 8.M Separate and Joint Continuity.- 9. Polish Groups.- 9.A Metrizable and Polish Groups.- 9.B Examples of Polish Groups.- 9.C Basic Facts about Baire Groups and Their Actions.- 9.D Universal Polish Groups.- II Borel Sets.- 10. Measurable Spaces and Functions.- 10.A Sigma-Algebras and Their Generators.- 10.B Measurable Spaces and Functions.- 11. Borel Sets and Functions.- 11.A Borel Sets in Topological Spaces.- 11.B The Borel Hierarchy.- 11.C Borel Functions.- 12. Standard Borel Spaces.- 12.A Borel Sets and Functions in Separable Metrizable Spaces.- 12.B Standard Borel Spaces.- 12.C The Effros Borel Space.- 12.D An Application to Selectors.- 12.E Further Examples.- 12.F Standard Borel Groups.- 13. Borel Sets as Clopen Sets.- 13.A Turning Borel into Clopen Sets.- 13.B Other Representations of Borel Sets.- 13.C Turning Borel into Continuous Functions.- 14. Analytic Sets and the Separation Theorem.- 14.A Basic Facts about Analytic Sets.- 14.B The Lusin Separation Theorem.- 14.C Sousliri’s Theorem.- 15. Borel Injections and Isomorphisms.- 15.A Borel Injective Images of Borel Sets.- 15.B The Isomorphism Theorem.- 15.C Homomorphisms of Sigma-Algebras Induced by Point Maps.- 15.D Some Applications to Group Actions.- 16. Borel Sets and Baire Category.- 16.A Borel Definability of Category Notions.- 16.B The Vaught Transforms.- 16.C Connections with Model Theory.- 16.D Connections with Cohen’s Forcing Method.- 17. Borel Sets and Measures.- 17.A General Facts on Measures.- 17.B Borel Measures.- 17.C Regularity and Tightness of Measures.- 17.D Lusin’s Theorem on Measurable Functions.- 17.E The Space of Probability Borel Measures.- 17.F The Isomorphism Theorem for Measures.- 18. Uniformization Theorems.- 18.A The Jankov, von Neumann Uniformization Theorem.- 18.B “Large Section” Uniformization Results.- 18.C “Small Section” Uniformization Results.- 18.D Selectors and Transversals.- 19. Partition Theorems.- 19.A Partitions with a Comeager or Non-meager Piece.- 19.B A Ramsey Theorem for Polish Spaces.- 19.C The Galvin-Prikry Theorem.- 19.D Ramsey Sets and the Ellentuck Topology.- 19.E An Application to Banach Space Theory.- 20. Borel Determinacy.- 20.A Infinite Games.- 20.B Determinacy of Closed Games.- 20.C Borel Determinacy.- 20.D Game Quantifiers.- 21. Games People Play.- 21.A The *-Games.- 21.B Unfolding.- 21.C The Banach-Mazur or **-Games.- 21.D The General Unfolded Banach-Mazur Games.- 21.E Wadge Games.- 21.F Separation Games and Hurewicz’s Theorem.- 21.G Turing Degrees.- 22. The Borel Hierarchy.- 22. A Universal Sets.- 22.B The Borel versus the Wadge Hierarchy.- 22.C Structural Properties.- 22.D Additional Results.- 22.E The Difference Hierarchy.- 23. Some Examples.- 23.A Combinatorial Examples.- 23.B Classes of Compact Sets.- 23.C Sequence Spaces.- 23.D Classes of Continuous Functions.- 23.E Uniformly Convergent Sequences.- 23.F Some Universal Sets.- 23.G Further Examples.- 24. The Baire Hierarchy.- 24.A The Baire Classes of Functions.- 24.B Functions of Baire Class 1.- III Analytic Sets.- 25. Representations of Analytic Sets.- 25.A Review.- 25.B Analytic Sets in the Baire Space.- 25.C The Souslin Operation.- 25.D Wellordered Unions and Intersections of Borel Sets.- 25. E Analytic Sets as Open Sets in Strong Choquet Spaces.- 26. Universal and Complete Sets.- 26.A Universal Analytic Sets.- 26.B Analytic Determinacy.- 26.C Complete Analytic Sets.- 26.D Classification up to Borel Isomorphism.- 27. Examples.- 27.A The Class of Ill-founded Trees.- 27.B Classes of Closed Sets.- 27.C Classes of Structures in Model Theory.- 27.D Isomorphism.- 27.E Some Universal Sets.- 27.F Miscellanea.- 28. Separation Theorems.- 28.A The Lusin Separation Theorem Revisited.- 28.B The Novilcov Separation Theorem.- 28.C Borel Sets with Open or Closed Sections.- 28.D Some Special Separation Theorems.- 28.E “Hurewicz-Type” Separation Theorems.- 29. Regularity Properties.- 29.A The Perfect Set Property.- 29.B Measure. Category, and Ramsey.- 29.C A Closure Property for the Souslin Operation.- 29.D The Class of C-Sets.- 29.E Analyticity of “Largeness” Conditions on Analytic Sets.- 30. Capacities.- 30.A The Basic Concept.- 30.B Examples.- 30.C The Choquet Capacitability Theorem.- 31. Analytic Well-founded Relations.- 31.A Bounds on Ranks of Analytic Well-founded Relations.- 31.B The Kunen-Martin Theorem.- IV Co-Analytic Sets.- 32. Review.- 32.A Basic Facts.- 32.B Representations of Co-Analytic Sets.- 32.C Regularity Properties.- 33. Examples.- 33.A Well-founded Trees and Wellorderings.- 33.B Classes of Closed Sets.- 33.C Sigma-ldoals of Compact Sets.- 33.D Differentiable Functions.- 33.E Everywhere Convergence.- 33.F Parametrizing Baire Class 1 Functions.- 33.G A Method for Proving Completeness.- 33.H Singular Functions.- 33.I Topological Examples.- 33.J Homeomorphisms of Compact Spaces.- 33.K Classes of Separable Banach Spaces.- 33.L Other Examples.- 34. Co-Analytic Ranks.- 34.A Ranks and Prewellorderings.- 34.B Ranked Classes.- 34.C Co-Analytic Ranks.- 34.D Derivatives.- 34.E Co-Analytic Ranks Associated with Borel Derivatives.- 34.F Examples.- 35. Rank Theory.- 35.A Basic Properties of Ranked Classes.- 35.B Parametrizing Bi-Analytic and Borel Sets.- 35.C Reflection Theorems.- 35.D Boundedness Properties of Ranks.- 35.E The Rank Method.- 35.F The Strategic Uniformization Theorem.- 35.G Co-Analytic Families of Closed Sets and Their Sigma-Ideals.- 35.H Borel Sots with F? and K? Sections.- 36. Scales and Uniformiiatiou.- 36.A Kappa-Souslin Sets.- 36.B Scales.- 36.C Sealed Classes and Urniformization.- 36.D The Novikov-Kondô Uniformization Theorem.- 36.E Regularity Properties of Uniformizing Functions.- 36.F Uniforniizing Co-Analytic Sets with Large Sections.- 36.G Examples of Co-Analytic Scales.- V Projective Sets.- 37. The Projective Hierarchy.- 37.A Basic Facts.- 37.B Examples.- 38. Projective Determinacy.- 38.A The Second Level of the Projective Hierarchy.- 38.B Projective Determinacy.- 38.C Regularity Properties.- 39. The Periodicity Theorems.- 39.A Periodicity in the Projective Hierarchy.- 39.B The First Periodicity Theorem.- 39.C The Second Periodicity Theorem.- 39.D The Third Periodicity Theorem.- 40. Epilogue.- 40.A Extensions of the Projective Hierarchy.- 40.B Effective Descriptive Set Theory.- 40.C Large Cardinals.- 40.D Connections to Other Areas of Mathematics.- Appendix A. Ordinals and Cardinals.- Appendix B. Well-founded Relations.- Appendix C. On Logical Notation.- Notes and Hints.- References.- Symbols and Abbreviations.
£48.59
Springer Mathematical Analysis
Book Synopsis1 Real Numbers.- 1.1 Sets, Relations, Functions.- 1.2 Numbers.- 1.3 Infinite Sets.- 1.4 Incommensurability.- 1.5 Ordered Fields.- 1.6 Functions on R.- 1.7 Intervals in R.- 1.8 Algebraic and Transcendental Numbers.- 1.9 Existence of R.- 1.10 Exercises.- 1.11 Notes.- 2 Sequences and Series.- 2.1 Sequences.- 2.2 Continued Fractions.- 2.3 Infinite Series.- 2.4 Rearrangements of Series.- 2.5 Unordered Series.- 2.6 Exercises.- 2.7 Notes.- 3 Continuous Functions on Intervals.- 3.1 Limits and Continuity.- 3.2 Two Fundamental Theorems.- 3.3 Uniform Continuity.- 3.4 Sequences of Functions.- 3.5 The Exponential function.- 3.6 Trigonometric Functions.- 3.7 Exercises.- 3.8 Notes.- 4 Differentiation.- 4.1 Derivatives.- 4.2 Derivatives of Some Elementary Functions.- 4.3 Convex Functions.- 4.4 The Differential Calculus.- 4.5 L'Hospital's Rule.- 4.6 Higher Order Derivatives.- 4.7 Analytic Functions.- 4.8 Exercises.- 4.9 Notes.- 5 The Riemann Integral.- 5.1 Riemann Sums.- 5.2 Existence Results.- 5.3 ProTrade ReviewThis is a very good textbook presenting a modern course in analysis both at the advanced undergraduate and at the beginning graduate level. It contains 14 chapters, a bibliography, and an index. At the end of each chapter interesting exercises and historical notes are enclosed.\par From the cover: ``The book begins with a brief discussion of sets and mappings, describes the real number field, and proceeds to a treatment of real-valued functions of a real variable. Separate chapters are devoted to the ideas of convergent sequences and series, continuous functions, differentiation, and the Riemann integral (of a real-valued function defined on a compact interval). The middle chapters cover general topology and a miscellany of applications: the Weierstrass and Stone-Weierstrass approximation theorems, the existence of geodesics in compact metric spaces, elements of Fourier analysis, and the Weyl equidistribution theorem. Next comes a discussion of differentiation of vector-valued functions of several real variables, followed by a brief treatment of measure and integration (in a general setting, but with emphasis on Lebesgue theory in Euclidean spaces). The final part of the book deals with manifolds, differential forms, and Stokes' theorem [in the spirit of M. Spivak's: ``Calculus on manifolds'' (1965; Zbl 141.05403)] which is applied to prove Brouwer's fixed point theorem and to derive the basic properties of harmonic functions, such as the Dirichlet principle''. ZENTRALBLATT MATH A. Browder Mathematical Analysis An Introduction "Everything needed is clearly defined and formulated, and there is a reasonable number of examples…. Anyone teaching a year course at this level to should seriously consider this carefully written book. In the reviewer's opinion, it would be a real pleasure to use this text with such a class."—MATHEMATICAL REVIEWSTable of Contents1 Real Functions 2 Sequences and Series 3 Continuous Functions on Intervals 4 Differentiation 5 The Riemann Integral 6 Topology 7 Function Spaces 8 Differentiable Maps 9 Measures 10 Integration 11 Manifolds 12 Multilinear Algebra 13 Differential Forms 14 Integration on Manifolds
£42.74
