Combinatorial Methods in Topology and Algebraic Geometry

Author: John R. Harper,Richard Mandelbaum

Publisher: American Mathematical Soc.

ISBN: 9780821850398

Category: Mathematics

Page: 349

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This collection marks the recent resurgence of interest in combinatorial methods, resulting from their deep and diverse applications both in topology and algebraic geometry. Nearly thirty mathematicians met at the University of Rochester in 1982 to survey several of the areas where combinatorial methods are proving especially fruitful: topology and combinatorial group theory, knot theory, 3-manifolds, homotopy theory and infinite dimensional topology, and four manifolds and algebraic surfaces. This material is accessible to advanced graduate students with a general course in algebraic topology along with some work in combinatorial group theory and geometric topology, as well as to established mathematicians with interests in these areas.For both student and professional mathematicians, the book provides practical suggestions for research directions still to be explored, as well as the aesthetic pleasures of seeing the interplay between algebra and topology which is characteristic of this field. In several areas the book contains the first general exposition published on the subject. In topology, for example, the editors have included M. Cohen, W. Metzler and K. Sauerman's article on 'Collapses of $K\times I$ and group presentations' and Metzler's 'On the Andrews-Curtis-Conjecture and related problems'. In addition, J. M. Montesino has provided summary articles on both 3 and 4-manifolds.

Low Dimensional Topology

Author: Samuel J. Lomonaco

Publisher: American Mathematical Soc.

ISBN: 0821850164

Category: Mathematics

Page: 346

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This volume arose from a special session on Low Dimensional Topology organized and conducted by Dr. Lomonaco at the American Mathematical Society meeting held in San Francisco, California, January 7-11, 1981.

Low-dimensional Topology and Kleinian Groups

Author: D. B. A. Epstein

Publisher: CUP Archive

ISBN: 9780521339056

Category: Mathematics

Page: 321

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Volume 2 is divided into three parts: the first 'Surfaces' contains an article by Thurston on earthquakes and by Penner on traintracks. The second part is entitled 'Knots and 3-Manifolds' and the final part 'Kleinian Groups'.

Elements of Point Set Topology

Author: John D. Baum

Publisher: Courier Corporation

ISBN: 0486668266

Category: Mathematics

Page: 150

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Topology continues to be a topic of prime importance in contemporary mathematics, but until the publication of this book there were few if any introductions to topology for undergraduates. This book remedied that need by offering a carefully thought-out, graduated approach to point set topology at the undergraduate level. To make the book as accessible as possible, the author approaches topology from a geometric and axiomatic standpoint; geometric, because most students come to the subject with a good deal of geometry behind them, enabling them to use their geometric intuition; axiomatic, because it parallels the student's experience with modern algebra, and keeps the book in harmony with current trends in mathematics. After a discussion of such preliminary topics as the algebra of sets, Euler-Venn diagrams and infinite sets, the author takes up basic definitions and theorems regarding topological spaces (Chapter 1). The second chapter deals with continuous functions (mappings) and homeomorphisms, followed by two chapters on special types of topological spaces (varieties of compactness and varieties of connectedness). Chapter 5 covers metric spaces. Since basic point set topology serves as a foundation not only for functional analysis but also for more advanced work in point set topology and algebraic topology, the author has included topics aimed at students with interests other than analysis. Moreover, Dr. Baum has supplied quite detailed proofs in the beginning to help students approaching this type of axiomatic mathematics for the first time. Similarly, in the first part of the book problems are elementary, but they become progressively more difficult toward the end of the book. References have been supplied to suggest further reading to the interested student.

Concepts of Modern Mathematics

Author: Ian Stewart

Publisher: Courier Corporation

ISBN: 0486134954

Category: Mathematics

Page: 368

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In this charming volume, a noted English mathematician uses humor and anecdote to illuminate the concepts of groups, sets, subsets, topology, Boolean algebra, and other mathematical subjects. 200 illustrations.

Topology of Algebraic Varieties and Singularities

Conference in Honor of Anatoly Libgober's 60th Birthday, June 22-26, 2009, Jaca, Huesca, Spain

Author: Anatoly Libgober

Publisher: American Mathematical Soc.

ISBN: 0821848909

Category: Mathematics

Page: 467

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This volume contains invited expository and research papers from the conference Topology of Algebraic Varieties, in honor of Anatoly Libgober's 60th birthday, held June 22-26, 2009, in Jaca, Spain. The volume contains four parts corresponding to the four main focal points of the conference: algebraic geometry and fundamental groups, braids and knots, hyperplane arrangements, and singularities. Together, the papers provide an overview of the current status of a broad range of topological questions in Algebraic Geometry.

Topology and Geometry in Dimension Three

Triangulations, Invariants, and Geometric Structures : Conference in Honor of William Jaco's 70th Birthday, June 4-6, 2010, Oklahoma State University, Stillwater, Oklahoma

Author: William H. Jaco

Publisher: American Mathematical Soc.

ISBN: 0821852957

Category: Mathematics

Page: 196

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This volume contains the proceedings of a conference held from June 4-6, 2010, at Oklahoma State University, in honor of William (Bus) Jaco's 70th birthday. His contributions to research in low dimensional geometry and topology and to the American mathematical community, especially through his work for the American Mathematical Society, were recognized during the conference. The focus of the conference was on triangulations and geometric structures for three-dimensional manifolds. The papers in this volume present significant new results on these topics, as well as in geometric group theory.

Compact Riemann Surfaces

An Introduction to Contemporary Mathematics

Author: Jürgen Jost

Publisher: Springer Science & Business Media

ISBN: 3662034468

Category: Mathematics

Page: 295

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This book is novel in its broad perspective on Riemann surfaces: the text systematically explores the connection with other fields of mathematics. The book can serve as an introduction to contemporary mathematics as a whole, as it develops background material from algebraic topology, differential geometry, the calculus of variations, elliptic PDE, and algebraic geometry. The book is unique among textbooks on Riemann surfaces in its inclusion of an introduction to Teichmüller theory. For this new edition, the author has expanded and rewritten several sections to include additional material and to improve the presentation.

Algebraic Invariants of Links

Author: Jonathan Arthur Hillman

Publisher: World Scientific

ISBN: 9814407399

Category: Mathematics

Page: 353

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This book serves as a reference on links and on the invariants derived via algebraic topology from covering spaces of link exteriors. It emphasizes the features of the multicomponent case not normally considered by knot-theorists, such as longitudes, the homological complexity of many-variable Laurent polynomial rings, the fact that links are not usually boundary links, free coverings of homology boundary links, the lower central series as a source of invariants, nilpotent completion and algebraic closure of the link group, and disc links. Invariants of the types considered here play an essential role in many applications of knot theory to other areas of topology. This second edition introduces two new chapters OCo twisted polynomial invariants and singularities of plane curves. Each replaces brief sketches in the first edition. Chapter 2 has been reorganized, and new material has been added to four other chapters.

Algebraic Invariants of Links

Author: Jonathan Hillman

Publisher: World Scientific

ISBN: 9814407402

Category: Mathematics

Page: 372

View: 9777

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This book serves as a reference on links and on the invariants derived via algebraic topology from covering spaces of link exteriors. It emphasizes the features of the multicomponent case not normally considered by knot-theorists, such as longitudes, the homological complexity of many-variable Laurent polynomial rings, the fact that links are not usually boundary links, free coverings of homology boundary links, the lower central series as a source of invariants, nilpotent completion and algebraic closure of the link group, and disc links. Invariants of the types considered here play an essential role in many applications of knot theory to other areas of topology. This second edition introduces two new chapters — twisted polynomial invariants and singularities of plane curves. Each replaces brief sketches in the first edition. Chapter 2 has been reorganized, and new material has been added to four other chapters. Sample Chapter(s) Chapter 1: Links (205 KB) Contents:Abelian Covers:LinksHomology and Duality in CoversDeterminantal InvariantsThe Maximal Abelian CoverSublinks and Other Abelian CoversTwisted Polynomial InvariantsApplications: Special Cases and Symmetries:Knot ModulesLinks with Two ComponentsSymmetriesSingularities of Plane Algebraic CurvesFree Covers, Nilpotent Quotients and Completion:Free CoversNilpotent QuotientsAlgebraic ClosureDisc Links Readership: Graduate students and academics in geometry and topology. Keywords:Algebraic Invariant;Links;Abelian Cover;Twisted Polynomial Invariant;Knot ModuleReviews: “The main change to the book is the addition of two chapters. These are timely and welcome additions to an already useful and comprehensive book.” Mathematical Reviews Reviews of the First Edition: “Jonathan Hillman has successfully filled an unfortunate gap in the literature of low-dimensional topology … With this up-to-date book we have a reference covering a range of topics that until now were available only in their original sources … Hillman has done an excellent job of referencing his book, both within the text and in the bibliography, with several hundred references included.” Mathematical Reviews “Algebraic Invariants of Links is masterful, offering a survey of work, much of which has not been summarized elsewhere. It is an essential reference for those interested in link theory … it is unique and valuable.” Bulletin of the American Mathematical Society “The author, who is one of the major experts on the topic, must be surely congratulated for this attractive book, written in a careful, very precise and quite readable style. It serves as an excellent self-contained and up-to-date monograph on the algebraic invariants of links … I strongly recommend this beautiful book to anyone interested in the algebraic theory of links and its applications.” Mathematics Abstracts

New Ideas In Low Dimensional Topology

Author: Manturov Vassily Olegovich,Kauffman Louis H

Publisher: World Scientific

ISBN: 9814630632

Category: Mathematics

Page: 540

View: 9484

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This book consists of a selection of articles devoted to new ideas and developments in low dimensional topology. Low dimensions refer to dimensions three and four for the topology of manifolds and their submanifolds. Thus we have papers related to both manifolds and to knotted submanifolds of dimension one in three (classical knot theory) and two in four (surfaces in four dimensional spaces). Some of the work involves virtual knot theory where the knots are abstractions of classical knots but can be represented by knots embedded in surfaces. This leads both to new interactions with classical topology and to new interactions with essential combinatorics.

Geometry and Topology Down Under

Author: Craig D. Hodgson,William H. Jaco,Martin G. Scharlemann,Stephan Tillmann

Publisher: American Mathematical Soc.

ISBN: 0821884808

Category: Mathematics

Page: 369

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This book contains the proceedings of the conference Geometry & Topology Down Under, held July 11-22, 2011, at the University of Melbourne, Parkville, Australia, in honour of Hyam Rubinstein. The main topic of the book is low-dimensional geometry and topology. It includes both survey articles based on courses presented at the conferences and research articles devoted to important questions in low-dimensional geometry. Together, these contributions show how methods from different fields of mathematics contribute to the study of 3-manifolds and Gromov hyperbolic groups. It also contains a list of favorite problems by Hyam Rubinstein.

The Wild World of 4-manifolds

Author: Alexandru Scorpan

Publisher: American Mathematical Soc.

ISBN: 0821837494

Category: Mathematics

Page: 609

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What a wonderful book! I strongly recommend this book to anyone, especially graduate students, interested in getting a sense of 4-manifolds. --MAA Reviews The book gives an excellent overview of 4-manifolds, with many figures and historical notes. Graduate students, nonexperts, and experts alike will enjoy browsing through it. -- Robion C. Kirby, University of California, Berkeley This book offers a panorama of the topology of simply connected smooth manifolds of dimension four. Dimension four is unlike any other dimension; it is large enough to have room for wild things to happen, but small enough so that there is no room to undo the wildness. For example, only manifolds of dimension four can exhibit infinitely many distinct smooth structures. Indeed, their topology remains the least understood today. To put things in context, the book starts with a survey of higher dimensions and of topological 4-manifolds. In the second part, the main invariant of a 4-manifold--the intersection form--and its interaction with the topology of the manifold are investigated. In the third part, as an important source of examples, complex surfaces are reviewed. In the final fourth part of the book, gauge theory is presented; this differential-geometric method has brought to light how unwieldy smooth 4-manifolds truly are, and while bringing new insights, has raised more questions than answers. The structure of the book is modular, organized into a main track of about two hundred pages, augmented by extensive notes at the end of each chapter, where many extra details, proofs and developments are presented. To help the reader, the text is peppered with over 250 illustrations and has an extensive index.

Low Dimensional Topology

Proceedings of a Conference on Low Dimensional Topology, January 12-17, 1998, Funchal, Madeira, Portugal

Author: Hanna Nencka

Publisher: American Mathematical Soc.

ISBN: 0821808842

Category: Mathematics

Page: 249

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This volume presents the proceedings from the conference on low dimensional topology held at the University of Madeira (Portugal). The event was attended by leading scientists in the field from the U.S., Asia, and Europe. The book has two main parts. The first is devoted to the Poincare conjecture, characterizations of $PL$-manifolds, covering quadratic forms of links and to categories in low dimensional topology that appear in connection with conformal and quantum field theory. The second part of the volume covers topological quantum field theory and polynomial invariants for rational homology 3-spheres, derived from the quantum $SU(2)$-invariants associated with the first cohomology class modulo two, knot theory, and braid groups. This collection reflects development and progress in the field and presents interesting and new results.

Classical Topology and Combinatorial Group Theory

Author: John Stillwell

Publisher: Springer Science & Business Media

ISBN: 1461243726

Category: Mathematics

Page: 336

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In recent years, many students have been introduced to topology in high school mathematics. Having met the Mobius band, the seven bridges of Konigsberg, Euler's polyhedron formula, and knots, the student is led to expect that these picturesque ideas will come to full flower in university topology courses. What a disappointment "undergraduate topology" proves to be! In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams. Pictures are kept to a minimum, and at the end the student still does nr~ understand the simplest topological facts, such as the rcason why knots exist. In my opinion, a well-balanced introduction to topology should stress its intuitive geometric aspect, while admitting the legitimate interest that analysts and algebraists have in the subject. At any rate, this is the aim of the present book. In support of this view, I have followed the historical development where practicable, since it clearly shows the influence of geometric thought at all stages. This is not to claim that topology received its main impetus from geometric recreations like the seven bridges; rather, it resulted from the l'isualization of problems from other parts of mathematics-complex analysis (Riemann), mechanics (Poincare), and group theory (Dehn). It is these connec tions to other parts of mathematics which make topology an important as well as a beautiful subject.

Riemann, Topology, and Physics

Author: Michael Monastyrsky

Publisher: Springer Science & Business Media

ISBN: 1489935142

Category: Mathematics

Page: 158

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Soviet citizens can buy Monastyrsky's biography of Riemann for eleven kopeks. This translated edition will cost considerably more, but it is still good value for the money. And we get Monastyrsky's monograph on topological methods in the bargain. It was a good idea of Birkhiiuser Boston to publish the two translations in one volume. The economics of publishing in a capitalist country make it impossible for us to produce the small cheap paperback booklets, low in quality of paper and high in quality of scholarship, at which the Soviet publishing industry excels. Monastyrsky's two booklets are out standing examples of the genre. By putting them together, Birkhiiuser has enabled them to fit into the Western book-marketing system. The two booklets were written separately and each is complete in itself, but they complement each other beautifully. The Riemann biography is short and terse, like Riemann's own writings. It describes in few words and fewer equations the revolutionary ideas which Riemann brought into mathematics and physics a hundred and twenty years ago. The topological methods booklet describes how some of these same ideas, after lying dormant for a century, found new and fruitful applications in the physics of our own time.

An Alpine Anthology of Homotopy Theory

Proceedings of the Second Arolla Conference on Algebraic Topology, August 24-29, 2004, Arolla, Switzerland

Author: Dominique Arlettaz,Kathryn Hess

Publisher: American Mathematical Soc.

ISBN: 082183696X

Category: Mathematics

Page: 209

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The second Arolla conference on algebraic topology brought together specialists covering a wide range of homotopy theory and $K$-theory. These proceedings reflect both the variety of talks given at the conference and the diversity of promising research directions in homotopy theory. The articles contained in this volume include significant contributions to classical unstable homotopy theory, model category theory, equivariant homotopy theory, and the homotopy theory of fusion systems, as well as to $K$-theory of both local fields and $C^*$-algebras.

Commutative Algebra and Algebraic Geometry

Joint International Meeting of the American Mathematical Society and the Indian Mathematical Society on Commutative Algebra and Algebraic Geometry, Bangalore, India, December 17-20, 2003

Author: Sudhir Ghorpade,Hema Srinivasan,Jugal Verma

Publisher: American Mathematical Soc.

ISBN: 0821836293

Category: Mathematics

Page: 173

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The first Joint AMS-India Mathematics Meeting was held in Bangalore (India). This book presents articles written by speakers from a special session on commutative algebra and algebraic geometry. Included are contributions from some leading researchers around the world in this subject area. The volume contains new and original research papers and survey articles suitable for graduate students and researchers interested in commutative algebra and algebraic geometry.