Book Description

A thorough introduction to a natural, geometric and intuitively appealing branch of mathematics! Topology is an excellent subject for one wanting to develop skills in writing clear and precise proofs. Written with the mathematically mature reader in mind, Patty's introductory treatment includes easy-to-follow, thorough proofs as well as clear and direct notation. Historical comments are dispersed throughout, and there are many exercises of varying degrees of difficulty. ... Read more

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This valuable reference text provides a thorough introduction to the theory of topological rings and modules -- presenting basic information and focusing on problems of topologization and extensions of ring topologies. ... Read more

Book Description

Topological and Algebraic Structures in Fuzzy Sets hasthese unique features: strategically located at the juncture of fuzzy sets, topology,algebra, lattices, foundations of mathematics; major studies inuniformities and convergence structures, fundamental examples inlattice-valued topology, modifications and extensions of sobriety,categorical aspects of lattice-valued subsets, logic and foundationsof mathematics, t-norms and associated algebraic and orderedstructures; internationally recognized authorities clarify deepmathematical aspects of fuzzy sets, particularly those topological oralgebraic in nature; comprehensive bibliographies and tutorialnature of longer chapters take readers to the frontier of each topic;extensively referenced introduction unifies volume and guidesreaders to chapters closest to their interests; annotated openquestions direct future research in the mathematics of fuzzy sets;suitable as a text for advanced graduate students. Audience: Scholars and students of fuzzy sets and their vitalinterplay with topology, algebra, and the foundations of mathematics. ... Read more

Book Description

For those working in singularity theory or other areas of complex geometry, this volume will open the door to the study of Frobenius manifolds. In the first part Hertling explains the theory of manifolds with a multiplication on the tangent bundle. He then presents a simplified explanation of the role of Frobenius manifolds in singularity theory along with all the necessary tools and several applications. Readers will benefit from this careful and sound study of the fundamental structures and results in this exciting branch of mathematics. ... Read more

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Hardbound. ... Read more

Book Description

The 20th Century brought the rise of General Topology. It arose from the effort to establish a solid base for Analysis and it is intimately related to the success of set theory. Many Valued Topology and Its Applications seeks to extend the field by taking the monadic axioms of general topology seriously and continuing the theory of topological spaces as topological space objects within an almost completely ordered monad in a given base category C. The richness of this theory is shown by the fundamental fact that the category of topological space objects in a complete and cocomplete (epi, extremal mono)-category C is topological over C in the sense of J. Adamek, H. Herrlich, and G.E. Strecker. Moreover, a careful, categorical study of the most important topological notions and concepts is given -- e.g., density, closedness of extremal subobjects, Hausdorff's separation axiom, regularity, and compactness. An interpretation of these structures, not only by the ordinary filter monad, but also by many valued filter monads, underlines the richness of the explained theory and gives rise to new concrete concepts of topological spaces -- so-called many valued topological spaces. Hence, many valued topological spaces play a significant role in various fields of mathematics -- e.g., in the theory of locales, convergence spaces, stochastic processes, and smooth Borel probability measures.

In its first part, the book develops the necessary categorical basis for general topology. In the second part, the previously given categorical concepts are applied to monadic settings determined by many valued filter monads. The third part comprises various applications of many valued topologies to probability theory and statistics as well as to non-classical model theory. These applications illustrate the significance of many valued topology for further research work in these important fields. ... Read more

Book Description

Topological defects formed at symmetry-breaking phasetransitions play an important role in many different fields ofphysics. They appear in many condensed matter systems at lowtemperature, including vortices in superfluid He-4, defects in He-3,quantized magnetic flux tubes in type-II superconductors, anddisclination lines and other defects in liquid crystals. In cosmology,unified gauge theories of particle interactions suggest a sequence ofphase transitions in the very early universe, some of which may leadto defect formation. In astrophysics, defects play an important rolein the dynamics of neutron stars. The present book begins with a Condensed Matter - High Energy &Cosmology dictionary, prepared by Grigory Volovik, which gives acommon basis to ideas pertaining to these branches of modern physics.The following papers, written by experts in their respective fields,provide the fundamental concepts of symmetry-breaking phasetransitions. The book represents the fruits of an exchange of ideasbetween particle physicists, cosmologists and condensed matterphysicists. ... Read more

Book Description

This volume contains the proceedings from the workshop on"Geometry, Topology and Dynamics" held at CRM at the Universityof Montreal. The event took place at a crucial time with respectto symplectic developments. During the previous year, Seibergand Witten had just introduced the famous gauge equations.Taubes then extracted new invariants that were shown to beequivalent in some sense to a particular form of Gromovinvariants for symplectic manifolds in dimension 4. WithGromov's deformation theory, this constitutes an importantadvance in symplectic geometry by furnishing existence criteria.

Meanwhile, contact geometry was rapidly developing. Using bothholomorphic arguments in symplectizations of contact manifoldsand ad hoc topological arguments---or even gauge theoreticmethods---several results were obtained on 3-dimensional contactmanifolds and new surprising facts were derived about theBennequin-Thurston invariant.

Furthermore, a fascinating relation exists between Hofer'sgeometry, pseudoholomorphic curves and the $K$-area recentlyintroduced by Gromov. Finally, longstanding conjectures on theflux were resolved in a substantial number of specific cases bycomparing various aspects of Floer-Novikov homology with Morsehomology.

The papers in this volume are written by leading experts and areall clear, comprehensive, and original. The work covers acomplete range of exciting new developments in symplectic andcontact geometries. ... Read more

Book Description

These are the proceedings of the joint seminar by M.I.T. and Harvard on the current Developments in mathematics for the year 2000. Established in 1995, this seminar has been continued on the third weekend of November every year. The organizing committee for the seminar consisted of distinguished mathematicians from the mathematics departments of both institutions: Barry Mazur, Wilfried Schmid, and Shing-Tung Yau from Harvard, and David Jerison, A. J. de Jong and George Lustig from M.I.T.. We trust that these proceedings will be of interest to many mathematicians, and will inspire future developments and research pursuits in mathematics. ... Read more

Book Description

This subject has been of great interest both to topologists and to number theorists. The first part of this book describes some of the work of Kuo-Tsai Chen on iterated integrals and the fundamental group of a manifold. The author attempts to make his exposition accessible to beginning graduate students. He then proceeds to apply Chen’s constructions to algebraic geometry, showing how this leads to some results on algebraic cycles and the Abel–Jacobi homomorphism. Finally, he presents a more general point of view relating Chen’s integrals to a generalization of the concept of linking numbers, and ends up with a new invariant of homology classes in a projective algebraic manifold. The book is based on a course given by the author at the Nankai Institute of Mathematics in the fall of 2001. ... Read more

Book Description

Excellent, well-written introductory text begins with the algebra of subsets and of rings and fields of sets. Placing the algebra of partial order within the context of topologic situations, it covers complementation and ideal theory in the distributive lattice, closure function, neighborhood topology, open and closed sets, topological maps, the derived set in T1-space and the topological product, and convergence in metrical space and convergence topology. Numerous exercises. 1960 ed. Prefaces. Index of Proper Names.
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Reviews (1)

A book for graduate math students
"Set Topology" is a book that bridges between books that only other PHD's read and ones that an engineer just interested in modern topology can read. I wouldn't tell a first time student to read it, but it is written so that if you have a background in topology, you can follow things like Holder's inequality. It has a useful section on convergence topology that applies to much of the applications of modern topology. I frankly wish I hadn't spent the money on it! ... Read more

Book Description

"A General Topology Workbook" presents elementary general topology in an unconventional way. Highly influenced by the legendary ideas of R.L. Moore, the author has taught several generations of mathematics students with these materials, proving again the usefulness and stimulation of the Moore method. The first part of the book gives a quick review of the basic definitions of the subject, interspersed with a large number of exercises, some of which are described as theorems. The second part contains complete solutions and complete proofs, providing students the opportunity to explore the details of solution and proof in comparison with what they have devised themselves. The book could be used for a moderately paced, one semester, upper division general topology course devoted to this method. ... Read more

Reviews (1)

A concentrated, intense way to learn general topology
This book brings back memories of a graduate course in general topology that I took as an undergraduate, which was taught via the "Moore method", after the late Robert Lee Moore, who invented it. Handouts were given to the class (there were only 3 of us), and each of us was expected to work out or prove every result in the handout, without consulting references or collaborating with other students. Theorems were to be proved, or counterexamples given, but we did not know a priori which item from the handout was actually true or false. Needless to say this took a lot of work, and all of us had to present our results on the blackboard for scrutiny by both classmates and instructor.

The Moore method has its defenders and detractors. It certainly encourages originality of thought and strict intellectual honesty. Students can find incredible reinforcment as they discover that they can indeed give original proofs of sometimes very difficult (and famous) results in general topology. The downside is that not as much material is covered as compared to a traditional course in general topology. Students who are hungry to get to the frontiers of research might become impatient because of this.

This book does not follow the strict methodologies that we followed in our class, but instead reveals to the reader which results are true and then encourages their proof. Readers are also lead through the construction of examples and counterexamples, allowing them to gain more of the intuition needed for a thorough understanding of general topology. It is also a good book to use for independent study, as the answers to the results are given in the book (and this actually is the major portion of its bulk). ... Read more

Book Description

This book describes those singularities encountered in different branches of mathematics. The distinguished mathematician, Vladimir Arnold, avoids giving difficult proofs of all the results in order to provide the reader with a concise and accessible overview of the many guises and areas in which singularities appear.Some of these areas include geometry and optics, optimal control theory and algebraic geometry, reflection groups theory, dynamical systems theory, and the classical and quantum catastrophe theory. ... Read more

Book Description

Geodesic flows are of considerable current interest since they are, perhaps, the most remarkable class of conservative dynamical systems. They provide a unified arena in which one can explore numerous interplays among several fields, including smooth ergodic theory, symplectic and Riemannian geometry, and algebraic topology.

The work begins with a concise introduction to the geodesic flow of a complete Riemannian manifold, emphasizing its symplectic properties and culminating with various applications, such as the non-existence of continuous invariant Lagrangian subbundles for manifolds with conjugate points. Subsequent chapters develop the relationship between the exponential growth rate of the average number of geodesic arcs between two points in the manifold and the topological entropy of the geodesic flow. A complete proof of Mae's formula relating these two quantities is presented. A final chapter explores the link between the topological entropy of the geodesic flow and the homology of the loop space of a manifold.

This self-contained monograph will be of interest to graduate students and researchers of dynamical systems and differential geometry. Numerous exercises and examples as well as a comprehensive bibliography and index make the work an excellent self-study resource or useful text for a one-semester course or seminar.

Series: Progress in Mathematics, Vol. 180 ... Read more

Book Description

Complex Polynomials explores the geometric theory of polynomials and rational functions in the plane. Early chapters build the foundations of complex variable theory, melding together ideas from algebra, topology, and analysis. Throughout the book, the author introduces a variety of ideas and constructs theories around them, incorporating much of the classical theory of polynomials as he proceeds. These ideas are used to study a number of unsolved problems. Several solutions to problems are given, including a comprehensive account of the geometric convolution theory. ... Read more

Book Description

This book deals with geometric and topological aspects ofdiscrete groups. The main topics are hyperbolic groups due toGromov, automatic group theory, invented and developed byEpstein, whose subjects are groups that can be manipulated bycomputers, and Kleinian group theory, which enjoys the longesttradition and the richest contents within the theory of discretesubgroups of Lie groups.

What is common among these three classes of groups is that whenseen as geometric objects, they have the properties of anegatively curved space rather than a positively curved space.As Kleinian groups are groups acting on a hyperbolic space ofconstant negative curvature, the technique employed to studythem is that of hyperbolic manifolds, typical examples ofnegatively curved manifolds. Although hyperbolic groups in thesense of Gromov are much more general objects than Kleiniangroups, one can apply for them arguments and techniques that arequite similar to those used for Kleinian groups. Automaticgroups are further general objects, including groups havingproperties of spaces of curvature 0. Still, relationshipsbetween automatic groups and hyperbolic groups are examined hereusing ideas inspired by the study of hyperbolic manifolds. Inall of these three topics, there is a "soul" of negativecurvature upholding the theory. The volume would make a finetextbook for a graduate-level course in discrete groups. ... Read more

Book Description

Researchers, teachers and graduate students in algebra and topology--familiar with the very basic notions of category theory--will welcome this categorical introduction to some of the key areas of modern mathematics, without being forced to study category theory. Rather, each of the eight largely independent chapters analyzes a particular subject, revealing the power and applicability of the categorical foundations in each case. ... Read more