This updated and revised edition of a widely acclaimed and successful text for undergraduates examines topology of recent compact surfaces through the development of simple ideas in plane geometry. Containing over 171 diagrams, the approach allows for a straightforward treatment of its subject area. It is particularly attractive for its wealth of applications and variety of interactions with branches of mathematics, linked with surface topology, graph theory, group theory, vector field theory, and plane Euclidean and non-Euclidean geometry.
Examines topology of recent compact surfaces through the development of simple ideas in plane geometryContains a wealth of applications and a variety of interactions with branches of mathematics, linked with surface topology, graph theory, group theory, vector field theory, and plane Euclidean and non-Euclidean geometry

The central theme of this book is the study of self-dual connections on four-manifolds. The author's aim is to present a lucid introduction to moduli space techniques (for vector bundles with SO (3) as structure group) and to apply them to four-manifolds. The authors have adopted a topologists' perspective. For example, they have included some explicit calculations using the Atiyah-Singer index theorem as well as methods from equivariant topology in the study of the topology of the moduli space. Results covered include Donaldson's Theorem that the only positive definite form which occurs as an intersection form of a smooth four-manifold is the standard positive definite form, as well as those of Fintushel and Stern which show that the integral homology cobordism group of integral homology three-spheres has elements of infinite order. Little previous knowledge of differential geometry is assumed and so postgraduate students and research workers will find this both an accessible and complete introduction to currently one of the most active areas of mathematical research.

The remarkable developments in differential topology and how these recent advances have been applied as a primary research tool in quantum field theory are presented here in a style reflecting the genuinely two-sided interaction between mathematical physics and applied mathematics. The author, following his previous work (Nash/Sen: Differential Topology for Physicists, Academic Press, 1983), covers elliptic differential and pseudo-differential operators, Atiyah-Singer index theory, topological quantum field theory, string theory, and knot theory. The explanatory approach serves to illuminate and clarify these theories for graduate students and research workers entering the field for the first time.
Key Features
* Treats differential geometry, differential topology, and quantum field theory
* Includes elliptic differential and pseudo-differential operators, Atiyah-Singer index theory, topological quantum field theory, string theory, and knot theory
* Tackles problems of quantum field theory using differential topology as a tool

This 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 those learners who opt to do it this way and for those instructors who would like to encourage this so-called 'Moore approach', even for a modest segment of the teaching term or for part of the class. In reality, most students simply do not have the combination of time, background and motivation needed to implement such a plan fully. The accessibility, flexibility and completeness of this text enable it to be used equally effectively for more conventional instructor-led courses. Critically, it furnishes a rich variety of exercises and examples, many of which have specimen solutions, through which to gain in confidence and competence.

The theory of buildings was introduced by J Tits in order to focus on geometric and combinatorial aspects of simple groups of Lie type. Since then the theory has blossomed into an extremely active field of mathematical research having deep connections with topics as diverse as algebraic groups, arithmetic groups, finite simple groups, and finite geometries, as well as with graph theory and other aspects of combinatorics. This volume is an up-to-date survey of the theory of buildings with special emphasis on its interaction with related geometries. As such it will be an invaluable guide to all those whose research touches on these themes. The articles presented here are by experts in their respective fields and are based on talks given at the 1988 Buildings and Related Geometries conference at Pingree Park, Colorado. Topics covered include the classification and construction of buildings, finite groups associated with building-like geometries, graphs and association schemes.

Over the last number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. The first edition of Introduction to Symplectic Topology was published in 1995. The book was the first comprehensive introduction to the subject and became a key text in the area. A significantly revised second edition was published in 1998 introducing new sections and updates on the fast-developing area. This new third edition includes updates and new material to bring the book right up-to-date.

The term "stereotype space" was introduced in 1995 and denotes a category of locally convex spaces with surprisingly elegant properties. Its study gives an unexpected point of view on functional analysis that brings this fi eld closer to other main branches of mathematics, namely, to algebra and geometry. This volume contains the foundations of the theory of stereotype spaces, with accurate definitions, formulations, proofs, and numerous examples illustrating the interaction of this discipline with the category theory, the theory of Hopf algebras, and the four big geometric disciplines: topology, differential geometry, complex geometry, and algebraic geometry.

This book contains selected chapters on recent research in topology. It bridges the gap between recent trends of topological theories and their applications in areas like social sciences, natural sciences, soft computing, economics, theoretical chemistry, cryptography, pattern recognitions and granular computing. There are 14 chapters, including two chapters on mathematical economics from the perspective of topology. The book discusses topics on function spaces, relator space, preorder, quasi-uniformities, bitopological dynamical systems, b-metric spaces and related fixed point theory. This book is useful to researchers, experts and scientists in studying the cutting-edge research in topology and related areas and helps them applying topology in solving real-life problems the society and science are facing these days.. 

Since Benoit Mandelbrot's pioneering work in the late 1970s, scores of research articles and books have been published on the topic of fractals. Despite the volume of literature in the field, the general level of theoretical understanding has remained low; most work is aimed either at too mainstream an audience to achieve any depth or at too specialized a community to achieve widespread use. Written by celebrated mathematician and educator A.A. Kirillov, A Tale of Two Fractals is intended to help bridge this gap, providing an original treatment of fractals that is at once accessible to beginners and sufficiently rigorous for serious mathematicians. The work is designed to give young, non-specialist mathematicians a solid foundation in the theory of fractals, and, in the process, to equip them with exposure to a variety of geometric, analytical, and algebraic tools with applications across other areas.

Quantum cohomology has its origins in symplectic geometry and algebraic geometry, but is deeply related to differential equations and integrable systems. This text explains what is behind the extraordinary success of quantum cohomology, leading to its connections with many existing areas of mathematics as well as its appearance in new areas such as mirror symmetry.
Certain kinds of differential equations (or D-modules) provide the key links between quantum cohomology and traditional mathematics; these links are the main focus of the book, and quantum cohomology and other integrable PDEs such as the KdV equation and the harmonic map equation are discussed within this unified framework.
Aimed at graduate students in mathematics who want to learn about quantum cohomology in a broad context, and theoretical physicists who are interested in the mathematical setting, the text assumes basic familiarity with differential equations and cohomology.

This book presents material in two parts. Part one provides an introduction to crossed modules of groups, Lie algebras and associative algebras with fully written out proofs and is suitable for graduate students interested in homological algebra. In part two, more advanced and less standard topics such as crossed modules of Hopf algebra, Lie groups, and racks are discussed as well as recent developments and research on crossed modules.

Noncommutative geometry studies an interplay between spatial forms and algebras with non-commutative multiplication. This book covers the key concepts of noncommutative geometry and its applications in topology, algebraic geometry, and number theory. Our presentation is accessible to the graduate students as well as nonexperts in the field. The second edition includes two new chapters on arithmetic topology and quantum arithmetic.

This book is devoted to group-theoretic aspects of topological dynamics such as studying groups using their actions on topological spaces, using group theory to study symbolic dynamics, and other connections between group theory and dynamical systems. One of the main applications of this approach to group theory is the study of asymptotic properties of groups such as growth and amenability. The book presents recently developed techniques of studying groups of dynamical origin using the structure of their orbits and associated groupoids of germs, applications of the iterated monodromy groups to hyperbolic dynamical systems, topological full groups and their properties, amenable groups, groups of intermediate growth, and other topics. The book is suitable for graduate students and researchers interested in group theory, transformations defined by automata, topological and holomorphic dynamics, and theory of topological groupoids. Each chapter is supplemented by exercises of various levels of complexity.

This monograph is the first and an initial introduction to the theory of bitopological spaces and its applications. In particular, different families of subsets of bitopological spaces are introduced and various relations between two topologies are analyzed on one and the same set; the theory of dimension of bitopological spaces and the theory of Baire bitopological spaces are constructed, and various classes of mappings of bitopological spaces are studied. The previously known results as well the results obtained in this monograph are applied in analysis, potential theory, general topology, and theory of ordered topological spaces. Moreover, a high level of modern knowledge of bitopological spaces theory has made it possible to introduce and study algebra of new type, the corresponding representation of which brings one to the special class of bitopological spaces.
It is beyond any doubt that in the nearest future the areas of essential applications will be the theories of linear topological spaces and topological groups, algebraic and differential topologies, the homotopy theory, not to mention other fundamental areas of modern mathematics such as geometry, mathematical logic, the probability theory and many other areas, including those of applied nature.
Key Features:
- First monograph is "Generalized Lattices"
* The first introduction to the theory of bitopological spaces and its applications.

The book is devoted to universality problems.
A new approach to these problems is given using some specific spaces. Since the construction of these specific spaces is set-theoretical, the given theory can be applied to different topics of Topology such as:
universal mappings, dimension theory, action of groups, inverse spectra, isometrical embeddings, and so on.
.Universal spaces
.Universal mappings
.Dimension theory
.Actions of groups
.Isometric Universal Spaces

I The fixed point theorems of Brouwer and Schauder.- 1 The fixed point theorem of Brouwer and applications.- 2 The fixed point theorem of Schauder and applications.- II Measures of noncompactness.- 1 The general notion of a measure of noncompactness.- 2 The Kuratowski and Hausdorff measures of noncompactness.- 3 The separation measure of noncompactness.- 4 Measures of noncompactness in Banach sequences spaces.- 5 Theorem of Darbo and Sadovskii and applications.- III Minimal sets for a measure of noncompactness.- 1 o-minimal sets.- 2 Minimalizable measures of noncompactness.- IV Convexity and smoothness.- 1 Strict convexity and smoothness.- 2 k-uniform convexity.- 3 k-uniform smoothness.- V Nearly uniform convexity and nearly uniform smoothness.- 1 Nearly uniformly convex Banach spaces.- 2 Nearly uniformly smooth Banach spaces.- 3 Uniform Opial condition.- VI Fixed points for nonexpansive mappings and normal structure.- 1 Existence of fixed points for nonexpansive mappings: Kirk's theorem.- 2 The coefficient N(X) and its connection with uniform convexity.- 3 The weakly convergent sequence coefficient.- 4 Uniform smoothness, near uniform convexity and normal structure.- 5 Normal structure in direct sum spaces.- 6 Computation of the normal structure coefficients in Lp-spaces.- VII Fixed point theorems in the absence of normal structure.- 1 Goebel-Karlovitz's lemma and Lin's lemma.- 2 The coefficient M(X) and the fixed point property.- VIII Uniformly Lipschitzian mappings.- 1 Lifshitz characteristic and fixed points.- 2 Connections between the Lifshitz characteristic and certain geometric coefficients.- 3 The normal structure coefficient and fixed points.- IX Asymptotically regular mappings.- 1 A fixed point theorem for asymptotically regular mappings.- 2 Connections between the ?-characteristic and some other geometric coefficients.- 3 The weakly convergent sequence coefficient and fixed points.- X Packing rates and o-contractiveness constants.- 1 Comparable measures of noncompactness.- 2 Packing rates of a metric space.- 3 Connections between the packing rates and the normal structure coefficients.- 4 Packing rates in lp-spaces.- 5 Packing rates in Lpspaces.- 6 Packing rates in direct sum spaces.- References.- List of Symbols and Notations.

This monograph focuses on the geometric theory of motivic integration, which takes its values in the Grothendieck ring of varieties. This theory is rooted in a groundbreaking idea of Kontsevich and was further developed by Denef & Loeser and Sebag. It is presented in the context of formal schemes over a discrete valuation ring, without any restriction on the residue characteristic. The text first discusses the main features of the Grothendieck ring of varieties, arc schemes, and Greenberg schemes. It then moves on to motivic integration and its applications to birational geometry and non-Archimedean geometry. Also included in the work is a prologue on p-adic analytic manifolds, which served as a model for motivic integration. With its extensive discussion of preliminaries and applications, this book is an ideal resource for graduate students of algebraic geometry and researchers of motivic integration. It will also serve as a motivation for more recent and sophisticated theories that have been developed since.

This book provides an introduction to topological groups and the structure theory of locally compact abelian groups, with a special emphasis on Pontryagin-van Kampen duality, including a completely self-contained elementary proof of the duality theorem. Further related topics and applications are treated in separate chapters and in the appendix.

There are many proposed aims for scientific inquiry - to explain or predict events, to confirm or falsify hypotheses, or to find hypotheses that cohere with our other beliefs in some logical or probabilistic sense. This book is devoted to a different proposal - that the logical structure of the scientist's method should guarantee eventual arrival at the truth, given the scientist's background assumptions. Interest in this methodological property, called "logical reliability", stems from formal learning theory, which draws its insights not from the theory of probability, but from the theory of computability. Kelly first offers an accessible explanation of formal learning theory, then goes on to develop and explore a systematic framework in which various standard learning-theoretic results can be seen as special cases of simpler and more general considerations. Finally, Kelly clarifies the relationship between the resulting framework and other standard issues in the philosophy of science, such as probability, causation, and relativism. Extensively illustrated with figures by the author, The Logic of Reliable Inquiry assumes only introductory knowledge of basic logic and computability theory. It is a major contribution to the literature and will be essential reading for scientists, statiticians, psychologists, linguists, logicians, and philosophers.