Topology Books

Book SynopsisOver the years, this book has become a standard reference and guide in the set theory community. It provides a comprehensive account of the theory of large cardinals from its beginnings and some of the direct outgrowths leading to the frontiers of contemporary research, with open questions and speculations throughout.Table of ContentsPreliminaries.- Beginnings.- Partition Properties.- Forcing and Sets of Reals.- Aspects of Measurability.- Strong Hypotheses.- Determinacy.

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Book SynopsisIt is a great satisfaction for a mathematician to witness the growth and expansion of a theory in which he has taken some part during its early years. When H. Weyl coined the words "classical groups", foremost in his mind were their connections with invariant theory, which his famous book helped to revive. Although his approach in that book was deliberately algebraic, his interest in these groups directly derived from his pioneering study of the special case in which the scalars are real or complex numbers, where for the first time he injected Topology into Lie theory. But ever since the definition of Lie groups, the analogy between simple classical groups over finite fields and simple classical groups over IR or C had been observed, even if the concept of "simplicity" was not quite the same in both cases. With the discovery of the exceptional simple complex Lie algebras by Killing and E. Cartan, it was natural to look for corresponding groups over finite fields, and already around 1900 this was done by Dickson for the exceptional Lie algebras G and E • However, a deep reason for this 2 6 parallelism was missing, and it is only Chevalley who, in 1955 and 1961, discovered that to each complex simple Lie algebra corresponds, by a uniform process, a group scheme (fj over the ring Z of integers, from which, for any field K, could be derived a group (fj(K).Table of ContentsNotation and Conventions.- 1. General Linear Groups, Steinberg Groups, and K-Groups.- 2. Linear Groups over Division Rings.- 3. Isomorphism Theory for the Linear Groups.- 4. Linear Groups over General Classes of Rings.- 5. Unitary Groups, Unitary Steinberg Groups, and Unitary K-Groups.- 6. Unitary Groups over Division Rings.- 7. Clifford Algebras and Orthogonal Groups over Commutative Rings.- 8. Isomorphism Theory for the Unitary Groups.- 9. Unitary Groups over General Classes of Form Rings.- Concluding Remarks.- Index of Concepts.- Index of Symbols.

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Book SynopsisA collection of five surveys on dynamical systems, indispensable for graduate students and researchers in mathematics and theoretical physics. Written in the modern language of differential geometry, the book covers all the new differential geometric and Lie-algebraic methods currently used in the theory of integrable systems.Table of ContentsContents: Nonholonomic Dynamical Systems, Geometry of Distributions and Variational Problems by A.M. Vershik, V.Ya. Gershkovich.- Integrable Systems and Infinite Dimensional Lie Algebras by M.A. Olshanetsky, M.A. Perelomov.- Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems by A.G. Reyman, M.A. Semenov-Tian-Shansky.- Quantization of Open Toda Lattices by M.A. Semenov-Tian-Shansky.- Geometric and Algebraic Mechanisms of the Integrability of Hamiltonian Systems on Homogeneous Spaces and Lie Algebras by V.V. Trofimov, A.T. Fomenko.

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Book SynopsisIn many areas of mathematics some “higher operations” are arising. These havebecome so important that several research projects refer to such expressions. Higher operationsform new types of algebras. The key to understanding and comparing them, to creating invariants of their action is operad theory. This is a point of view that is 40 years old in algebraic topology, but the new trend is its appearance in several other areas, such as algebraic geometry, mathematical physics, differential geometry, and combinatorics. The present volume is the first comprehensive and systematic approach to algebraic operads. An operad is an algebraic device that serves to study all kinds of algebras (associative, commutative, Lie, Poisson, A-infinity, etc.) from a conceptual point of view. The book presents this topic with an emphasis on Koszul duality theory. After a modern treatment of Koszul duality for associative algebras, the theory is extended to operads. Applications to homotopy algebra are given, for instance the Homotopy Transfer Theorem. Although the necessary notions of algebra are recalled, readers are expected to be familiar with elementary homological algebra. Each chapter ends with a helpful summary and exercises. A full chapter is devoted to examples, and numerous figures are included. After a low-level chapter on Algebra, accessible to (advanced) undergraduate students, the level increases gradually through the book. However, the authors have done their best to make it suitable for graduate students: three appendices review the basic results needed in order to understand the various chapters. Since higher algebra is becoming essential in several research areas like deformation theory, algebraic geometry, representation theory, differential geometry, algebraic combinatorics, and mathematical physics, the book can also be used as a reference work by researchers.Trade ReviewFrom the reviews:“It is a welcome addition to the existing literature and will, no doubt, become a standard reference for many authors working in this quickly developing field. … it is an impressive piece of work, which gives a comprehensive account of the foundations of the theory of algebraic operads, starting from the most basic notions, such as associative algebras and modules. It will be of interest to a broad swath of mathematicians: from undergraduate students to experts in the field.” (Andrey Yu. Lazarev, Mathematical Reviews, March, 2013)Table of ContentsPreface.- 1.Algebras, coalgebras, homology.- 2.Twisting morphisms.- 3.Koszul duality for associative algebras.- 4.Methods to prove Koszulity of an algebra.- 5.Algebraic operad.- 6 Operadic homological algebra.- 7.Koszul duality of operads.- 8.Methods to prove Koszulity of an operad.- 9.The operads As and A\infty.- 10.Homotopy operadic algebras.- 11.Bar and cobar construction of an algebra over an operad.- 12.(Co)homology of algebras over an operad.- 13.Examples of algebraic operads.- Apendices: A.The symmetric group.- B.Categories.- C.Trees.- References.- Index.- List of Notation.

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Book SynopsisSignal processing is the discipline of extracting information from collections of measurements. To be effective, the measurements must be organized and then filtered, detected, or transformed to expose the desired information. Distortions caused by uncertainty, noise, and clutter degrade the performance of practical signal processing systems.In aggressively uncertain situations, the full truth about an underlying signal cannot be known. This book develops the theory and practice of signal processing systems for these situations that extract useful, qualitative information using the mathematics of topology -- the study of spaces under continuous transformations. Since the collection of continuous transformations is large and varied, tools which are topologically-motivated are automatically insensitive to substantial distortion. The target audience comprises practitioners as well as researchers, but the book may also be beneficial for graduate students.Trade ReviewFrom the book reviews:“This text provides a nice exposition of the topological ideas used to extract information from signals and the practical details of signal processing. … Robinson’s intended audience is first year graduate students in both engineering and mathematics, and advanced undergraduates. … Throughout the text there are numerous examples and diagrams. Each chapter also ends with some open questions. These features make the book quite readable.” (Michele Intermont, MAA Reviews, February, 2015)“Three major goals for this book: firstly to show that topological invariants provide qualitative information about signals that is both relevant and practical, second to show that the signal processing concepts of filtering, detection, and noise correspond respectively to the concepts of sheaves, functoriality and sequences, and third to advocate for the use of sheaf theory in signal processing. … The target audience is practitioners so that the theoretical notions are covered with the practitioner in mind with motivations emphasized.” (Jonathan Hodgson, zbMATH, Vol. 1294, 2014)Table of ContentsIntroduction and informal discussion.- Parametrization.- Signals.- Detection.- Transforms.- Noise.

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Book SynopsisJürgen Beetz führt zuerst in den Ursprung der erdachten Geschichten der Mathematik aus der Steinzeit ein. Im Anschluss daran stellt er die zentrale Fragestellung der „Infinitesimalrechnung“ anhand eines einfachen Beispiels dar. Dann erläutert der Autor die Grundproblematik des Differenzierens: die Steigung (d. h. die Richtung der Tangente) an einer beliebigen Stelle einer Funktion y=f(x) festzustellen. Als praktische Beispiele des Differenzierens behandelt er die Hyperbel und die Sinusfunktion. Ein eigenes Kapitel widmet Jürgen Beetz den Besonderheiten der Exponentialfunktion.Table of ContentsDas Maß für Veränderung.- Die Praxis der Differentialrechnung.- Die Exponentialfunktion beweist ihre königliche Eigenschaft.

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Book SynopsisDieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.Table of ContentsErster Teil. Grundbegriffe der mengentheoretischen Topologie.- Erstes Kapitel: Topologische und metrische Räume.- Zweites Kapitel: Kompakte Räume.- Zweiter Teil. Topologie der Komplexe.- Drittes Kapitel: Polyeder und ihre Zellenzerlegungen.- Viertes Kapitel: Eckpunkt- und Koeffizientenbereiche.- Fünftes Kapitel: Bettische Gruppen.- Sechstes Kapitel: Zerspaltungen und Unterteilungen von Komplexen.- Siebentes Kapitel: Spezielle Fragen aus der Theorie der Komplexe.- Dritter Teil. Topologische Invarianzsätze und anschließende Begriffsbildungen.- Achtes Kapitel: Simpliziale Approximationen stetiger Abbildungen. Stetige Zyklen.- Neuntes Kapitel: Kanonische Verschiebungen. Nochmals Invarianz der Dimensionszahl und der Bettischen Gruppen. Allgemeiner Dimensionsbegriff.- Zehntes Kapitel: Der Zerlegungssatz für den Euklidischen Raum. Weitere Invarianzsätze.- Vierter Teil. Verschlingungen im Euklidischen Raum. Stetige Abbildungen von Polyedern.- Elftes Kapitel: Verschlingungstheorie. Der Alexandersche Dualitätssatz.- Zwölftes Kapitel: Der Brouwersche Abbildungsgrad. Die Kroneckersche Charakteristik.- Dreizehntes Kapitel: Homotopie- und Erweiterungssätze für Abbildungen.- Vierzehntes Kapitel: Fixpunkte.- Anhang I. Abelsche Gruppen.- § 1 Allgemeine Begriffe und Sätze.- § 2. Moduln (Freie Gruppen).- § 4. Gruppen mit endlich-vielen Erzeugenden.- § 5. Charaktere.- § 2. Konvexe Mengen.- § 3. Konvexe und baryzentrische Hüllen. Simplexe.- § 4. Konvexe Raumstücke. Konvexe Zellen.- 1. Nachtrag: Zentralprojektion.- 2. Nachtrag: Der Schwerpunkt.- Verzeichnis der topologischen Bücher.

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Book SynopsisSome years ago a conference on l-adic cohomology in Oberwolfach was held with the aim of reaching an understanding of Deligne's proof of the Weil conjec­ tures. For the convenience of the speakers the present authors - who were also the organisers of that meeting - prepared short notes containing the central definitions and ideas of the proofs. The unexpected interest for these notes and the various suggestions to publish them encouraged us to work somewhat more on them and fill out the gaps. Our aim was to develop the theory in as self­ contained and as short a manner as possible. We intended especially to provide a complete introduction to etale and l-adic cohomology theory including the monodromy theory of Lefschetz pencils. Of course, all the central ideas are due to the people who created the theory, especially Grothendieck and Deligne. The main references are the SGA-notes [64-69]. With the kind permission of Professor J. A. Dieudonne we have included in the book that finally resulted his excellent notes on the history of the Weil conjectures, as a second introduction. Our original notes were written in German. However, we finally followed the recommendation made variously to publish the book in English. We had the good fortune that Professor W. Waterhouse and his wife Betty agreed to translate our manuscript. We want to thank them very warmly for their willing involvement in such a tedious task. We are very grateful to the staff of Springer-Verlag for their careful work.Table of ContentsI. The Essentials of Etale Cohomology Theory.- II. Rationality of Weil ?-Functions.- III. The Monodromy Theory of Lefschetz Pencils.- IV. Deligne’s Proof of the Weil Conjecture.- Appendices.- A I. The Fundamental Group.- A II. Derived Categories.- A III. Descent.

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Book SynopsisThis book is designed for graduate students to acquire knowledge of dimension theory, ANR theory (theory of retracts), and related topics. These two theories are connected with various fields in geometric topology and in general topology as well. Hence, for students who wish to research subjects in general and geometric topology, understanding these theories will be valuable. Many proofs are illustrated by figures or diagrams, making it easier to understand the ideas of those proofs. Although exercises as such are not included, some results are given with only a sketch of their proofs. Completing the proofs in detail provides good exercise and training for graduate students and will be useful in graduate classes or seminars.Researchers should also find this book very helpful, because it contains many subjects that are not presented in usual textbooks, e.g., dim X × I = dim X + 1 for a metrizable space X; the difference between the small and large inductive dimensions; a hereditarily infinite-dimensional space; the ANR-ness of locally contractible countable-dimensional metrizable spaces; an infinite-dimensional space with finite cohomological dimension; a dimension raising cell-like map; and a non-AR metric linear space. The final chapter enables students to understand how deeply related the two theories are.Simplicial complexes are very useful in topology and are indispensable for studying the theories of both dimension and ANRs. There are many textbooks from which some knowledge of these subjects can be obtained, but no textbook discusses non-locally finite simplicial complexes in detail. So, when we encounter them, we have to refer to the original papers. For instance, J.H.C. Whitehead's theorem on small subdivisions is very important, but its proof cannot be found in any textbook. The homotopy type of simplicial complexes is discussed in textbooks on algebraic topology using CW complexes, but geometrical arguments using simplicial complexes are rather easy.Trade ReviewFrom the book reviews:“This excellent book is designed for different graduate courses in geometric topology, as well as in general topology. At the same time it contains complete proofs of results interesting also for the specialist in geometric topology … .” (Vesko Valov, Mathematical Reviews, September, 2014)

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Book SynopsisThis book provides an accessible introduction to algebraic topology, a field at the intersection of topology, geometry and algebra, together with its applications. Moreover, it covers several related topics that are in fact important in the overall scheme of algebraic topology. Comprising eighteen chapters and two appendices, the book integrates various concepts of algebraic topology, supported by examples, exercises, applications and historical notes. Primarily intended as a textbook, the book offers a valuable resource for undergraduate, postgraduate and advanced mathematics students alike.Focusing more on the geometric than on algebraic aspects of the subject, as well as its natural development, the book conveys the basic language of modern algebraic topology by exploring homotopy, homology and cohomology theories, and examines a variety of spaces: spheres, projective spaces, classical groups and their quotient spaces, function spaces, polyhedra, topological groups, Lie groups and cell complexes, etc. The book studies a variety of maps, which are continuous functions between spaces. It also reveals the importance of algebraic topology in contemporary mathematics, theoretical physics, computer science, chemistry, economics, and the biological and medical sciences, and encourages students to engage in further study.Trade Review“Adhikari’s work is an excellent resource for any individual seeking to learn more about algebraic topology. By no means will this text feel like an introduction to algebraic topology, but it does offer much for both beginners and experts. … the text will be a valuable reference on the bookshelf of any reader with an interest in algebraic topology. Summing Up: Recommended. Upper-division undergraduates and above; researchers and faculty.” (A. Misseldine, Choice, Vol. 54 (9), May, 2017)“I am pretty enthusiastic about this book. … it shows very good taste on the author’s part as far as what he’s chosen to do and how he’s chosen to do it. … Wow! What a nice book. I’m glad I have a copy.” (Michael Berg, MAA Reviews, maa.org, February, 2017)“This is a comprehensive textbook on algebraic topology. … accessible to students of all levels of mathematics, so suitable for anyone wanting and needing to learn about algebraic topology. It can also offer a valuable resource for advanced students with a specialized knowledge in other areas who want to pursue their interest in this area. … further readings are provided at the end of each of them, which also enables students to study the subject discussed therein in more depth.” (Haruo Minami, zbMATH 1354.55001, 2017)Table of ContentsPrerequisite Concepts and Notations.- Basic Homotopy.- The Fundamental Groups.-Covering Spaces.- Fibre Bundles, Vector Bundles and K-theory.- Geometry of Simplicial Complexes and Fundamental Groups.- Higher Homotopy Groups.- Products in Higher Homotopy Groups.- CW-complexes and Homotopy.- Eilenberg-MacLane Spaces.- Homology and Cohomology Theories.- Eilenberg-Steenrod Axioms for Homology and Cohomology Theories.- Consequences of the Eilenberg-Steenrod Axioms.- Some Applications of Homology Theory.- Spectral Homology and Cohomology Theories.- Obstruction Theory.- More Relations Between Homotopy and Homology Groups.- A Brief Historical Note.

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Book SynopsisThis book deals with the qualitative theory of dynamical systems and is devoted to the study of flows and cascades in the vicinity of a smooth invariant manifold. Its main purpose is to present, as completely as possible, the basic results concerning the existence of stable and unstable local manifolds and the recent advancements in the theory of finitely smooth normal forms of vector fields and diffeomorphisms in the vicinity of a rest point and a periodic trajectory. A summary of the results obtained so far in the investigation of dynamical systems near an arbitrary invariant submanifold is also given.Table of ContentsTopological properties of flows and cascades in the vicinity of a rest point and a periodic trajectory; finitely smooth normal forms of vector fields and diffeo-morphisms; linear extensions of dynamical systems; invariant subbundles of weakly non-linear extensions; invariant manifolds; normal forms in the vicinity of an invariant manifold.

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Book SynopsisThe book is an introduction to the foundations of Mathematics. The use of the constructive method in Arithmetic and the axiomatic method in Geometry gives a unitary understanding of the backgrounds of geometry, of its development and of its organic link with the study of real numbers and algebraic structures.Table of ContentsElements of set theory; arithmetics; axiomatic theories; algebraic background of geometry; bases of Euclidean geometry; Birkhoff's axiomatics; geometrical transformations; Erlangen programme; Bachmann's axiomatics.

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Book SynopsisThese lecture notes are based on a series of lectures given at the Nankai Institute of Mathematics in the fall of 1998. They provide an overview of the work of the author and the late Chih-Han Sah on various aspects of Hilbert's Third Problem: Are two Euclidean polyhedra with the same volume “scissors-congruent”, i.e. can they be subdivided into finitely many pairwise congruent pieces? The book starts from the classical solution of this problem by M Dehn. But generalization to higher dimensions and other geometries quickly leads to a great variety of mathematical topics, such as homology of groups, algebraic K-theory, characteristic classes for flat bundles, and invariants for hyperbolic manifolds. Some of the material, particularly in the chapters on projective configurations, is published here for the first time.Table of ContentsIntroduction and history; scissors congruence group and homology; homology of flag complexes; translational scissors congruences; Euclidean scissors congruences; Sydler's theorem and non-commutative differential forms; spherical scissors congruences; hyperbolic scissors congruences; homology of Lie groups made discrete; invariants; simplices in spherical and hyperbolic 3-space; rigidity of Cheeger-Chern-Simons invariants; projective configurations and homology of the projective linear group; homology of indecomposable configurations; the case of PGI(3,F).

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Book SynopsisThese lecture notes are based on a series of lectures given at the Nankai Institute of Mathematics in the fall of 1998. They provide an overview of the work of the author and the late Chih-Han Sah on various aspects of Hilbert's Third Problem: Are two Euclidean polyhedra with the same volume “scissors-congruent”, i.e. can they be subdivided into finitely many pairwise congruent pieces? The book starts from the classical solution of this problem by M Dehn. But generalization to higher dimensions and other geometries quickly leads to a great variety of mathematical topics, such as homology of groups, algebraic K-theory, characteristic classes for flat bundles, and invariants for hyperbolic manifolds. Some of the material, particularly in the chapters on projective configurations, is published here for the first time.Table of ContentsIntroduction and history; scissors congruence group and homology; homology of flag complexes; translational scissors congruences; Euclidean scissors congruences; Sydler's theorem and non-commutative differential forms; spherical scissors congruences; hyperbolic scissors congruences; homology of Lie groups made discrete; invariants; simplices in spherical and hyperbolic 3-space; rigidity of Cheeger-Chern-Simons invariants; projective configurations and homology of the projective linear group; homology of indecomposable configurations; the case of PGI(3,F).

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Book SynopsisThis invaluable book is based on the notes of a graduate course on differential geometry which the author gave at the Nankai Institute of Mathematics. It consists of two parts: the first part contains an introduction to the geometric theory of characteristic classes due to Shiing-shen Chern and André Weil, as well as a proof of the Gauss-Bonnet-Chern theorem based on the Mathai-Quillen construction of Thom forms; the second part presents analytic proofs of the Poincaré-Hopf index formula, as well as the Morse inequalities based on deformations introduced by Edward Witten.Table of ContentsChern-Weil theory for characteristic classes; Bott and Duistermaat-Heckman formulas; Gauss-Bonnet-Chern theorem; Poincar -Hopf index formula - an analytic proof; morse inequalities - an analytic proof; Thom-Smale and Witten complexes; Atiyah theorem on Kervaire Semi-characteristic.

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Book SynopsisThis book collects select papers presented at the International Workshop and Conference on Topology & Applications, held in Kochi, India, from 9–11 December 2018. The book discusses topics on topological dynamical systems and topological data analysis. Topics are ranging from general topology, algebraic topology, differential topology, fuzzy topology, topological dynamical systems, topological groups, linear dynamics, dynamics of operator network topology, iterated function systems and applications of topology. All contributing authors are eminent academicians, scientists, researchers and scholars in their respective fields, hailing from around the world. The book is a valuable resource for researchers, scientists and engineers from both academia and industry.Table of ContentsH. Bruin, An Overview of Unimodal Inverse Limit Spaces.- B. Barany, M. Rams, K. Simon, Dimension Theory of Some Non Markovian Rapellers Part I: A General Introduction.- B. Barany, M. Rams, K. Simon, Dimension Theory of Some Non Markovian Repellers: Part II: Dynamically Defined Function Graphs.- K. Lesniak, Iterated Function Systems – A Topological Approach Attractors.- H. Kato, Zero Dimensional Covers of Dynamical Systems.- H. Kato, Chaotic Continua in Chaotic Dynamical Systems.- R. L. Devaney, S. M. Marotta, Mandelpinski Necklaces in the Parameter Planes of Rational Maps.- Kit C Chan, Some Examples of Hypercyclic Operators and Universal Sequences of Operators.- Kit C Chan, Some Basic Properties of Hypercyclic Operators.- Kit C Chan, The Testing Ground of Weighted Shift Operators for Hypercyclicity.- D. Drozdov, M. Samuel, A. Tetenov, On -deformations of Polygonal Dendrites.- A. Tetenov, K. Kamalutdinov, V. Aseev, General Position Theorems and its Applications.- A. Raj P, V. Kumar P B, The nth iterate of a map with dense orbit.- Aswathy R K, S. Mathew, Finite Products of Irregular Iterated Function Systems and Their Separation Properties.- A. Akbar, Mubeena T, Periodic Points of N-dimensional Toral Automorphisms.- S. Jose, V. Kumar P B, Julia Sets in Topological Spaces.- K U Sreeja, V. Kumar P B, Ramkumar P B, Julia, Sets of Some Graphs Using Independence Polynomials.- P. Frosini, An Introduction to the Notion of Natural Pseudo Distance in Topological Data Analysis.- A. Cerri, P. Frosini, A Brief Introduction to Multidimensional Persistent Betti Numbers.- N. Quercioli, Some New Methods to Build Group Equivariant Non Expansive Operators in TDA.- Y. Dabaghian, Topological Stability of the Hippocampal Spatial Map and Synaptic Transience.- A. Jacob, Ramkumar P B, Intuitionistic Fuzzy Graph Morphological Topology.- A. G. Pillai, Ramkumar P B, Some Properties of the Bitopological Space Associated with the 3-Uniform Semigraph of Cycle graph.- D. Chandran R, Ramkumar P B, Hypergraph Topology.

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Book SynopsisThis first of the three-volume book is targeted as a basic course in topology for undergraduate and graduate students of mathematics. It studies metric spaces and general topology. It starts with the concept of the metric which is an abstraction of distance in the Euclidean space. The special structure of a metric space induces a topology that leads to many applications of topology in modern analysis and modern algebra, as shown in this volume. This volume also studies topological properties such as compactness and connectedness. Considering the importance of compactness in mathematics, this study covers the Stone–Cech compactification and Alexandroff one-point compactification. This volume also includes the Urysohn lemma, Urysohn metrization theorem, Tietz extension theorem, and Gelfand–Kolmogoroff theorem. The content of this volume is spread into eight chapters of which the last chapter conveys the history of metric spaces and the history of the emergence of the concepts leading to the development of topology as a subject with their motivations with an emphasis on general topology. It includes more material than is comfortably covered by beginner students in a one-semester course. Students of advanced courses will also find the book useful. This book will promote the scope, power, and active learning of the subject, all the while covering a wide range of theories and applications in a balanced unified way.Trade Review“With an encyclopedic range of topics and terse exposition, Basic Topology 1 may make a reasonable reference for self-motivated learners … .” (Timothy Clark, MAA Reviews, March 20, 2023)Table of Contents1. Prerequisites: Sets, Algebraic Systems, and Classical Analysis.- 2. Metric Spaces and Normed Linear Spaces.- 3. Topological Spaces and Continuous Maps.- 4. Separation Axioms.- 5. Compactness and Connectedness.- 6. Real-valued Continuous Functions.- 7. Countability, Separability and Embedding.- 8. Brief History of General Topology.

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Book SynopsisThis third of the three-volume book is targeted as a basic course in algebraic topology and topology for fiber bundles for undergraduate and graduate students of mathematics. It focuses on many variants of topology and its applications in modern analysis, geometry, and algebra. Topics covered in this volume include homotopy theory, homology and cohomology theories, homotopy theory of fiber bundles, Euler characteristic, and the Betti number. It also includes certain classic problems such as the Jordan curve theorem along with the discussions on higher homotopy groups and establishes links between homotopy and homology theories, axiomatic approach to homology and cohomology as inaugurated by Eilenberg and Steenrod. It includes more material than is comfortably covered by beginner students in a one-semester course. Students of advanced courses will also find the book useful. This book will promote the scope, power and active learning of the subject, all the while covering a wide range of theory and applications in a balanced unified way.Table of Contents1. Prerequisite Concepts of Topology, Algebra and Category Theory.- 2. Homotopy Theory: Fundamental and Higher Homotopy Groups.- 3. Homology and Cohomology Theories: An Axiomatic Approach with Consequences.- 4. Topology of Fiber Bundles.- 5. Homotopy Theory of Bundles.- 6. Some Applications of Algebraic Topology.- 7. Brief History on Algebraic Topology and Fiber Bundles.

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Book SynopsisThis second of the three-volume book is targeted as a basic course in topology for undergraduate and graduate students of mathematics. It focuses on many variants of topology and its applications in modern analysis, geometry, algebra, and the theory of numbers. Offering a proper background on topology, analysis, and algebra, this volume discusses the topological groups and topological vector spaces that provide many interesting geometrical objects which relate algebra with geometry and analysis. This volume follows a systematic and comprehensive elementary approach to the topology related to manifolds, emphasizing differential topology. It further communicates the history of the emergence of the concepts leading to the development of topological groups, manifolds, and also Lie groups as mathematical topics with their motivations. This book will promote the scope, power, and active learning of the subject while covering a wide range of theories and applications in a balanced unified way.Table of Contents1. Background on Topology, Analysis and Algebra.- 2. Topological Groups.- 3. Topology of Manifolds.- 4. Lie Groups and Lie Algebra.- 5. Brief History of Topological Groups, Manifold and Lie Groups.

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Book SynopsisThis volume offers an introduction to recent developments in several active topics of research at the interface between geometry, topology and quantum field theory. These include Hopf algebras underlying renormalization schemes in quantum field theory, noncommutative geometry with applications to index theory on one hand and the study of aperiodic solids on the other, geometry and topology of low dimensional manifolds with applications to topological field theory, Chern-Simons supergravity and the anti de Sitter/conformal field theory correspondence. It comprises seven lectures organized around three main topics, noncommutative geometry, topological field theory, followed by supergravity and string theory, complemented by some short communications by young participants of the school.Table of ContentsNoncommutative Geometry: Hopf Algebras in Noncommutative Geometry (J C Varilly); The Noncommutative Geometry of Aperiodic Solids (J Bellissard); Noncommutative Geometry and Abstract Integration Theory (M-T Benameur); Topological Field Theory: Introduction to Quantum Invariants of 3-Manifolds, Topological Quantum Field Theories and Modular Categories (C Blanchet); An Introduction to Donaldson-Witten Theory (M Marino); Supergravity and String Theory: (Super)-Gravities Beyond 4 Dimensions (J Zanelli); Introductory Lectures on String Theory and the AdS/CFT Correspondence (A Pankiewicz & S Theisen); Short Communications: Group Contractions and Its Consequences Upon Representations of Different Spatial Symmetry Groups (M Ayala-Sanchez & R W Haase); Phase Anomalies as Trace Anomalies in Chern-Simons Theory (A Cardona); Deligne Cohomology for Orbifolds, Discrete Torsion and B-Fields (E Lupercio & B Uribe).

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Book SynopsisThe first edition of this influential book, published in 1970, opened up a completely new field of invariant metrics and hyperbolic manifolds. The large number of papers on the topics covered by the book written since its appearance led Mathematical Reviews to create two new subsections “invariant metrics and pseudo-distances” and “hyperbolic complex manifolds” within the section “holomorphic mappings”. The invariant distance introduced in the first edition is now called the “Kobayashi distance”, and the hyperbolicity in the sense of this book is called the “Kobayashi hyperbolicity” to distinguish it from other hyperbolicities. This book continues to serve as the best introduction to hyperbolic complex analysis and geometry and is easily accessible to students since very little is assumed. The new edition adds comments on the most recent developments in the field.Trade Review"This book continues to serve as a fine introduction to hyperbolic complex analysis at a very elementary level." Zentralblatt MATH "A student with some background in complex differential geometry will find this to be an accessible, yet comprehensive, introduction to the subject." Mathematical ReviewsTable of Contents* The Schwarz Lemma and Its Generalizations * Volume Elements and the Schwarz Lemma * Distance and the Schwarz Lemma * Invariant Distances on Complex Manifolds * Holomorphic Mappings into Hyperbolic Manifolds * The Big Picard Theorem and Extension of Holomorphic Mappings * Generalization to Complex Spaces * Hyperbolic Manifolds and Minimal Models

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Book SynopsisThis book is devoted to the structure of the Mandelbrot set — a remarkable and important feature of modern theoretical physics, related to chaos and fractals and simultaneously to analytical functions, Riemann surfaces, phase transitions and string theory. The Mandelbrot set is one of the bridges connecting the world of chaos and order.The authors restrict consideration to discrete dynamics of a single variable. This restriction preserves the most essential properties of the subject, but drastically simplifies computer simulations and the mathematical formalism.The coverage includes a basic description of the structure of the set of orbits and pre-orbits associated with any map of an analytic space into itself. A detailed study of the space of orbits (the algebraic Julia set) as a whole, together with related attributes, is provided. Also covered are: moduli space in the space of maps and the classification problem for analytic maps, the relation of the moduli space to the bifurcations (topology changes) of the set of orbits, a combinatorial description of the moduli space (Mandelbrot and secondary Mandelbrot sets) and the corresponding invariants (discriminants and resultants), and the construction of the universal discriminant of analytic functions in terms of series coefficients. The book concludes by solving the case of the quadratic map using the theory and methods discussed earlier.Table of ContentsNotions and Notation; Summary; Fragments of Theory; Map f(x) = X2 + c: From Standard Example to General Conclusions.

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Book Synopsis'The book is an engaging and influential collection of significant contributions from an assembly of world expert leaders and pioneers from different fields, working at the interface between topology and physics or applications of topology to physical systems … The book explores many interesting and novel topics that lie at the intersection between gravity, quantum fields, condensed matter, physical cosmology and topology … A rich, well-organized, and comprehensive overview of remarkable and insightful connections between physics and topology is here made available to the physics reader.'Contemporary PhysicsSince its birth in Poincaré's seminal 1894 'Analysis Situs', topology has become a cornerstone of mathematics. As with all beautiful mathematical concepts, topology inevitably — resonating with that Wignerian principle of the effectiveness of mathematics in the natural sciences — finds its prominent role in physics. From Chern-Simons theory to topological quantum field theory, from knot invariants to Calabi-Yau compactification in string theory, from spacetime topology in cosmology to the recent Nobel Prize winning work on topological insulators, the interactions between topology and physics have been a triumph over the past few decades.In this eponymous volume, we are honoured to have contributions from an assembly of grand masters of the field, guiding us with their world-renowned expertise on the subject of the interplay between 'Topology' and 'Physics'. Beginning with a preface by Chen Ning Yang on his recollections of the early days, we proceed to a novel view of nuclei from the perspective of complex geometry by Sir Michael Atiyah and Nick Manton, followed by an entrée toward recent developments in two-dimensional gravity and intersection theory on the moduli space of Riemann surfaces by Robbert Dijkgraaf and Edward Witten; a study of Majorana fermions and relations to the Braid group by Louis H Kauffman; a pioneering investigation on arithmetic gauge theory by Minhyong Kim; an anecdote-enriched review of singularity theorems in black-hole physics by Sir Roger Penrose; an adventure beyond anyons by Zhenghan Wang; an aperçu on topological insulators from first-principle calculations by Haijun Zhang and Shou-Cheng Zhang; finishing with synopsis on quantum information theory as one of the four revolutions in physics and the second quantum revolution by Xiao-Gang Wen. We hope that this book will serve to inspire the research community.

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Book SynopsisThe book is the first systematic research completely devoted to a comprehensive study of virtual knots and classical knots as its integral part. The book is self-contained and contains up-to-date exposition of the key aspects of virtual (and classical) knot theory.Virtual knots were discovered by Louis Kauffman in 1996. When virtual knot theory arose, it became clear that classical knot theory was a small integral part of a larger theory, and studying properties of virtual knots helped one understand better some aspects of classical knot theory and encouraged the study of further problems. Virtual knot theory finds its applications in classical knot theory. Virtual knot theory occupies an intermediate position between the theory of knots in arbitrary three-manifold and classical knot theory.In this book we present the latest achievements in virtual knot theory including Khovanov homology theory and parity theory due to V O Manturov and graph-link theory due to both authors. By means of parity, one can construct functorial mappings from knots to knots, filtrations on the space of knots, refine many invariants and prove minimality of many series of knot diagrams.Graph-links can be treated as “diagramless knot theory”: such “links” have crossings, but they do not have arcs connecting these crossings. It turns out, however, that to graph-links one can extend many methods of classical and virtual knot theories, in particular, the Khovanov homology and the parity theory.Table of ContentsBasic Definitions and Notions; Virtual Knots and Three-Dimensional Topology; Quandles (Distributive Groupoids) in Virtual Knot Theory; The Jones Polynomial. Atoms; Khovanov Homology; Virtual Braids; Vassiliev's Invariants; Parity in Knot Theory. Free-Knots. Cobordisms; Theory of Graph-Links.

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Book SynopsisChapter 1 Closure Functions Induced by * and psi Operators.- Chapter 2 PNDP Manifolds.- Chapter 3 A Unified Study of Normal Spaces.

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Book SynopsisChapter 1 Introduction.- Chapter 2 A glimpse of progress of di?erential topology.- Chapter 3 Spin geometry.- Chapter 4 Seiberg–Witten theory.- Chapter 5  Bauer–Furuta theory.-  Chapter 6 ??^?? cohomology.- Chapter 7 ??^2-Betti number and von Neumann trace.- Chapter 8 Aspherical 10/8 -inequality and Singer’s conjecture.- Solutions.- References.- Index.

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