• 1 Introduction 1.1 Scope of the Thesis 1.2 Outline of the Thesis 1.3 Quantum Hall States 1.4 Topological Insulators 1.5 Weyl and Dirac Semimetals 1.6 -(BEDT-TTF)2I3 1.7 Topological Mott Insulators 1.8 Topological Crystalline Insulators 1.9 Classification of Topological States of Matter 2 Interacting Dirac Fermions in (3+1) Dimensions 2.1 Model 2.2 Renormalization Group Analysis 2.3 Density of States 2.4 Electromagnetic Properties 2.5 Spectral Function 2.6 Electric Conductivity 2.7 Energy Gap 2.8 Discussions and Summary 3 Tilted Dirac Cones in Two Dimensions 3.1 Model 3.2 Perturbative Renormalization Group Analysis 3.3 Spin Susceptibility 3.4 Discussions and Summary 4 Generalized Hund's Rule for Two-Atom Systems 4.1 Model 4.2 Results 4.3 Perturbative Calculation 4.4 Entanglement Entropy 4.5 Symmetry 4.6 Discussions and Summary 5 Interacting Topological Crystalline Insulators 5.1 Classification in Two Dimensions 5.2 Interacting TCIs in Three Dimensions 5.3 Discussions and Summary 6 Conclusions and Prospects.
• (source: Nielsen Book Data)9789811037429 20170612

This thesis elucidates electron correlation effects in topological matter whose electronic states hold nontrivial topological properties robust against small perturbations. In addition to a comprehensive introduction to topological matter, this thesis provides a new perspective on correlated topological matter. The book comprises three subjects, in which electron correlations in different forms are considered. The first focuses on Coulomb interactions for massless Dirac fermions. Using a perturbative approach, the author reveals emergent Lorentz invariance in a low-energy limit and discusses how to probe the Lorentz invariance experimentally. The second subject aims to show a principle for synthesizing topological insulators with common, light elements. The interplay between the spin-orbit interaction and electron correlation is considered, and Hund's rule and electron filling are consequently found to play a key role for a strong spin-orbit interaction important for topological insulators. The last subject is classification of topological crystalline insulators in the presence of electron correlation. Unlike non-interacting topological insulators, such two- and three-dimensional correlated insulators with mirror symmetry are demonstrated to be characterized, respectively, by the Z4 and Z8 group by using the bosonization technique and a geometrical consideration.
(source: Nielsen Book Data)9789811037429 20170612

• Topological Homogeneity.- Some Recent Progress Concerning Topology of Fractals.- A biased view of topology as a tool in functional analysis.- Large scale versus small scale.- Descriptive aspects of Rosenthal compacta.- Minimality conditions in topological groups.- Set-Theoretic update on Topology.- Topics in Dimension Theory.- Representations of dynamical systems on Banach spaces.- Generalized metrizable spaces.- Permanence in Coarse Geometry.- Selections and Hyperspaces.- Continuum Theory.- Almost disjoint families and topology.- Some Topics in Geometric Topology II.- Topological aspects of dynamics of pairs, tuples and sets.- Continuous selections of multivalued mappings.- The combinatorics of open covers.- Covering properties.- Paratopological and semitopological groups vs topological groups.
• (source: Nielsen Book Data)9789462390232 20160614

The book presents surveys describing recent developments in most of the primary subfields of General Topology, and its applications to Algebra and Analysis during the last decade, following the previous editions (North Holland, 1992 and 2002). The book was prepared in connection with the Prague Topological Symposium, held in 2011. During the last 10 years the focus in General Topology changed and therefore the selection of topics differs from that chosen in 2002. The following areas experienced significant developments: Fractals, Coarse Geometry/Topology, Dimension Theory, Set Theoretic Topology and Dynamical Systems.
(source: Nielsen Book Data)9789462390232 20160614

• Fundamentals What Is Topology? First Definitions Mappings The Separation Axioms Compactness Homeomorphisms Connectedness Path-Connectedness Continua Totally Disconnected Spaces The Cantor Set Metric Spaces Metrizability Baire's Theorem Lebesgue's Lemma and Lebesgue Numbers Advanced Properties of Topological Spaces Basis and Sub-Basis Product Spaces Relative Topology First Countable, Second Countable, and So Forth Compactifications Quotient Topologies Uniformities Morse Theory Proper Mappings Paracompactness An Application to Digital Imaging Basic Algebraic Topology Homotopy Theory Homology Theory Covering Spaces The Concept of Index Mathematical Economics Manifold Theory Basic Concepts The Definition Moore-Smith Convergence and Nets Introductory Remarks Nets Function Spaces Preliminary Ideas The Topology of Pointwise Convergence The Compact-Open Topology Uniform Convergence Equicontinuity and the Ascoli-Arzela Theorem The Weierstrass Approximation Theorem Knot Theory What Is a Knot? The Alexander Polynomial The Jones Polynomial Graph Theory Introduction Fundamental Ideas of Graph Theory Application to the Konigsberg Bridge Problem Coloring Problems The Traveling Salesman Problem Dynamical Systems Flows Planar Autonomous Systems Lagrange's Equations Appendix 1: Principles of Logic Truth "And" and "Or" "Not" "If - Then" Contrapositive, Converse, and "Iff" Quantifiers Truth and Provability Appendix 2: Principles of Set Theory Undefinable Terms Elements of Set Theory Venn Diagrams Further Ideas in Elementary Set Theory Indexing and Extended Set Operations Countable and Uncountable Sets Appendix 3: The Real Numbers The Real Number System Construction of the Real Numbers Appendix 4: The Axiom of Choice and Its Implications Well Ordering The Continuum Hypothesis Zorn's Lemma The Hausdorff Maximality Principle The Banach-Tarski Paradox Appendix 5: Ideas from Algebra Groups Rings Fields Modules Vector Spaces Solutions of Selected Exercises Bibliography Index Exercises appear at the end of each chapter.
• (source: Nielsen Book Data)9781420089752 20160616

Brings Readers Up to Speed in This Important and Rapidly Growing Area Supported by many examples in mathematics, physics, economics, engineering, and other disciplines, Essentials of Topology with Applications provides a clear, insightful, and thorough introduction to the basics of modern topology. It presents the traditional concepts of topological space, open and closed sets, separation axioms, and more, along with applications of the ideas in Morse, manifold, homotopy, and homology theories. After discussing the key ideas of topology, the author examines the more advanced topics of algebraic topology and manifold theory. He also explores meaningful applications in a number of areas, including the traveling salesman problem, digital imaging, mathematical economics, and dynamical systems. The appendices offer background material on logic, set theory, the properties of real numbers, the axiom of choice, and basic algebraic structures. Taking a fresh and accessible approach to a venerable subject, this text provides excellent representations of topological ideas. It forms the foundation for further mathematical study in real analysis, abstract algebra, and beyond.
(source: Nielsen Book Data)9781420089752 20160616

• Preface-- Part I. Fundamentals: 1.1. What is topology?-- 1.2. First definitions-- 1.3 Mappings-- 1.4. The separation axioms-- 1.5. Compactness-- 1.6. Homeomorphisms-- 1.7. Connectedness-- 1.8. Path-connectedness-- 1.9. Continua-- 1.10. Totally disconnected spaces-- 1.11. The Cantor set-- 1.12. Metric spaces-- 1.13. Metrizability-- 1.14. Baire's theorem-- 1.15. Lebesgue's lemma and Lebesgue numbers-- Part II. Advanced Properties: 2.1 Basis and subbasis-- 2.2. Product spaces-- 2.3. Relative topology-- 2.4. First countable and second countable-- 2.5. Compactifications-- 2.6. Quotient topologies-- 2.7. Uniformities-- 2.8. Morse theory-- 2.9. Proper mappings-- 2.10. Paracompactness-- Part III. Moore-Smith Convergence and Nets: 3.1. Introductory remarks-- 3.2. Nets-- Part IV. Function Spaces: 4.1. Preliminary ideas-- 4.2. The topology of pointwise convergence-- 4.3. The compact-open topology-- 4.4. Uniform convergence-- 4.5. Equicontinuity and the Ascoli-Arzela theorem-- 4.6. The Weierstrass approximation theorem-- Table of notation-- Glossary-- Bibliography-- Index.
• (source: Nielsen Book Data)9780883853467 20160603

A Guide to Topology is an introduction to basic topology for graduate or advanced undergraduate students. It covers point-set topology, Moore-Smith convergence and function spaces. It treats continuity, compactness, the separation axioms, connectedness, completeness, the relative topology, the quotient topology, the product topology, and all the other fundamental ideas of the subject. The book is filled with examples and illustrations. Students studying for exams will find this book to be a concise, focused and informative resource. Professional mathematicians who need a quick review of the subject, or need a place to look up a key fact, will find this book to be a useful resource too.
(source: Nielsen Book Data)9780883853467 20160603

• Homeomorphic Sets.- Topological Properties.- Equivalent Subsets.- Surfaces and Spaces.- Polyhedra.- Winding Number.
• (source: Nielsen Book Data)9781848009127 20160605

Topologyhasbeenreferredtoas"rubber-sheetgeometry".Thenameisapt , for the subject is concerned with properties of an object that would be preserved, no matter how much it is stretched, squashed, or distorted, so long as it is not in any way torn apart or glued together. One's ?rst reaction might be that such animprecise-soundingsubjectcouldhardlybepartofseriousmathematics , and wouldbeunlikelytohaveapplicationsbeyondtheamusementofsimpleparlo ur games. This reaction could hardly be further from the truth. Topology is one of the most important and broad-ranging disciplines of modern mathematics. It is a subject of great precision and of breadth of development. It has vastly many applications, some of great importance, ranging from particle physics to cosmology, and from hydrodynamics to algebra and number theory. It is also a subject of great beauty and depth. To appreciate something of this, it is not necessary to delve into the more obscure aspects of mathematical formalism. For topology is, at least initially, a very visual subject. Some of its concepts apply to spaces of large numbers of dimensions, and therefore do not easily submit to reasoning that depends upon direct pictorial representation. But even in such cases, important insights can be obtained from the visual - rusal of a simple geometrical con?guration. Although much modern topology depends upon ?nely tuned abstract algebraic machinery of great mathematical sophistication, the underlying ideas are often very simple and can be appre- ated by the examination of properties of elementary-looking drawings.
(source: Nielsen Book Data)9781848009127 20160605

• Part 1. General Topology 1. Selected ordered space problems (H. Bennett and D. Lutzer) 2. Problems on star-covering properties (M. Bonanzinga and M. Matveev) 3. Function space topologies (D.N. Georgiou, S.D. Iliadis and F. Mynard) 4. Spaces and mappings: special networks (C. Liu and Y. Tanaka) 5. Extension problems of real-valued continuous functions (H. Ohta and K. Yamazaki) 6. LE(k)-spaces (O. Okunev) 7. Problems on (ir) resolvability (O. Pavlov) 8. Topological games and Ramsey theory (M. Scheepers) 9. Selection principles and special sets of reals (B. Tsaban) Part 2. Set-theoretic Topology 10. Introduction: Twenty problems in set-theoretic topology (M. Hrusak and J.T. Moore) 11. Thin-tall spaces and cardinal sequences (J. Bagaria) 12. Sequential order (A. Dow) 13. On D-spaces (T. Eisworth) 14. The fourth head of BN (I. Farah) 15. Are stratifiable spaces M1? (G. Gruenhage) 16. Perfect compacta and basis problems in topology (G. Gruenhage and J.T. Moore) 17. Selection problems for hyperspaces (V. Gutev and T. Nogura) 18. Efimov's problem (K.P. Hart) 19. Completely separable MAD families (M. Hrusak and P. Simon) 20. Good, splendid and Jakovlev (I. Juhasz and W.A.R. Weiss) 21. Homogeneous compacta (J. van Mill) 22. Compact spaces with hereditarily normal squares (J.T. Moore) 23. The metrization problem for Frechet groups (J.T. Moore and S. Todorcevic) 24. Cech-Stone remainders of discrete spaces (P.J. Nyikos) 25. First countable, countably compact, noncompact spaces (P.J. Nyikos) 26. Linearly Lindelof problems (E. Pearl) 27. Small Dowker spaces (P.J. Szeptycki) 28. Reflection of topological properties to N1 (F.D. Tall) 29. The Scarborough-Stone problem (J.E. Vaughan) Part 3. Continuum Theory 30. Questions in and out of context (D.P. Bellamy) 31. An update on the elusive fixed-point property (C.L. Hagopian) 32. Hyperspaces of continua (A. Ilanes) 33. Inverse limits and dynamical systems (W.T. Ingram) 34. Indecomposable continua (W. Lewis) 35. Open problems on dendroids (V. Martinez-de-la-Vega and J.M. Martinez-Montejano) 36. -Homogeneous continua (S.B. Nadler, Jr.) 37. Thirty open problems in the theory of homogeneous continua (J.R. Prajs) Part 4. Topological Algebra 38. Problems about the uniform structures of topological groups (A. Bouziad and J-P. Troallic) 39. On some special classes of continuous maps (M.M. Clementino and D. Hofmann) 40. Dense subgroups of compact groups (W.W. Comfort) 41. Selected topics from the structure theory of topological groups (D. Dikranjan and D. Shakhmatov) 42. Recent results and open questions relating Chu duality and Bohr compactifications of locally compact groups (J. Galindo, S. Hernandez and T-S. Wu) 43. Topological transformation groups: selected topics (M. Megrelishvili) 44. Forty-plus annotated questions about large topological groups (V. Pestov) Part 5. Dynamical Systems 45. Minimal flows (W.F. Basener, K. Parwani and T. Wiandt) 46. The dynamics of tiling spaces (A. Clark) 47. Open problems in complex dynam.
• (source: Nielsen Book Data)9780444522085 20160528

This volume is a collection of surveys of research problems in topology and its applications. The topics covered include general topology, set-theoretic topology, continuum theory, topological algebra, dynamical systems, computational topology and functional analysis. It contains new surveys of research problems in topology. It gives new perspectives on classic problems and presents representative surveys of research groups from all around the world.
(source: Nielsen Book Data)9780444522085 20160528

• A Crisis in the Experimental Method Archetype-- Orbit Organization in R2 x S1-- Braids as Indicators of Phase-Space Dynamics-- Braids and the Poincare Section-- Reconstruction of Phase-Space Dynamics -- Basic Course-- Reconstruction of Phase-Space Dynamics -- Advanced Course.
• (source: Nielsen Book Data)9789812703804 20160528

This book presents the development and application of some topological methods in the analysis of data coming from 3D dynamical systems (or related objects). The aim is to emphasize the scope and limitations of the methods, what they provide and what they do not provide. Braid theory, the topology of surface homeomorphisms, data analysis and the reconstruction of phase-space dynamics are thoroughly addressed.
(source: Nielsen Book Data)9789812703804 20160528

• PART I: A GEOMETRIC INTRODUCTION TO TOPOLOGY-- 1. Basic point set topology-- 2. The classification of surfaces-- 3. The fundamental group and its applications-- PART II: COVERING SPACES, CW COMPLEXES AND HOMOLOGY-- 4. Covering spaces-- 5. CW complexes-- 6. Homology-- Selected solutions-- References-- Index.
• (source: Nielsen Book Data)9780199202485 20160528

This new-in-paperback introduction to topology emphasizes a geometric approach with a focus on surfaces. A primary feature is a large collection of exercises and projects, which fosters a teaching style that encourages the student to be an active class participant. A wide range of material at different levels supports flexible use of the book for a variety of students. Part I is appropriate for a one-semester or two-quarter course, and Part II (which is problem based) allows the book to be used for a year-long course which supports a variety of syllabuses. The over 750 exercises range from simple checks of omitted details in arguments, to reinforce the material and increase student involvement, to the development of substantial theorems that have been broken into many steps. The style encourages an active student role. Solutions to selected exercises are included as an appendix, with solutions to all exercises available to the instructor on a companion website.
(source: Nielsen Book Data)9780199202485 20160528

• PART I: A GEOMETRIC INTRODUCTION TO TOPOLOGY-- 1. Basic point set topology-- 2. The classification of surfaces-- 3. The fundamental group and its applications-- PART II: COVERING SPACES, CW COMPLEXES AND HOMOLOGY-- 4. Covering spaces-- 5. CW complexes-- 6. Homology-- Selected solutions-- References-- Index.
• (source: Nielsen Book Data)9780199202485 20160528

This new-in-paperback introduction to topology emphasizes a geometric approach with a focus on surfaces. A primary feature is a large collection of exercises and projects, which fosters a teaching style that encourages the student to be an active class participant. A wide range of material at different levels supports flexible use of the book for a variety of students. Part I is appropriate for a one-semester or two-quarter course, and Part II (which is problem based) allows the book to be used for a year-long course which supports a variety of syllabuses. The over 750 exercises range from simple checks of omitted details in arguments, to reinforce the material and increase student involvement, to the development of substantial theorems that have been broken into many steps. The style encourages an active student role. Solutions to selected exercises are included as an appendix, with solutions to all exercises available to the instructor on a companion website.
(source: Nielsen Book Data)9786610904259 20160527

• Preface.- Introduction.- Set Theory.- Metric Spaces.- Set Theoretic Topology.- Systems of Continuous Functions.- Basic Algebraic Topology.- The Classical Mittag-Leffler Theorem Derived from Bourbaki?s.- Failure of the Heine-Borel Theorem in Infinite-Dimensional Spaces.- The Arzela-Ascoli Theorem.- References.- List of Symbols.- Index.
• (source: Nielsen Book Data)9780387257907 20160528

If mathematics is a language, then taking a topology course at the undergraduate level is cramming vocabulary and memorizing irregular verbs: a necessary, but not always exciting exercise one has to go through before one can read great works of literature in the original language. The present book grew out of notes for an introductory topology course at the University of Alberta. It provides a concise introduction to set theoretic topology (and to a tiny little bit of algebraic topology). It is accessible to undergraduates from the second year on, but even beginning graduate students can benefit from some parts. Great care has been devoted to the selection of examples that are not self-serving, but already accessible for students who have a background in calculus and elementary algebra, but not necessarily in real or complex analysis.In some points, the book treats its material differently than other texts on the subject: Baire's theorem is derived from Bourbaki's Mittag-Leffler theorem; nets are used extensively, in particular for an intuitive proof of Tychonoff's theorem; and, a short and elegant, but little known proof for the Stone-Weierstrass theorem is given.
(source: Nielsen Book Data)9780387257907 20160528

• Topics in transformation groups (A. Adem and J.F .Davis). Piecewise linear topology (J.L. Bryant). Infinite dimensional topology and shape theory (A. Chigogidze). Nonpositive curvature and reflection groups (M.W. Davis). Nielsen fixed point theory (R. Geoghegan). Mapping class groups (N.V. Ivanov). Seifert manifolds (Kyung Bai Lee and F. Raymond). Quantum invariants of 3-manifolds and CW-complexes (W. Lueck). Hyperbolic manifolds (J.G .Ratcliffe). Flows with knotted closed orbits (J. Franks and M.C. Sullivan). Heegaard splittings of compact 3-manifolds (M. Scharlemann). Representations of 3-manifold groups (P.B. Schalen). Homology manifolds (S. Weinberger). R-trees in topology, geometry, and group theory (F. Bonathon). Dehn surgery on knots (S. Boyer). Geometric group theory (J. Cannon). Cohomological dimension theory (J. Dydak). Metric spaces of curvature greater than or equal to k (C. Plaut). Topological rigidity theorems (C.W. Stark).
• (source: Nielsen Book Data)9780444824325 20160528

Geometric Topology is a foundational component of modern mathematics, involving the study of spacial properties and invariants of familiar objects such as manifolds and complexes. This volume, which is intended both as an introduction to the subject and as a wide ranging resource for those already grounded in it, consists of 21 expository surveys written by leading experts and covering active areas of current research. They provide the reader with an up-to-date overview of this flourishing branch of mathematics.
(source: Nielsen Book Data)9780444824325 20160528