Topology Books

Book SynopsisOffers a panorama of the topology of simply connected smooth manifolds of dimension four. Dimension four is unlike any other dimension; it is large enough to have room for wild things to happen, but small enough so that there is no room to undo the wildness.Trade ReviewThe book gives an excellent overview of 4-manifolds, with many figures and historical notes. Graduate students, nonexperts, and experts alike will enjoy browsing through it. -- Robion C. Kirby, University of California, Berkeley What a wonderful book! I strongly recommend this book to anyone, especially graduate students, interested in getting a sense of 4-manifolds."" —MAA Reviews""The author records many spectacular results in the subject ... (the author) gives the reader a taste of the techniques involved in the proofs, geometric topology, gauge theory and complex and symplectic structures. The book has a large and up-to-date collection of references for the reader wishing to get a more detailed or rigorous knowledge of a specific topic. The exposition is user-friendly, with a large number of illustrations and examples."" —Mathematical ReviewsTable of Contents Introduction Front matter Part I: Background scenery: Introduction Contents of part I Chapter 1: Higher dimensions and the $h$-cobordism theorem Chapter 2: Topological 4-manifolds and $h$-cobordisms Part II: Smooth 4-manifolds and intersection forms: Introduction Contents of part II Chapter 3: Getting acquainted with intersection forms Chapter 4: Intersection forms and topology Chapter 5: Classifications and counterclassifications Part III: A survey of complex surfaces: Introduction Contents of part III Chapter 6: Running through complex geometry Chapter 7: The Enriques-Kodaira classification Chapter 8: Elliptic surfaces Part IV: Gauge theory on 4-manifolds: Introduction Contents of part IV Chapter 9: Prelude, and the Donaldson invariants Chapter 10: The Seiberg-Witten invariants Chapter 11: The minimum genus of embedded surfaces Chapter 12: Wildness unleashed: The Fintushel-Stern surgery Epilogue List of figures and tables Bibliography Index Errata

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Book SynopsisFeatures plane curves to illustrate many deep and inspiring results in the field in an elementary and accessible way. After an introduction to the basic properties of plane curves, the authors introduce a number of complex and beautiful topics, including the rotation number, rotation index, Jordan curve theorem, and isoperimetric inequality.Table of Contents Plane curves Winding number Rotation index Jordan curve theorem Isoperimetric inequality Convex curves The four-vertex theorem Curve-shortening flow Appendix A: The class $\mathcal{C}^\infty$ convergence of the curvature function under the curve-shortening flow Appendix B: Answers to selected exercises Bibliography Index

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Book SynopsisA collection of 24 classroom modules (PSPs) produced by TRIUMPHS that incorporate the reading of primary source excerpts to teach core mathematical topics. The selected excerpts are intertwined with thoughtfully designed student tasks that prompt students to actively engage with and explore the source material.Trade ReviewPrimary sources provide motivation in the words of the original discoverers of new mathematics, draw attention to subtleties, encourage reflection on today's paradigms, and enhance students' ability to participate equally, regardless of their background. These beautifully written primary source projects that adopt an ``inquiry'' approach are rich in features lacking in modern textbooks. Prompted by the study of historical sources, students will grapple with uncertainties, ask questions, interpret, conjecture, and compare multiple perspectives, resulting in a unique and vivid guided learning experience. -- David Pengelley, Oregon State UniversityTable of ContentsJ. H. Barnett, D. K. Ruch, and N. A. Scoville, Contents; Introduction: J. H. Barnett, D. K. Ruch, and N. A. Scoville, Teaching and Learning with Primary Source Projects; J. H. Barnett, D. K. Ruch, and N. A. Scoville, PSP Summaries: The Collection at a Glance; J. H. Barnett, Historical Overview; Real Analysis: J. H. Barnett, Why Be So Critical? Nineteenth-Century Mathematics and the Origins of Analysis; D. Ruch, Investigations into Bolzano's Bounded Set Theorem; M. P. Saclolo, Stitching Dedekind Cuts to Construct the Real Numbers; D. Ruch, Investigations into d'Alembert's Definition of Limit; D. Ruch, Bolzano on Continuity and the Intermediate Value Theorem; N. Somasunderam, Understanding Compactness: Early Work, Uniform Continuity to the Heine-Borel Theorem; D. Ruch, An Introduction to a Rigorous Definition of Derivative; J. H. Barnett, Rigorous Debates over Debatable Rigor: Monster Functions in Introductory Analysis; D. Ruch, The Mean Value Theorem; D. Ruch, Euler's Rediscovery of $e$; D. Ruch, Abel and Cauchy on a Rigorous Approach to Infinite Series; D. Ruch, The Definite Integrals of Cauchy and Riemann; J. H. Barnett, Henri Lebesgue and the Development of the Integral Concept; Topology: N. A. Scoville, The Cantor Set before Cantor; N. A. Scoville, Topology from Analysis; N. A. Scoville, Nearness without Distance; N. A. Scoville, Connectedness: Its Evolution and Applications; N. A. Scoville, Connecting Connectedness; N. A. Scoville, From Sets to Metric Spaces to Topological Spaces; N. A. Scoville, The Closure Operation as the Foundation of Topology; N. A. Scoville, A Compact Introduction to a Generalized Extreme Value Theorem; Complex Variables: D. Klyve, The Logarithm of $-1$; D. Ruch, Riemann's Development of the Cauchy-Riemann Equations; D. Ruch, Gauss and Cauchy on Complex Integration.

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Book SynopsisFocuses on new and existing connections between three types of compactifications, thereby setting the stage for further research. The book draws on the discipline-specific expertise of all contributors, and at the same time gives a unified, self-contained reference for compactifications and related constructions in different contexts.Table of ContentsA. Balibanu, A quasi-Poisson structure on the multiplicative Grothendieck-Springer resolution; P. Brosnan, Volumes of definable sets in o-minimal expansions and affine GAGA theorems; P. Crooks and R. Roser, Hessenberg varieties and Poisson slices; G. Denham and A. Steiner, Geometry of logarithmic derivations of hyperplane arrangements; I. Halacheva, Shift of argument algebras and de Concini-Procesi spaces; B. Knudsen, Projection spaces and twisted Lie algebras; A. I. Suciu, Cohomology, Bocksteins, and resonance varieties in characteristic 2.

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Book SynopsisPuts together main approaches to study amenability. A novel feature of the book is that the exposition of the material starts with examples which introduce a method rather than illustrating it. This allows the reader to quickly move on to meaningful material without learning and remembering a lot of additional definitions and preparatory results.

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Book SynopsisDefines and studies multidimensional residues via analytic continuation for holomorphic bundle-valued current maps. This point of view offers versatility and flexibility to the tools and constructions proposed, allowing these residues to be defined and studied outside the classical case of complete intersection.Table of ContentsResidue calculus in one variable; Residue currents: A multiplicative approach; Residue currents: A bundle approach; Bochner-Martinelli kernels and weights; Integral closure, Briancon-Skoda type theorems; Residue calculus and trace formulae; Miscellaneous applications: Intersection, division; Complex manifolds and analytic spaces; Holomorphic bundles over complex analytic spaces; Positivity on complex analytic spaces; Various concepts in algebraic or analytic geometry; Bibliography; Index.

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Book SynopsisFocuses on 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.Table of Contents Dynamical systems Group actions Groupoids Iterated monodromy groups Groups from groupoids Growth and amenability Bibliography Index

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Book SynopsisFocuses on several successful classification problems. Namely, fix a geometry in the sense of Klein and a topological manifold. Then the different ways of locally putting the geometry on the manifold lead to a moduli space'.Table of Contents Part 1. Affine and projective geometry: Affine geometry Projective geometry Duality and non-Euclidean geometry Convexity Part 2. Geometric manifolds: Locally homogeneous geometric structures Examples of geometric structures Classification Completeness Part 3. Affine and projective structures: Affine structures on surfaces and the Euler characteristic Affine Lie groups Parallel volume and completeness Hyperbolicity Projective structures on surfaces Complex-projective structures Geometric structures on 3-manifolds Appendices: Appendix A. Transformation groups Appendix B. Affine connections Appendix C. Representations of nilpotent groups Appendix D. 4-dimensional filiform nilpotent Lie algebras Appendix E. Semicontinuous functions Appendix F. $\mathsf{SL}(2,\mathbb{C})$ and $O(3,1)$ Appendix G. Lagrangian foliations of symplectic manifolds Bibliography Index

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Book SynopsisRicci flow is a powerful analytic method for studying the geometry and topology of manifolds. This book is an introduction to Ricci flow for graduate students and mathematicians interested in working in the subject. To this end, the first chapter is a review of the relevant basics of Riemannian geometry. For the benefit of the student, the text includes a number of exercises of varying difficulty. The book also provides brief introductions to some general methods of geometric analysis and other geometric flows. Comparisons are made between the Ricci flow and the linear heat equation, mean curvature flow, and other geometric evolution equations whenever possible. Several topics of Hamilton's program are covered, such as short time existence, Harnack inequalities, Ricci solitons, Perelman's no local collapsing theorem, singularity analysis, and ancient solutions. A major direction in Ricci flow, via Hamilton's and Perelman's works, is the use of Ricci flow as an approach to solving the Poincare conjecture and Thurston's geometrization conjecture.Trade Review“The style of the book is very pleasant, including lots of motivations and background material, course outlines and exercises (with solutions), the bibliography is rather comprehensive. This work is certain to become one of the main references in this field of great current interest.” - M. Kunzinger“This book is a very well written introduction to and resource for study of the Ricci flow. It is quite self-contained, but relevant references are provided at appropriate points. The style of the book renders it accessible to graduate students (suggested course outlines and many relevant further references are provided), while its substance provides an essential resource for background, key concepts and fundamental ideas for further study in the area.” - James McCoy, Mathematical Reviews Table of Contents Riemannian geometry Fundamentals of the Ricci flow equation Closed 3-manifolds with positive Ricci curvature Ricci solitons and special solutions Isoperimetric estimates and no local collapsing Preparation for singularity analysis High-dimensional and noncompact Ricci flow Singularity analysis Ancient solutions Differential Harnack estimates Space-time geometry Appendix A. Geometric analysis related to Ricci flow Appendix B. Analytic techniques for geometric flows Appendix S. Solutions to selected exercises Bibliography Index

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Book SynopsisFocuses on Ricci solitons, shedding light on their role in understanding singularity formation in Ricci flow and formulating surgery-based Ricci flow, which holds potential applications in topology.Table of Contents Ricci flow singularity formation The Ricci soliton equation The $2$-dimensional classification Estimates for shrinking Ricci solitons Classification of $3$-dimensional shrinkers The Bryant soliton Expanding and steady GRS and the flying wing Brendle's theorem on the uniqueness of $3$-dimensional steadies Geometric preliminaries Analytic preliminaries Bibliography Index

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Book SynopsisWilliam Thurston's work has had a profound influence on mathematics. He connected whole mathematical subjects in entirely new ways and changed the way mathematicians think about geometry, topology, foliations, group theory, dynamical systems, and the way these areas interact. This four-part collection brings together Thurston's major writings.

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Book SynopsisWilliam Thurston's work has had a profound influence on mathematics. He connected whole mathematical subjects in entirely new ways and changed the way mathematicians think about geometry, topology, foliations, group theory, dynamical systems, and the way these areas interact. This four-part collection brings together Thurston's major writings.

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Table of Contents1. Basic definitions.- 2. The invariant bilinear form and the generalized Casimir operator.- 3. Integrable representations and the Weyl group of a Kac-Moody algebra.- 4. Some properties of generalized Cartan matrices.- 5. Real and imaginary roots.- 6. Affine Lie algebras: the normalized invariant bilinear form, the root system and the Weyl group.- 7. Affine Lie algebras: the realization (case k = 1).- 8. Affine Lie algebras: the realization (case k = 2 or 3). Application to the classification of finite order automorphisms.- 9. Highest weight modules over the Lie algebra g(A).- 10. Integrable highest weight modules: the character formula.- 11. Integrable highest weight modules: the weight system, the contravariant Hermitian form and the restriction problem.- 12. Integrable highest weight modules over affine Lie algebras. Application to ?-function identities.- 13. Affine Lie algebras, theta functions and modular forms.- 14. The principal realization of the basic representation. Application to the KdV-type hierarchies of non-linear partial differential equations.- Index of notations and definitions.- References.

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Book SynopsisUncover the Useful Interactions of Fixed Point Theory with Topological StructuresNonlinear Functional Analysis in Banach Spaces and Banach Algebras: Fixed Point Theory under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications is the first book to tackle the topological fixed point theory for block operator matrices with nonlinear entries in Banach spaces and Banach algebras. The book provides researchers and graduate students with a unified survey of the fundamental principles of fixed point theory in Banach spaces and algebras. The authors present several extensions of Schauder's and Krasnosel'skii's fixed point theorems to the class of weakly compact operators acting on Banach spaces and algebras, particularly on spaces satisfying the DunfordPettis property. They also address under which conditions a 2×2 block operator matrix with single- and multi-valued nonlinear entries will have a fixed point.Table of ContentsFixed Point Theory: Fundamentals. Fixed Point Theory under Weak Topology. Fixed Point Theory in Banach Algebras. Fixed Point Theory for BOM on Banach Spaces and Banach Algebras. Applications in Mathematical Physics and Biology: Existence of Solutions for Transport Equations. Exsistence of Solutions for Nonlinear Integral Equations. Two-Dimensional Boundary Value Problems.

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Book SynopsisThe theory of linear topological dynamics is a rapidly growing field of research over the last three decades or so. This book presents a survey of recent results of the author obtained in this field during the period 2016-2019. Without any doubt, this is the first research monograph concerning the topological dynamics of multivalued operators and binary relations, especially, multivalued linear woperators, simple graphs, digraphs and tournaments (we feel duty bound to say that multivalued topological dynamics is still a very undeveloped field of investigation, full of open problems and possible for further expansion). Asiede from that, the main purpose of this monograph is to consider topologically dynamical properties of linear single-valued operators in Frechet spaces and abstract fractional differential equations in Frechet spaces, which could be degenerate or non-degenerate in time variable. In this monograph, we use only two types of fractional derivatives, namely the Caputo time-fractional derivatives and Weyl time-fractional derivatives. However, most results on dynamics of differential equations are given to the abstract differential equations with integer order derivatives, especially those of first and second order in time. The monograph is consistsed of two chapters; the first chapter is further broken down into nine sections, while the second chapter is broken down into seven sections. It is not of introductory character to linear topological dynamics and it is not written in a traditional manner. As in all my previously published monographs, the numbering of definitions, theorems, propositions, remarks, lemmas, corollaries, definitions, etc., are by chapter and section; the bibliography is by author in alphabetic order. Concerning target audience, wWe deeply believe that the book could be of invaluable help to experts in linear topological dynamics, researchers in abstract partial differential equations but and also to PhD students and advanced graduate students in mathematics as well. A potential reader should be familiar with backgrounds including elementary functional analysis, measure and integration theory as well as the basic theory of abstract (degenerate) Volterra integro-differential equations. At some places, the knowledge of graph theory is preferable but not demandedable. This monograph is not intended to be a comprehensive review of current trends; albeit includes several recent results from the field of linear topological dynamics and has more than 450 titles, our reference list is far from being exhaustively complete.Table of ContentsPrefaceIntroductionDynamical Properties of Linear Operators and Abstract Differential EquationsDisjoint Dynamical Properties of Linear Operators and Abstract Differential EquationsBibliography.

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Book SynopsisTopological, Projective & Combinatorial Properties Of Spaces

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Book SynopsisThis book pursues optimal design from the perspective of mechanical properties and resistance to failure caused by cracks and fatigue. The book abandons the scale separation hypothesis and takes up phase-field modeling, which is at the cutting edge of research and is of high industrial and practical relevance. Part 1 starts by testing the limits of the homogenization-based approach when the size of the representative volume element is non-negligible compared to the structure. The book then introduces a non-local homogenization scheme to take into account the strain gradient effects. Using a phase field method, Part 2 offers three significant contributions concerning optimal placement of the inclusion phases. Respectively, these contributions take into account fractures in quasi-brittle materials, interface cracks and periodic composites. The topology optimization proposed has significantly increased the fracture resistance of the composites studied.Table of ContentsIntroduction ix Part 1. Multiscale Topology Optimization in the Context of Non-separated Scales 1 Chapter 1. Size Effect Analysis in Topology Optimization for Periodic Structures Using the Classical Homogenization 3 1.1. The classical homogenization method 4 1.1.1. Localization problem 4 1.1.2. Definition and computation of the effective material properties 7 1.1.3. Numerical implementation for the local problem with PER 9 1.2. Topology optimization model and procedure 10 1.2.1. Optimization model and sensitivity number 10 1.2.2. Finite element meshes and relocalization scheme 12 1.2.3. Optimization procedure 14 1.3. Numerical examples 16 1.3.1. Doubly clamped elastic domain 17 1.3.2. L-shaped structure 19 1.3.3. MBB beam 24 1.4. Concluding remarks 25 Chapter 2. Multiscale Topology Optimization of Periodic Structures Taking into Account Strain Gradient 29 2.1. Non-local filter-based homogenization for non-separated scales 30 2.1.1. Definition of local and mesoscopic fields through the filter 30 2.1.2. Microscopic unit cell calculations 33 2.1.3. Mesoscopic structure calculations 39 2.2. Topology optimization procedure 41 2.2.1. Model definition and sensitivity numbers 41 2.2.2. Overall optimization procedure 42 2.3. Validation of the non-local homogenization approach 43 2.4. Numerical examples 45 2.4.1. Cantilever beam with a concentrated load 46 2.4.2. Four-point bending lattice structure 52 2.5. Concluding remarks 55 Chapter 3. Topology Optimization of Meso-structures with Fixed Periodic Microstructures 57 3.1. Optimization model and procedure 58 3.2. Numerical examples 61 3.2.1. A double-clamped beam 61 3.2.2. A cantilever beam 64 3.3. Concluding remarks 66 Part 2. Topology Optimization for Maximizing the Fracture Resistance 67 Chapter 4. Topology Optimization for Optimal Fracture Resistance of Quasi-brittle Composites 69 4.1. Phase field modeling of crack propagation 71 4.1.1. Phase field approximation of cracks 71 4.1.2. Thermodynamics of the phase field crack evolution 72 4.1.3. Weak forms of displacement and phase field problems 75 4.1.4. Finite element discretization 76 4.2. Topology optimization model for fracture resistance 78 4.2.1. Model definitions 78 4.2.2. Sensitivity analysis 80 4.2.3. Extended BESO method 85 4.3. Numerical examples 87 4.3.1. Design of a 2D reinforced plate with one pre-existing crack notch 88 4.3.2. Design of a 2D reinforced plate with two pre-existing crack notches 93 4.3.3. Design of a 2D reinforced plate with multiple pre-existing cracks 96 4.3.4. Design of a 3D reinforced plate with a single pre-existing crack notch surface 98 4.4. Concluding remarks 101 Chapter 5. Topology Optimization for Optimal Fracture Resistance Taking into Account Interfacial Damage 103 5.1. Phase field modeling of bulk crack and cohesive interfaces 104 5.1.1. Regularized representation of a discontinuous field 104 5.1.2. Energy functional 106 5.1.3. Displacement and phase field problems 108 5.1.4. Finite element discretization and numerical implementation 111 5.2. Topology optimization method 114 5.2.1. Model definitions 114 5.2.2. Sensitivity analysis 116 5.3. Numerical examples 119 5.3.1. Design of a plate with one initial crack under traction 120 5.3.2. Design of a plate without initial cracks for traction loads 123 5.3.3. Design of a square plate without initial cracks in tensile loading 125 5.3.4. Design of a plate with a single initial crack under three-point bending 128 5.3.5. Design of a plate containing multiple inclusions 130 5.4. Concluding remarks 133 Chapter 6. Topology Optimization for Maximizing the Fracture Resistance of Periodic Composites 135 6.1. Topology optimization model 136 6.2. Numerical examples 138 6.2.1. Design of a periodic composite under three-point bending 138 6.2.2. Design of a periodic composite under non-symmetric three-point bending 146 6.3. Concluding remarks 151 Conclusion 153 References 157 Index 173

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Book SynopsisAn introduction to geometry in the plane, both Euclidean and hyperbolic, this book is designed for an undergraduate course in geometry. With its patient approach, and plentiful illustrations, it will also be a stimulating read for anyone comfortable with the language of mathematical proof. While the material within is classical, it brings together topics that are not generally found together in books at this level, such as: parametric equations for the pseudosphere and its geodesics; trilinear and barycentric coordinates; Euclidean and hyperbolic tilings; and theorems proved using inversion. The book is divided into four parts, and begins by developing neutral geometry in the spirit of Hilbert, leading to the Saccheri–Legendre Theorem. Subsequent sections explore classical Euclidean geometry, with an emphasis on concurrence results, followed by transformations in the Euclidean plane, and the geometry of the Poincaré disk model.Trade Review…The author succeeds in elevating Euclidean geometry in particular to the level of advanced undergraduate study and in so doing presents it as a counterpart to a first analysis course. At the same time, some readers may prefer to move on to new ideas and results more quickly; to that end, Harvey gives an indication of several paths through the book in his preface." - Choice"To my mind, this book stands out from the crowd partly because of the way in which its imaginatively devised illustrations are used to stage the introduction of basic concepts and to guide the reader through various steps in a proof. It's not just a matter of pretty pictures, however, because Matthew Harvey's written commentary is easy going and yet mathematically precise. He has written a truly lovely book, which is now top of my reading list as an introduction to geometry at this level." - Peter RuaneTable of ContentsAxioms and models; Part I. Neutral Geometry: 1. The axioms of incidence and order; 2. Angles and triangles; 3. Congruence verse I: SAS and ASA; 4. Congruence verse II: AAS; 5. Congruence verse III: SSS; 6. Distance, length and the axioms of continuity; 7. Angle measure; 8. Triangles in neutral geometry; 9. Polygons; 10. Quadrilateral congruence theorems; Part II. Euclidean Geometry: 11. The axiom on parallels; 12. Parallel projection; 13. Similarity; 14. Circles; 15. Circumference; 16. Euclidean constructions; 17. Concurrence I; 18. Concurrence II; 19. Concurrence III; 20. Trilinear coordinates; Part III. Euclidean Transformations: 21. Analytic geometry; 22. Isometries; 23. Reflections; 24. Translations and rotations; 25. Orientation; 26. Glide reflections; 27. Change of coordinates; 28. Dilation; 29. Applications of transformations; 30. Area I; 31. Area II; 32. Barycentric coordinates; 33. Inversion I; 34. Inversion II; 35. Applications of inversion; Part IV. Hyperbolic Geometry: 36. The search for a rectangle; 37. Non-Euclidean parallels; 38. The pseudosphere; 39. Geodesics on the pseudosphere; 40. The upper half-plane; 41. The Poincaré disk; 42. Hyperbolic reflections; 43. Orientation preserving hyperbolic isometries; 44. The six hyperbolic trigonometric functions; 45. Hyperbolic trigonometry; 46. Hyperbolic area; 47. Tiling; Bibliography; Index.

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Book SynopsisWhile the architecture of present-day parallel supercomputers is largely based on the concept of a shared memory, with its attendant limitations of common access, advances in semicoductor technology have led to the development of highly parellel computer architectures with decentralized storage and limited connections in which each processor possesses high bandwidth local memory connected to a small number of such architectures, enabling cost-effective high-speed parallel processing for large volumes of data, with ultra-high throughput rates. Algorithms suitable for implementation on systolic arrays find applications in areas such as signal and image processing, pattern matching, linear algebra, recurrence algorithms and graph problems. This book provides an insight into the implementation of systolic arrays and gives a comprehensive overview of the techniques and theories contributing to the design of systolic algorithms.Table of ContentsPREFACE 1. INTRODUCTION Systolic Algorithms 2. POLYNOMIAL AND ROOT FINDING METHODS 3. SYSTOLIC MATRIX OPERATIONS 4. QUADRATURE AND DIFFERENTIAL EQUATIONS 5 . SOLUTION OF LINEAR SYSTEMS 6. EIGENV ALOE-EIGENVECTOR COMPUTATIONS 7. LINEAR AND DYNAMIC PROGRAMMING

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Book SynopsisThis monograph could be used for a graduate course on symplectic geometry as well as for independent study.The monograph starts with an introduction of symplectic vector spaces, followed by symplectic manifolds and then Hamiltonian group actions and the Darboux theorem. After discussing moment maps and orbits of the coadjoint action, symplectic quotients are studied. The convexity theorem and toric manifolds come next and we give a comprehensive treatment of Equivariant cohomology. The monograph also contains detailed treatment of the Duistermaat-Heckman Theorem, geometric quantization, and flat connections on 2-manifolds. Finally, there is an appendix which provides background material on Lie groups. A course on differential topology is an essential prerequisite for this course. Some of the later material will be more accessible to readers who have had a basic course on algebraic topology. For some of the later chapters, it would be helpful to have some background on representation theory and complex geometry.Trade Review“The target audience is graduate students; ... this monograph could easily be used by researchers interested in learning the subject at a fast pace. It is a perfect text for a seminar course. ... the book's material is presented in a crisp and abridged manner. ... This makes the presentation short and highly valuable.” (Eduardo A. Gonzalez, Mathematical Reviews, December, 2020)Table of ContentsSymplectic vector spaces.- Hamiltonian group actions.- The Darboux-Weinstein Theorem.- Elementary properties of moment maps.- The symplectic structure on coadjoint orbits.- Symplectic Reduction.- Convexity.- Toric Manifolds.- Equivariant Cohomology.- The Duistermaat-Heckman Theorem.- Geometric Quantization.- Flat connections on 2-manifolds.

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Book SynopsisThis book aims to provide a friendly introduction to non-commutative geometry. It studies index theory from a classical differential geometry perspective up to the point where classical differential geometry methods become insufficient. It then presents non-commutative geometry as a natural continuation of classical differential geometry. It thereby aims to provide a natural link between classical differential geometry and non-commutative geometry. The book shows that the index formula is a topological statement, and ends with non-commutative topology.Trade Review“The present book is well written. It is very useful to researchers in differential geometry who are interested in non-commutative geometry. It provides motivations for tudying non commutative geometry.” (Ion Mihai, zbMATH 1458.58001, 2021)Table of Contents1. Part I Spaces, bundles and characteristic classes in differential geometry.- 2. Part II Non-commutative differential geometry.- 3. Part III Index Theorems.- 4. Part IV Prospects in Index Theory. Part V.- 5. Non-commutative topology.

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Book SynopsisThis book presents, in a clear and structured way, the set function \mathcal{T} and how it evolved since its inception by Professor F. Burton Jones in the 1940s. It starts with a very solid introductory chapter, with all the prerequisite material for navigating through the rest of the book. It then gradually advances towards the main properties, Decomposition theorems, \mathcal{T}-closed sets, continuity and images, to modern applications.The set function \mathcal{T} has been used by many mathematicians as a tool to prove results about the semigroup structure of the continua, and about the existence of a metric continuum that cannot be mapped onto its cone or to characterize spheres. Nowadays, it has been used by topologists worldwide to investigate open problems in continuum theory.This book can be of interest to both advanced undergraduate and graduate students, and to experienced researchers as well. Its well-defined structure make this book suitable not only for self-study but also as support material to seminars on the subject. Its many open problems can potentially encourage mathematicians to contribute with further advancements in the field.Table of ContentsPreliminaries.- The Set Function T.- Decomposition Theorems.- T-Closed Sets.- Continuity of T.- Images of T.- Applications.- Questions.- References.- Index.

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Book SynopsisThis book is devoted to geometric problems of foliation theory, in particular those related to extrinsic geometry, modern branch of Riemannian Geometry. The concept of mixed curvature is central to the discussion, and a version of the deep problem of the Ricci curvature for the case of mixed curvature of foliations is examined. The book is divided into five chapters that deal with integral and variation formulas and curvature and dynamics of foliations. Different approaches and methods (local and global, regular and singular) in solving the problems are described using integral and variation formulas, extrinsic geometric flows, generalizations of the Ricci and scalar curvatures, pseudo-Riemannian and metric-affine geometries, and 'computable' Finsler metrics.The book presents the state of the art in geometric and analytical theory of foliations as a continuation of the authors' life-long work in extrinsic geometry. It is designed for newcomers to the field as well as experienced geometers working in Riemannian geometry, foliation theory, differential topology, and a wide range of researchers in differential equations and their applications. It may also be a useful supplement to postgraduate level work and can inspire new interesting topics to explore.Trade Review“The reader is assumed to have some background in topology and differential geometry. The book is a continuation of the authors’ work in extrinsic geometry and thus provides a useful reference for researchers in this field.” (Emanuel-Ciprian Cismaş, zbMATH 1479.53002, 2022)Table of ContentsPreface.- 1. Preliminaries.- 2. Integral formulas.- 3. Prescribing the mean curvature.- 4. Variational formulae.- 5. Extrinsic Geometric flows.- References.- Index.

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Book SynopsisThis book gives a brief treatment of the equivariant cohomology of the classical configuration space F(ℝ^d,n) from its beginnings to recent developments. This subject has been studied intensively, starting with the classical papers of Artin (1925/1947) on the theory of braids, and progressing through the work of Fox and Neuwirth (1962), Fadell and Neuwirth (1962), and Arnol'd (1969). The focus of this book is on the mod 2 equivariant cohomology algebras of F(ℝ^d,n), whose additive structure was described by Cohen (1976) and whose algebra structure was studied in an influential paper by Hung (1990). A detailed new proof of Hung's main theorem is given, however it is shown that some of the arguments given by him on the way to his result are incorrect, as are some of the intermediate results in his paper.This invalidates a paper by three of the authors, Blagojević, Lück and Ziegler (2016), who used a claimed intermediate result in order to derive lower bounds for the existence of k-regular and ℓ-skew embeddings. Using the new proof of Hung's main theorem, new lower bounds for the existence of highly regular embeddings are obtained: Some of them agree with the previously claimed bounds, some are weaker.Assuming only a standard graduate background in algebraic topology, this book carefully guides the reader on the way into the subject. It is aimed at graduate students and researchers interested in the development of algebraic topology in its applications in geometry.Trade Review“The book is well written. … The book will be important for those who study the cohomology rings of configuration spaces.” (Shintarô Kuroki, Mathematical Reviews, November, 2022)Table of Contents​- 1. Snapshots from the History. - Part I Mod 2 Cohomology of Configuration Spaces. - 2. The Ptolemaic Epicycles Embedding. - 3. The Equivariant Cohomology of Pe(Rd, 2m). - 4. Hu’ng’s Injectivity Theorem. - Part II Applications to the (Non-)Existence of Regular and SkewEmbeddings. - 5. On Highly Regular Embeddings: Revised. - 6. More Bounds for Highly Regular Embeddings. - Part III Technical Tools. - 7. Operads. - 8. The Dickson Algebra. - 9. The Stiefel–Whitney Classes of the Wreath Square of a Vector Bundle. - 10. Miscellaneous Calculations.

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Book SynopsisThis book presents fixed point theory, one of the crucial tools in applied mathematics, functional analysis, and topology, which has been used to solve distinct real-world problems in computer science, engineering, and physics. The authors begin with an overview of the extension of metric spaces. Readers are introduced to general fixed-point theorems while comparing and contrasting important and insignificant metric spaces. The book is intended to be self-contained and serves as a unique resource for researchers in various disciplines.Table of ContentsMetric Spaces.- Extension of Metric Spaces.- Fixed Point Theorems on Extended Metric Spaces.

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Book SynopsisThis volume celebrates the 100th birthday of Professor Chen-Ning Frank Yang (Nobel 1957), one of the giants of modern science and a living legend. Starting with reminiscences of Yang's time at the research centre for theoretical physics at Stonybrook (now named C. N. Yang Institute) by his successor Peter van Nieuwenhuizen, the book is a collection of articles by world-renowned mathematicians and theoretical physicists. This emphasizes the Dialogue Between Physics and Mathematics that has been a central theme of Professor Yang’s contributions to contemporary science. Fittingly, the contributions to this volume range from experimental physics to pure mathematics, via mathematical physics. On the physics side, the contributions are from Sir Anthony Leggett (Nobel 2003), Jian-Wei Pan (Willis E. Lamb Award 2018), Alexander Polyakov (Breakthrough Prize 2013), Gerard 't Hooft (Nobel 1999), Frank Wilczek (Nobel 2004), Qikun Xue (Fritz London Prize 2020), and Zhongxian Zhao (Bernd T. Matthias Prize 2015), covering an array of topics from superconductivity to the foundations of quantum mechanics. In mathematical physics there are contributions by Sir Roger Penrose (Nobel 2022) and Edward Witten (Fields Medal 1990) on quantum twistors and quantum field theory, respectively. On the mathematics side, the contributions by Vladimir Drinfeld (Fields Medal 1990), Louis Kauffman (Wiener Gold Medal 2014), and Yuri Manin (Cantor Medal 2002) offer novel ideas from knot theory to arithmetic geometry.Inspired by the original ideas of C. N. Yang, this unique collection of papers b masters of physics and mathematics provides, at the highest level, contemporary research directions for graduate students and experts alike.Table of Contents1 Frank Yang at Stony Brook and the Beginning of Supergravity.- 2. A Stacky Approach to Crystals.- 3 The Potts Model, the Jones Polynomial and Link Homology.- 4 The Penrose–Onsager–Yang Approach to Superconductivity and Superfluidity.- 5 Quantum Operads.- 6 Quantum computational complexity withphotons and linear optics.- 7 Quantized Twistors, G2*, and the Split Octonions.- 8 Kronecker Anomalies and Gravitational Striction.- 9 Projecting Local and Global Symmetries to the Planck Scale.- 10 Gauge Symmetry in Shape Dynamics.- 11 Why Does Quantum Field Theory In Curved Spacetime Make Sense? And What Happens To The Algebra of Observables In The Thermodynamic Limit?.- 12 Quantum Anomalous Hall Effect.- 13 Magic Superconducting States in Cuprates.

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Book SynopsisThe first of three volumes on partial differential equations, this one introduces basic examples arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, in particular Fourier analysis, distribution theory, and Sobolev spaces. These tools are then applied to the treatment of basic problems in linear PDE, including the Laplace equation, heat equation, and wave equation, as well as more general elliptic, parabolic, and hyperbolic equations. The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.The third edition further expands the material by incorporating new theorems and applications throughout the book, and by deepening connections and relating concepts across chapters. In includes new sections on rigid body motion, on probabilistic results related to random walks, on aspects of operator theory related to quantum mechanics, on overdetermined systems, and on the Euler equation for incompressible fluids. The appendices have also been updated with additional results, ranging from weak convergence of measures to the curvature of Kahler manifolds.Michael E. Taylor is a Professor of Mathematics at the University of North Carolina, Chapel Hill, NC.Review of first edition: “These volumes will be read by several generations of readers eager to learn the modern theory of partial differential equations of mathematical physics and the analysis in which this theory is rooted.”(Peter Lax, SIAM review, June 1998)Table of ContentsContents of Volumes II and III.- Preface.- 1 Basic Theory of ODE and Vector Fields.- 2 The Laplace Equation and Wave Equation.- 3 Fourier Analysis, Distributions, and Constant-Coefficient Linear PDE.- 4 Sobolev Spaces.- 5 Linear Elliptic Equation.- 6 Linear Evolution Equations.- A Outline of Functional Analysis.- B Manifolds, Vector Bundles, and Lie Groups.- Index.

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Book SynopsisUntil the mid-twentieth century, topological studies were focused on the theory of suitable structures on sets of points. The concept of open set exploited since the twenties offered an expression of the geometric intuition of a "realistic" place (spot, grain) of non-trivial extent.Imitating the behaviour of open sets and their relations led to a new approach to topology flourishing since the end of the fifties.It has proved to be beneficial in many respects. Neglecting points, only little information was lost, while deeper insights have been gained; moreover, many results previously dependent on choice principles became constructive. The result is often a smoother, rather than a more entangled, theory.No monograph of this nature has appeared since Johnstone's celebrated Stone Spaces in 1983. The present book is intended as a bridge from that time to the present. Most of the material appears here in book form for the first time or is presented from new points of view. Two appendices provide an introduction to some requisite concepts from order and category theories.Trade ReviewFrom the reviews:“The book covers, in a comprehensive and self-contained way, the development of point-free topology in the last 30 years … . The book ends with two short appendices containing those facts about partially ordered sets and categories needed in the book. In conclusion, this is a very good book; it is nicely written and is highly recommended for anyone wishing to gain an overview of point-free topology.” (Javier Gutiérrez García, Mathematical Reviews, Issue 2012 j)“The book starts with a recount of sobriety and the TD-axiom, which is then quickly followed by the definition of the category of locales, various facets of spectra, the usual adjunctions and several criteria for spatiality. … this book is an erudite account of the current status of pointfree topology, written beautifully in the authors’ inimitable style. It is both easily accessible to a complete beginner, and an excellent source of reference for the mature pointfree practitioner.” (Themba Dube, Zentralblatt MATH, Vol. 1231, 2012)“In a total of fifteen chapters the reader is taken from spaces and lattices of open sets, through a thorough discussion of the notions of frames and locales … . book ends (modulo its appendices) with a discussion of localic groups … together with frames, that’s what this entire book is ultimately all about. … is quite-well written: it’s a clear and thorough treatment of what is really rather an important subject, which is to say, a greatly underappreciated one.” (Michael Berg, The Mathematical Association of America, April, 2012)Table of ContentsPreface.- Introduction.- I. Spaces and lattices of open sets.- II. Frames and locales. Spectra.- III. Sublocales.- IV. Structure of localic morphisms. The categories Loc and Frm.- V. Separation axioms.- VI. More on sublocales.-VII. Compactness and local compactness.- VIII. (Symmetric) uniformity and nearness.- IX. Paracompactness.- X. More about completion.- XI. Metric frames.- XII. Entourages, non-symmetric uniformity.- XIII. Connectedness.- XIV. The frame of reals and real functions.- XV. Localic groups.- Appendix I: Posets.- Appendix II: Categories.- Bibliography.- Index of Notation.- Index.

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Book SynopsisThis book is an introduction to several active research topics in Foliation Theory and its connections with other areas. It contains expository lectures showing the diversity of ideas and methods converging in the study of foliations. The lectures by Aziz El Kacimi Alaoui provide an introduction to Foliation Theory with emphasis on examples and transverse structures. Steven Hurder's lectures apply ideas from smooth dynamical systems to develop useful concepts in the study of foliations: limit sets and cycles for leaves, leafwise geodesic flow, transverse exponents, Pesin Theory and hyperbolic, parabolic and elliptic types of foliations. The lectures by Masayuki Asaoka compute the leafwise cohomology of foliations given by actions of Lie groups, and apply it to describe deformation of those actions. In his lectures, Ken Richardson studies the properties of transverse Dirac operators for Riemannian foliations and compact Lie group actions, and explains a recently proved index formula. Besides students and researchers of Foliation Theory, this book will be interesting for mathematicians interested in the applications to foliations of subjects like Topology of Manifolds, Differential Geometry, Dynamics, Cohomology or Global Analysis.Trade Review“This book contains the lecture notes of four courses on several topics with rather different flavor, which are linked by their relation with Foliation Theory. … the courses will be very helpful for any reader that wants to get quickly introduced to any of these lines of research.” (Jesus A. Álvarez López, zbMATH 1318.57001, 2015)Table of ContentsFundamentals of Foliation Theory.- Foliation Dynamics.- Deformation of Locally Free Actions and Leafwise Cohomology.- Transversal Dirac Operators on Distributions, Foliations, and G-Manifolds.​

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Book SynopsisThis textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring minimal prerequisites. In particular, the theory of matrix Lie groups and their Lie algebras is developed using only linear algebra, and more motivation and intuition for proofs is provided than in most classic texts on the subject.In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including: a treatment of the Baker–Campbell–Hausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebras motivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C) an unconventional definition of semisimplicity that allows for a rapid development of the structure theory of semisimple Lie algebras a self-contained construction of the representations of compact groups, independent of Lie-algebraic arguments The second edition of Lie Groups, Lie Algebras, and Representations contains many substantial improvements and additions, among them: an entirely new part devoted to the structure and representation theory of compact Lie groups; a complete derivation of the main properties of root systems; the construction of finite-dimensional representations of semisimple Lie algebras has been elaborated; a treatment of universal enveloping algebras, including a proof of the Poincaré–Birkhoff–Witt theorem and the existence of Verma modules; complete proofs of the Weyl character formula, the Weyl dimension formula and the Kostant multiplicity formula.Review of the first edition:This is an excellent book. It deserves to, and undoubtedly will, become the standard text for early graduate courses in Lie group theory ... an important addition to the textbook literature ... it is highly recommended.— The Mathematical GazetteTrade Review“The first edition of this book was very good; the second is even better, and more versatile. This text remains one of the most attractive sources available from which to learn elementary Lie group theory, and is highly recommended.” (Mark Hunacek, The Mathematical Gazette, Vol. 101 (551), July, 2017)Table of ContentsPart I: General Theory.-Matrix Lie Groups.- The Matrix Exponential.- Lie Algebras.- Basic Representation Theory.- The Baker–Campbell–Hausdorff Formula and its Consequences.- Part II: Semisimple Lie Algebras.- The Representations of sl(3;C).-Semisimple Lie Algebras.- Root Systems.- Representations of Semisimple Lie Algebras.- Further Properties of the Representations.- Part III: Compact lie Groups.- Compact Lie Groups and Maximal Tori.- The Compact Group Approach to Representation Theory.- Fundamental Groups of Compact Lie Groups.- Appendices.

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Book SynopsisThis book, intended for postgraduate students and researchers, presents many results of historical importance on pseudocompact spaces. In 1948, E. Hewitt introduced the concept of pseudocompactness which generalizes a property of compact subsets of the real line. A topological space is pseudocompact if the range of any real-valued, continuous function defined on the space is a bounded subset of the real line. Pseudocompact spaces constitute a natural and fundamental class of objects in General Topology and research into their properties has important repercussions in diverse branches of Mathematics, such as Functional Analysis, Dynamical Systems, Set Theory and Topological-Algebraic structures. The collection of authors of this volume include pioneers in their fields who have written a comprehensive explanation on this subject. In addition, the text examines new lines of research that have been at the forefront of mathematics. There is, as yet, no text that systematically compiles and develops the extensive theory of pseudocompact spaces, making this book an essential asset for anyone in the field of topology.Table of Contents1. Basic and Classic Results on Pseudocompact Spaces.- 2. Pseudocompact Topological Groups.- 3. Pseudocompactness and Ultrafilters.- 4. Bounded Subsets of Tychonoff Spaces: A Survey of Results and Problems.- 5. Weakly Pseudocompact Spaces.- 6. Maximal Pseudocompact Spaces.- 7. Pseudocompactness in the Realm of Topological Transformation Groups.- 8. Topology of Mrówka-Isbell Spaces.

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Book SynopsisThis book explains and helps readers to develop geometric intuition as it relates to differential forms. It includes over 250 figures to aid understanding and enable readers to visualize the concepts being discussed. The author gradually builds up to the basic ideas and concepts so that definitions, when made, do not appear out of nowhere, and both the importance and role that theorems play is evident as or before they are presented. With a clear writing style and easy-to- understand motivations for each topic, this book is primarily aimed at second- or third-year undergraduate math and physics students with a basic knowledge of vector calculus and linear algebra.Trade Review “The reviewer recommends young mathematics and physics majors to open the book and to keep it on their bookshelves. Indeed, the reviewer even envies young students who can study differential forms with such a fascinating book.” (Hirokazu Nishimura, zbMath 1419.58001, 2019)Table of Contents

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Book SynopsisHauptgegenstand des Buches sind Homologie-, Kohomologietheorien und Mannigfaltigkeiten. In den ersten acht Kapiteln werden Begriffe wie Homologie, CW-Komplexe, Produkte und Poincaré Dualität eingeführt und deren Anwendungen diskutiert. In den davon unabhängigen Kapiteln 9 bis 13 werden Differentialformen und der Satz von Stokes auf Mannigfaltigkeiten behandelt. Die in Kapitel 14 und 15 behandelte de Rham Kohomologie und der Satz von de Rham verbinden diese beiden Teile.Table of ContentsHomologie - Singuläre Homologie - CW-Komplexe - Euler-Charakteristik und Lefschetz-Zahlen - Kohomologie - Homologische Algebra - Produkte - Dualität - Glatte Mannigfaltigkeiten und ihr Tangentialbündel - Elementare Lineare Algebra - Parametrisierte Lineare Algebra - Differentialformen - Der Satz von Stokes - De Rham Kohomologie - Der Satz von de Rham

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Table of ContentsStatement of the theorem.- Applications of the index theorem.- Outline of the proof.- The atiyah-singer fixed point theorem.- Applications of the fixed point theorem.

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Book SynopsisAus den Rezensionen: "Was das Buch vor allem auszeichnet, ist die unkonventionelle Darstellungsweise. Hier wird Mathematik nicht im trockenen Definition-Satz-Beweis-Stil geboten, sondern sie wird dem Leser pointiert und mit viel Humor schmackhaft gemacht. In ungewöhnlich fesselnder Sprache geschrieben, ist die Lektüre dieses Buches auch ein belletristisches Vergnügen. Fast 200 sehr instruktive und schöne Zeichnungen unterstützen das Verständnis, motivieren die behandelten Aussagen, modellieren die tragenden Beweisideen heraus. Ungewöhnlich ist auch das Register, das unter jedem Stichwort eine Kurzdefinition enthält und somit umständliches Nachschlagen erspart". Wiss. Zeitschrift der TU Dresden Jetzt in der achten Auflage des bewährten Lehrbuches!Trade Review"... Es handelt sich um ein sehr überschaubares Werk, dessen Qualität wohl gerade darin liegt, daß es trotz Beschränkung im Umfang (gut 200 kleinformatige Seiten) einen schönen Einblick in die wichtigsten Grundideen der Topologie vermittelt. ... Sehr reizvoll in Jänichs Buch sind auch die zahlreichen suggestiven Abbildungen. Vor allem jene zu Beginn der Kapitel werden jeden Topologieliebhaber durch ihre manchmal geradezu poetische Originalität erfreuen."Internationale Mathematische Nachrichten ÖsterreichTable of ContentsDie Grundbegriffe.- Topologische Vektorräume.- Die Quotiententopologie.- Vervollständigung metrischer Räume.- Homotopie.- Die beiden Abzählbarkeitsaxiome.- CW-Komplexe.- Konstruktion von stetigen Funktionen auf topologischen Räumen.- Überlagerungen.- Der Satz von Tychonoff.- Letztes Kapitel. Mengenlehre.

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Book SynopsisCategories and sheaves appear almost frequently in contemporary advanced mathematics. This book covers categories, homological algebra and sheaves in a systematic manner starting from scratch and continuing with full proofs to the most recent results in the literature, and sometimes beyond. The authors present the general theory of categories and functors, emphasizing inductive and projective limits, tensor categories, representable functors, ind-objects and localization.Trade ReviewFrom the reviews: "This book of Kashiwara and Schapira, recognized specialists in algebraic analysis, is a detailed full-scale exposition of categories, homological algebra and sheaves. These notions are presented from scratch up to the most recent (sometimes new) results … ." (Corrado Marastoni, Mathematical Reviews, Issue 2006 k)Table of ContentsThe Language of Categories.- Limits.- Filtrant Limits.- Tensor Categories.- Generators and Representability.- Indization of Categories.- Localization.- Additive and Abelian Categories.- ?-accessible Objects and F-injective Objects.- Triangulated Categories.- Complexes in Additive Categories.- Complexes in Abelian Categories.- Derived Categories.- Unbounded Derived Categories.- Indization and Derivation of Abelian Categories.- Grothendieck Topologies.- Sheaves on Grothendieck Topologies.- Abelian Sheaves.- Stacks and Twisted Sheaves.

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Book SynopsisThis book offers a comprehensive introduction to the general theory of C*-algebras and von Neumann algebras. Beginning with the basics, the theory is developed through such topics as tensor products, nuclearity and exactness, crossed products, K-theory, and quasidiagonality. The presentation carefully and precisely explains the main features of each part of the theory of operator algebras; most important arguments are at least outlined and many are presented in full detail.Trade ReviewFrom the reviews: "The volume under review is supposed to cover basics on operator algebras … . Blackadar’s book is very well written and pleasant to read. It is especially suited to readers who already know the basics of operator algebras but who need a reference for some result or who wish to have a unified approach to topics treated by them." (Paul Jolissaint, Mathematical Reviews, Issue 2006 k) "This volume is an important and useful contribution to the literature on C*-algebras and von Neumann algebras. … The book is extremely well written. It can be recommended as a reference to graduate students working in operator algebra theory and to other mathematicians who want to bring themselves up-to-date on the subject." (V. M. Manuilov, Zentralblatt MATH, Vol. 1092 (18), 2006)Table of ContentsOperators on Hilbert Space.- C*-Algebras.- Von Neumann Algebras.- Further Structure.- K-Theory and Finiteness.

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Book SynopsisLes Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce livre est le cinquième du traité ; il est consacré aux bases de l’analyse fonctionnelle. Il contient en particulier le théorème de Hahn-Banach et le théorème de Banach-Steinhaus. Il comprend les chapitres: -1. Espaces vectoriels topologiques sur un corps value; -2. Ensembles convexes et espaces localement convexes; -3. Espaces d’applications linéaires continues; -4. La dualité dans les espaces vectoriels topologiques; -5. Espaces hilbertiens (théorie élémentaire). Il contient également des notes historiques. Ce volume a été publié en 1981.Table of ContentsEspaces vectoriels topologiques sur un corps valué.- Ensembles convexes et espaces localement convexes.- Espaces d'applications linéaires continues.- La dualité dans les espaces vectoriels topologiques.- Espaces hilbertiens (théorie élémentaire).

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Book SynopsisFollowing Quillen's approach to complex cobordism, the authors introduce the notion of oriented cohomology theory on the category of smooth varieties over a fixed field. They prove the existence of a universal such theory (in characteristic 0) called Algebraic Cobordism. The book also contains some examples of computations and applications.Table of ContentsCobordism and oriented cohomology.- The definition of algebraic cobordism.- Fundamental properties of algebraic cobordism.- Algebraic cobordism and the Lazard ring.- Oriented Borel-Moore homology.- Functoriality.- The universality of algebraic cobordism.

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Book SynopsisSince its inception by von Neumann 70 years ago, the theory of operator algebras has become a rapidly developing area of importance for the understanding of many areas of mathematics and theoretical physics. Accessible to the non-specialist, this first part of a three-volume treatise provides a clear, carefully written survey that emphasizes the theory's analytical and topological aspects.Trade Review"... These three bulky volumes [EMS 124, 125, 127], written by one of the most prominent researchers of the area, provide an introduction to this repidly developing theory. ... These books can be warmly recommended to every graduate student who wants to become acquainted with this exciting branch of matematics. Furthermore, they should be on the bookshelf of every researcher of the area." (László Kérchy, Acta Scientiarum Mathematicarum, Vol. 69, 2003)Table of ContentsFundaments of Banach Algebras and C*-Algebras.- Topologies and Density Theorems in Operator Algebras.- Conjugate Spaces.- Tensor Products of Operator Algebras and Direct Integrals.- Types of von Neumann Algebras and Traces.- Appendix: Polish Spaces and Standard Borel Spaces.

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Book SynopsisThe monograph is a study of the local bifurcations of multiparameter symplectic maps of arbitrary dimension in the neighborhood of a fixed point.The problem is reduced to a study of critical points of an equivariant gradient bifurcation problem, using the correspondence between orbits ofa symplectic map and critical points of an action functional. New results onsingularity theory for equivariant gradient bifurcation problems are obtained and then used to classify singularities of bifurcating period-q points. Of particular interest is that a general framework for analyzing group-theoretic aspects and singularities of symplectic maps (particularly period-q points) is presented. Topics include: bifurcations when the symplectic map has spatial symmetry and a theory for the collision of multipliers near rational points with and without spatial symmetry. The monograph also includes 11 self-contained appendices each with a basic result on symplectic maps. The monograph will appeal to researchers and graduate students in the areas of symplectic maps, Hamiltonian systems, singularity theory and equivariant bifurcation theory.Table of ContentsGeneric bifurcation of periodic points.- Singularity theory for equivariant gradient bifurcation problems.- Classification of Zq-equivariant gradient bifurcation problems.- Period-3 points of the generalized standard map.- Classification of Dq-equivariant gradient bifurcation problems.- Reversibility and degenerate bifurcation of period-q points of multiparameter maps.- Periodic points of equivariant symplectic maps.- Collision of multipliers at rational points for symplectic maps.- Equivariant maps and the collision of multipliers.

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Book SynopsisEinstein's equations stem from General Relativity. In the context of Riemannian manifolds, an independent mathematical theory has developed around them. This is the first book which presents an overview of several striking results ensuing from the examination of Einstein’s equations in the context of Riemannian manifolds. Parts of the text can be used as an introduction to modern Riemannian geometry through topics like homogeneous spaces, submersions, or Riemannian functionals.Trade ReviewFrom the reviews: "[...] an efficient reference book for many fundamental techniques of Riemannian geometry. [...] despite its length, the reader will have no difficulty in getting the feel of its contents and discovering excellent examples of all interaction of geometry with partial differential equeations, topology, and Lie groups. Above all, the book provides a clear insight into the scope and diversity of problems posed by its title."S.M. Salamon in MathSciNet 1988 "It seemed likely to anyone who read the previous book by the same author, namely Manifolds all of whose geodesic are closed, that the present book would be one of the most important ever published on Riemannian geometry. This prophecy is indeed fulfilled."T.J. Wilmore in Bulletin of the London Mathematical Society 1987 "Einstein Manifolds is accordingly described as Besse’s second book … . there is no doubt that Einstein Manifolds is a magnificient work of mathematical scholarship. … It is truly a seminal work on an incomparably fascinating and important subject." (Michael Berg, MathDL, March, 2008) "The present book is intended to be a complete reference book. … The book under review serves several purposes. It is an efficient reference for many fundamental techniques of Riemannian geometry as well as excellent examples of the interaction of geometry with partial differential equations, topology and Lie groups. Certainly the monograph provides a clear insight into the scope and diversity of problems posed by its title." (Adela-Gabriela Mihai, Zentralblatt MATH, Vol. 1147, 2008)Table of ContentsBasic Material.- Basic Material (Continued): Kähler Manifolds.- Relativity.- Riemannian Functionals.- Ricci Curvature as a Partial Differential Equation.- Einstein Manifolds and Topology.- Homogeneous Riemannian Manifolds.- Compact Homogeneous Kähler Manifolds.- Riemannian Submersions.- Holonomy Groups.- Kähler-Einstein Metrics and the Calabi Conjecture.- The Moduli Space of Einstein Structures.- Self-Duality.- Quaternion-Kähler Manifolds.- A Report on the Non-Compact Case.- Generalizations of the Einstein Condition.

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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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