Topology Books

Book SynopsisIt has been realized that Hilbert schemes originally studied in algebraic geometry are closely related to several branches of mathematics, such as singularities, symplectic geometry, representation theory - even theoretical physics. This book reflects this feature of Hilbert schemes.Table of ContentsIntroduction Hilbert scheme of points Framed moduli space of torsion free sheaves on $\mathbb{P}^2$ Hyper-Kahler metric on $(\mathbb{C}^2)^{[n]}$ Resolution of simple singularities Poincare polynomials of the Hilbert schemes (1) Poincare polynomials of Hilbert schemes (2) Hilbert scheme on the cotangent bundle of a Riemann surface Homology group of the Hilbert schemes and the Heisenberg algebra Symmetric products of an embedded curve, symmetric functions and vertex operators Bibliography Index.

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Book SynopsisPresents the proceedings of the conference on 'Trends in Mathematical Physics' held at the University of Tennessee. The conference drew international experts from mathematical and computational physics. The following topics were addressed: superstrings and quantum gravity, pattern formation, and crystallographic topology.Table of ContentsBerry's connection in a quantum relativistic description of the curved space analogue of development by R. R. Aldinger Heat-kernel asymptotics of the Gilkey-Smith boundary-value problem by I. G. Avramidi and G. Esposito On ellipticity and gauge invariance in Euclidean quantum gravity by I. G. Avramidi and G. Esposito Undercompressive waves in driven thin film flow: Theory, computation, and experiment by A. L. Bertozzi, A. Munch, and M. Shearer Internal time peculiarities as a cause of bifurcations arising in classical trajectory problem and quantum chaos creation in three-body systems by A. V. Bogdanov, A. S. Gevorkyan, and A. G. Grigoryan Random motion of quantum harmonic oscillator-Thermodynamics of nonrelativistic vacuum by A. V. Bogdanov, A. S. Gevorkyan, and A. G. Grigoryan A new formulation of Lax pairs for generalized Calogero-Moser models by A. J. Bordner Four-point correlation functions in the AdS/CFT correspondence by G. Chalmers and K. Schalm Gravitational instantons and moduli spaces by S. A. Cherkis Superdualities and twisted self-duality by E. Cremmer, B. Julia, H. Lu, and C. N. Pope Boomerons in field theory by I. Dasgupta A layman's guide to $M$-theory by M. J. Duff Family symmetry, the anomalous $U(1)$, and neutrino mixing by J. K. Elwood Stochastic quantization approach to $c \leq 1$ open-closed string field theories by D. Ennyu, H. Kawabe, and N. Nakazawa Gauge invariance and effective actions at finite temperature by C. D. Fosco Superconformal deformations and space-time symmetries by I. Giannakis Nonlinear patterns in microstructures by M. E. Glicksman Non-thermalizability of a quantum field theory by C. R. Hagen Field theory for chemical spaces by R. A. Hefferlin From functions to matrices by J. Hoppe Duality and construction of quantum integrable systems by E. Horozov and A. Kasman Geometric concepts for the mass in general relativity by G. Huisken Crystallographic topology 2: Overview and work in progress by C. K. Johnson Gravitationally dressed RG flows, zigzag symmetry and zero-tension strings by I. I. Kogan and O. A. Soloviev Einstein metrics and the Yamabe problem by C. LeBrun Numerical evolution in time of curvature perturbations in Kerr black holes by R. Lopez-Aleman Nonlinearity and self-similarity: Wavelets and compactons on a physical background by A. Ludu and J. P. Draayer On analytic approach to perturbative quantum chromodynamics by B. A. Magradze Lattice gauge theory by C. Michael Integrable models with boundary by R. I. Nepomechie Mathematical aspects of chiral gauge theories on the lattice by H. Neuberger The statistical entropy of black holes and the $AdS_3$ geometry by L. A. Pando Zayas Canonical treatment of regular Lagrangians with holonomic constraints as singular systems by E. M. Rabei Algebraic shape invariant models by C. Rasinariu, U. P. Sukhatme, and A. Gangopadhyaya Molecular configurations and Euclidean distance matrices by W. Rivera-Gallego Derivative expansion of the one-loop effective action in QED by I. A. Shovkovy Nonlinear generalizations of a 3-manifold's Dirac operator by C. H. Taubes Dynamical symmetry approach to quantum many-body problems by C.-L. Wu.

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Book SynopsisPresents a comprehensive exposition of the theory of linear operators on Banach spaces and Banach lattices. This book offers a presentation that includes various developments in operator theory and draws together results that are spread over the vast literature. It contains over 600 exercises to help students master the material developed.Table of ContentsOdds and ends Basic operator theory Operators on $AL$- and $AM$-spaces Special classes of operators Integral operators Spectral properties Some special spectra Positive matrices Irreducible operators Invariant subspaces The Daugavet equation Bibliography Index.

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Book SynopsisPresents the study of Riemannian manifolds through specializing the theory of connections on principle bundles and affine connections. This work emphasizes the geometry of Riemannian manifolds, so that the theorems of Hopf-Rinow, Hadamard-Cartan, and Cartan's local isometry theorem are included, but no elliptic operator theory.Table of ContentsManifolds Lie groups Fibre bundles Differential forms Connexions Affine connexions Riemannian manifolds Geodesics and complete Riemannian manifolds Riemannian curvature Immersions and the second fundamental form Second variation of arc length Theorems on differential equations Bibliography Subject index.

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Book SynopsisGroup-theoretic methods have taken an increasingly prominent role in analysis. Some of this change has been due to the writings of Sigurdur Helgason. This book is an introduction to such methods on spaces with symmetry given by the action of a Lie group.Trade ReviewFrom reviews of the original edition:""The book is excellent both as a text and as a reference work; it will clearly become another instant classic."" —American Scientist""This volume makes an excellent companion to the author's Differential Geometry, Lie Groups, and Symmetric Spaces, putting to work many of the abstract concepts developed in the earlier volume. The introductory material and large number of exercises (with answers!) will make the book quite appropriate for students. Researchers will find numerous useful references on geometric analysis, along with proofs, connections with other parts of mathematics, and valuable historical remarks. This book, like the author's previous work on differential geometry, will no doubt inspire considerable further research and become the standard text on the subjects it covers."" —Mathematical Reviews""Few treatises today can lay claim to being ""aere perennius"", but all of Helgason's books certainly do with a vengeance ... [He] sets a model of style and clarity that has not been matched since Enriques's Geometria proiettiva. This is the kind of mathematics that will live forever."" —The Bulletin of Mathematics Books""A most valuable contribution to Lie theory and to the interplay between geometry and analysis. It is remarkable that the beautiful theory in Chapter IV can be presented in a textbook form with complete proofs."" —Bulletin of the London Mathematical Society""The diversity of subjects treated is great. Nevertheless the author has managed to achieve coherence of presentation by clearly putting forward a few main themes and basic problems. The first third of the book is suitable as a text for beginning graduate students; the book is also an excellent source of reference for experts. No doubt it will become a new standard in the field."" —CWI QuarterlyTable of Contents Geometric Fourier analysis on spaces of constant curvature Integral geometry and Radon transforms Invariant differential operators Invariants and harmonic polynomials Spherical functions and spherical transforms Analysis on compact symmetric spaces Appendix Some details Bibliography Symbols frequently used Index Errata

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Book SynopsisPresents an exposition of spherical functions on compact symmetric spaces, from the viewpoint of Cartan-Selberg. This title treats compact symmetric pairs, spherical representations for compact symmetric pairs, the fundamental groups of compact symmetric spaces, and the radial part of an invariant differential operator.Table of ContentsIntroduction Spherical functions Compact symmetric pairs Spherical functions on spheres and on complex projective spaces Appendix References Subject index Notation index.

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Book SynopsisTrade ReviewIn the introduction the authors give a good outline of the proof so the reader can catch the spirit of such a complex proof. In the course of proving the conjecture, the authors apply very difficult tools reviewed in the book. They give a good survey on Ricci flows with surgery on 3-dimensional manifolds and they discuss in details the properties of the Hausdorff–Gromov distance and the theory of Alexandrov spaces." - János Kincses, Acta Sci. Math.Table of Contents Introduction Geometric and analytic results for Ricci flow with surgery Ricci flow with surger Limits as t?? Local results valid for large time Proofs of the three propositions Locally volume collapsed 3-manifolds Introduction to part II The collapsing theorem Overview of the rest of the argument Basics of Gromov-Hausdorff convergence Basics of Alexandrov spaces 2-dimensional Alexandrov spaces 3-dimensional analogues The global result The equivariant case The equivariant case Bibliography Glossary of symbols Index

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Book SynopsisPresents an exposition of fixed point theory. This work focuses on the problem of whether a closed interval, square, disk, or sphere has the fixed point property. It aims to show how fixed point theory uses combinatorial ideas related to decomposition of figures into distinct parts called faces, which adjoin each other in a regular fashion.Table of ContentsContinuous mappings of a closed interval and a square First combinatorial lemma Second combinatorial lemma, or walks through the rooms in a house Sperner's lemma Continuous mappings, homeomorphisms, and the fixed point property Compactness Proof of Brouwer's Theorem for a closed interval, the intermediate value theorem, and applications Proof of Brouwer's Theorem for a square The iteration method Retraction Continuous mappings of a circle, homotopy, and degree of a mapping Second definition of the degree of a mapping Continuous mappings of a sphere Lemma on equality of degrees.

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Book SynopsisSpecialized as it might be, continuum theory is one of the most intriguing areas in mathematics. However, despite being popular journal fare, few books have thoroughly explored this interesting aspect of topology. In Topics on Continua, Sergio Macías, one of the field's leading scholars, presents four of his favorite continuum topics: inverse limits, Jones's set function T, homogenous continua, and n-fold hyperspaces, and in doing so, presents the most complete set of theorems and proofs ever contained in a single topology volume. Many of the results presented have previously appeared only in research papers, and some appear here for the first time. After building the requisite background and exploring the inverse limits of continua, the discussions focus on Professor Jones''s set function T and continua for which T is continuous. An introduction to topological groups and group actions lead to a proof of Effros''s Theorem, followed by a presentTable of ContentsPreliminaries, including an introduction to Product Topology. Inverse Limits and Related Topics. Jones Set Function T. A Theorem of E.G. Effros. Decomposition Theorems. n-Fold Hyperspaces. Questions.

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Book SynopsisThis book explains how to use simple 2-dimensional pictures to understand the geometry and topology of 4-dimensional spaces. These spaces are of relevance in Hamiltonian dynamics, in algebraic geometry, and in mathematical string theory. It is suitable for graduate students and researchers in geometry and topology.Trade Review'Lagrangian torus fibrations are an interesting source of examples in symplectic geometry, since their symplectic features are encoded by the geometry of certain half-dimensional base diagrams. Enriched by many pictures and exercises with solutions, this book provides an accessible and well-written introduction to this topic, which is of interest to a broad audience through its connections with integrable systems and algebraic geometry. This work will be appreciated by students and experts alike, since it fills a crucial gap in the literature by giving an excellent discussion of almost toric fibrations, which have attracted a lot of attention in recent years.' Felix Schlenk, Université de Neuchatel'This is a lucid and engaging introduction to the fascinating world of (almost) toric geometry, in which one can understand the properties of Lagrangian and symplectic submanifolds in four dimensions simply by drawing suitable two-dimensional diagrams. The book has many illustrations and intricate examples.' Dusa McDuff, Barnard College, Columbia UniversityTable of Contents1. The Arnold–Liouville theorem; 2. Lagrangian fibrations; 3. Global action-angle coordinates and torus actions; 4. Symplectic reduction; 5. Visible Lagrangian submanifolds; 6. Focus-focus singularities; 7. Examples of focus-focus systems; 8. Almost toric manifolds; 9. Surgery; 10. Elliptic and cusp singularities; A. Symplectic linear algebra; B. Lie derivatives; C. Complex projective spaces; D. Cotangent bundles; E. Moser's argument; F. Toric varieties revisited; G. Visible contact hypersurfaces and Reeb flows; H. Tropical Lagrangian submanifolds; I. Markov triples; J. Open problems; References; Index.

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Book SynopsisCarl Friedrich Gauss, the foremost of mathematicians, was a land surveyor. Measuring and calculating geodetic networks on the curved Earth was the inspiration for some of his greatest mathematical discoveries. This is just one example of how mathematics and geodesy, the science and art of measuring and mapping our world, have evolved together throughout history.This text is for students and professionals in geodesy, land surveying, and geospatial science who need to understand the mathematics of describing the Earth and capturing her in maps and geospatial data: the discipline known as mathematical geodesy. Map of the World: An Introduction to Mathematical Geodesy aims to provide an accessible introduction to this area, presenting and developing the mathematics relating to maps, mapping, and the production of geospatial data. Described are the theory and its fundamental concepts, its application for processing, analyzing, transforming, and projecting geospatial data, and how these are used in producing charts and atlases. Also touched upon are the multitude of cross-overs into other sciences sharing in the adventure of discovering what our world really looks like.FEATURESâ Written in a fluid and accessible style, replete with exercises; adaptable for courses on different levels.â Suitable for students and professionals in the mapping sciences, but also for lovers of maps and map making.Trade Review"Map of the World: An Introduction to Mathematical Geodesy is organized, written and presented in an impressively accessible style that is replete with exercises -- making it highly adaptable textbook for curriculum courses on different levels. Especially and unreservedly recommended for students and professionals in the mapping sciences, Map of the World will prove to be an ideal and instructive source for non-specialist readers with an interest in maps and map making. While a critically important addition to college and university library collections, it should be noted for personal reading lists that Maps of the World is also available in a digital book format."—Midwest Book Review"This is a textbook covering mathematics applied to geodesy: the measuring and mapping of our ellipsoid spheroid earth that includes an overview of earth measurement and mapping back to remote times. The mathematics of describing the Earth through maps and geospatial data is covered from underpinnings to application. [. . .] This textbook, including some exercises (without solutions), is aimed at students and practitioners in geodesy, land surveying, and geospatial science. It is easy to see this as a reference work. [. . .] this is a concise review of the theory and development of coordinate reference systems."—Tom Schulte, MAA Reviews ". . .(T)his text, by a geodesist (Vermeer) and a mathematician (Rasila), focuses primarily on the mathematics enabling map projections, coordinate systems, and transformation of three-dimensional coordinate representations, ranging from Euclidean to Reimannian geometries. Although the geometry is beyond what most geography students would need to address, the detailed mathematics offers a bridge for integration of collaborative teaching appropriate for upper-level mathematics and physics students, with applications to both cartography and geophysics. Each chapter concludes with exercises that provide an opportunity for learning the explicit mathematics behind the calculation presented. Interesting historical anecdotes about mathematicians and the evolution of geodesy are also included throughout. Students and readers of mathematics and geophysics as well as scientists working in the interdisciplinary area of geodesy will appreciate this book."– Choice Review, C. A. Badurek, SUNY Cortland"Map of the World: An Introduction to Mathematical Geodesy is organized, written and presented in an impressively accessible style that is replete with exercises -- making it highly adaptable textbook for curriculum courses on different levels. Especially and unreservedly recommended for students and professionals in the mapping sciences, Map of the World will prove to be an ideal and instructive source for non-specialist readers with an interest in maps and map making. While a critically important addition to college and university library collections, it should be noted for personal reading lists that Maps of the World is also available in a digital book format."—Midwest Book Review"This is a textbook covering mathematics applied to geodesy: the measuring and mapping of our ellipsoid spheroid earth that includes an overview of earth measurement and mapping back to remote times. The mathematics of describing the Earth through maps and geospatial data is covered from underpinnings to application. [. . .] This textbook, including some exercises (without solutions), is aimed at students and practitioners in geodesy, land surveying, and geospatial science. It is easy to see this as a reference work. [. . .] this is a concise review of the theory and development of coordinate reference systems."—Tom Schulte, MAA Reviews Table of Contents1. A Brief History of Mapping. 2. Popular Conformal Map Projections. 3. The Complex Plane and Conformal Mappings. 4. Complex Analysis. 5. Conformal Mappings. 6. Transversal Mercator Projections. 7. Sperical Trigonometry. 8. The Geometry of the Ellipsoid of Revolution. 9. Three-dimensional Co-ordinates and Transformations. 10. Co-ordinate Reference Systems. 11. Co-ordinates of Heaven and Earth. 12. The Orbital Motion of Satellites. 13. The Surface Theory of Gauss. 14. Riemann Surfaces and Charts. 15. Map Projections in the Light of Surface Theory. 16. Appendices

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Book SynopsisDifferential Geometry and Its Visualization is suitable for graduate level courses in differential geometry, serving both students and teachers. It can also be used as a supplementary reference for research in mathematics and the natural and engineering sciences.Differential geometry is the study of geometric objects and their properties using the methods of mathematical analysis. The classical theory of curves and surfaces in three-dimensional Euclidean space is presented in the first three chapters. The abstract and modern topics of tensor algebra, Riemannian spaces and tensor analysis are studied in the last two chapters. A great number of illustrating examples, visualizations and genuine figures created by the authors' own software are included to support the understanding of the presented concepts and results, and to develop an adequate perception of the shapes of geometric objects, their properties and the relations between them.FeatuTable of Contents1. Curves in Three–dimensional Euclidean Space. 1.1. Points and Vectors. 1.2. Vector–valued Functions of a Real Variable. 1.3. The General Concept of Curves. 1.4. Some Examples of Planar Curves. 1.5. The Arc Length of a Curve. 1.6. The Vectors of the Trihedron of a Curve. 1.7. Frenet’s Formulae. 1.8. The Geometric Significance of Curvature and Torsion. 1.9. Osculating Circles and Spheres. 1.10. Involutes and Evolutes. 1.11. The Fundamental Theorem of Curves. 1.12. Lines of Constant Slope. 1.13. Spherical Images of a Curve. 2. Surfaces in Three–dimensional Euclidean Space. 2.1. Surfaces and Curves on Surfaces. 2.2. The Tangent Planes and Normal Vectors of a Surface. 2.3. The Arc Length, Angles and Gauss’s First Fundamental Coefficients. 2.4. the Curvature of Curves on Surfaces, Geodesic and Normal Curvature. 2.5. The Normal, Principal, Gaussian and Mean Curvature. 2.6. The Shape of a Surface in the Neighbourhood of a Point. 2.7. Dupin’s Indicatrix. 2.8. Lines of Curvature and Asymptotic Lines. 2.9. Triple Orthogonal Systems. 2.10. the Weingarten Equations. 3. The Intrinsic Geometry of Surfaces. 3.1. the Christoffel Symbols. 3.2. Geodesic Lines. 3.3. Geodesic Lines on Surfaces with Orthogonal Parameters. 3.4. Geodesic Lines on Surfaces of Revolution. 3.5. the Minimum Property of Geodesic Lines. 3.6. Orthogonal and Geodesic Parameters. 3.7. Levi–civitá Parallelism. 3.8. Theorema Egregium. 3.9. Maps Between Surfaces. 3.10. the Gauss–bonnet Theorem. 3.11. Minimal Surfaces. 4. Tensor Algebra and Riemannian Geometry. 4.1. Differentiable Manifolds. 4.2. Transformation of Bases. 4.3. Linear Functionals and Dual Spaces. 4.4. Tensors of Second Order. 4.5. Symmetric Bilinear Forms and Inner Products. 4.6. Tensors of Arbitary Order. 4.7. Symmetric and Anti–symmetric Tensors. 4.8. Riemann Spaces. 4.9. the Christoffel Symbols. 5. Tensor Analysis. 5.1. Covariant Differentiation. 5.2. the Covariant Derivative of an (R, S)–tensor. 5.3. the Interchange of Order for Covariant Differentiation and Ricci’s Identity. 5.4. Bianchi’s Identities for the Covariant Derivative of the Tensors of Curvature. 5.5. Beltrami’s Differentiators. 5.6. a Geometric Meaning of the Covariant Differentiation, the Levi–civitá Parallelism. 5.7. The Fundamental Theorem for Surfaces. 5.8. A Geometric Meaning of the Riemann Tensor of Curvature. 5.9. Spaces With Vanishing Tensor of Curvature. 5.10. An Extension of Frenet’s Formulae. 5.11. Riemann Normal Coordinates and the Curvature of Spaces.

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Book SynopsisProviding a succinct yet comprehensive treatment of the essentials of modern differential geometry and topology, this book's clear prose and informal style make it accessible to advanced undergraduate and graduate students in mathematics and the physical sciences. It features over 250 detailed exercises and discusses a variety of applications.Table of ContentsPreface; 1. Linear algebra; 2. Multilinear algebra; 3. Differentiation on manifolds; 4. Homotopy and de Rham cohomology; 5. Elementary homology theory; 6. Integration on manifolds; 7. Vector bundles; 8. Geometric manifolds; 9. The degree of a smooth map; Appendixes; References; Index.

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Book SynopsisMonoidal Topology describes an active research area that, after various past proposals on how to axiomatize 'spaces' in terms of convergence, began to emerge at the beginning of the millennium. It combines Barr's relational presentation of topological spaces in terms of ultrafilter convergence with Lawvere's interpretation of metric spaces as small categories enriched over the extended real half-line. Hence, equipped with a quantale V (replacing the reals) and a monad T (replacing the ultrafilter monad) laxly extended from set maps to V-valued relations, the book develops a categorical theory of (T,V)-algebras that is inspired simultaneously by its metric and topological roots. The book highlights in particular the distinguished role of equationally defined structures within the given lax-algebraic context and presents numerous new results ranging from topology and approach theory to domain theory. All the necessary pre-requisites in order and category theory are presented in the book.Table of ContentsPreface; 1. Introduction Robert Lowen and Walter Tholen; 2. Monoidal structures Gavin J. Seal and Walter Tholen; 3. Lax algebras Dirk Hofmann, Gavin J. Seal and Walter Tholen; 4. Kleisli monoids Dirk Hofmann, Robert Lowen, Rory Lucyshyn-Wright and Gavin J. Seal; 5. Lax algebras as spaces Maria Manuel Clementino, Eva Colebunders and Walter Tholen; Bibliography; Tables; Index.

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Book SynopsisThis detailed yet accessible text introduces the advanced mathematical methods at the core of theoretical physics. Based on a course for senior undergraduate students of physics, it is written in a clear, pedagogical style and would also be valuable to students in other areas of science and engineering.Trade Review'The recent explosive development of topological quantum matter requires a deep systematic understanding of modern mathematics. Quantum many-body entanglement in topological quantum matter is a new phenomenon that requires new mathematical language to describe. This is a rare book that provides systematic and in-depth coverage of some of the most important mathematical concepts, such as groups, geometry, topology and algebra, among others. Many abstract mathematical notions are explained in an easy, explicit fashion. This is an in-depth, friendly book on modern mathematics. Very timely and highly recommended.' Xiao-Gang Wen, Massachusetts Institute of TechnologyTable of Contents1. Introduction; 2. Group theory; 3. Representation theory of groups; 4. Differentiable manifolds; 5. Riemannian geometry; 6. Semisimple Lie algebras and their unitary representations; Appendix A; References; Index.

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Book SynopsisThis text is an introduction to some of the mathematical wonders of Maxwell''s equations. These equations led to the prediction of radio waves, the realization that light is a type of electromagnetic wave, and the discovery of the special theory of relativity. In fact, almost all current descriptions of the fundamental laws of the universe can be viewed as deep generalizations of Maxwell''s equations. Even more surprising is that these equations and their generalizations have led to some of the most important mathematical discoveries of the past thirty years. It seems that the mathematics behind Maxwell''s equations is endless. The goal of this book is to explain to mathematicians the underlying physics behind electricity and magnetism and to show their connections to mathematics. Starting with Maxwell''s equations, the reader is led to such topics as the special theory of relativity, differential forms, quantum mechanics, manifolds, tangent bundles, connections, and curvature.Table of Contents1. A brief history; 2. Maxwell's equations; 3. Electromagnetic waves; 4. Special relativity; 5. Mechanics and Maxwell's equations; 6. Mechanics, Lagrangians, and the calculus of variations; 7. Potentials; 8. Lagrangians and electromagnetic forces; 9. Differential forms; 10. The Hodge * operator; 11. The electromagnetic two-form; 12. Some mathematics needed for quantum mechanics; 13. Some quantum mechanical thinking; 14. Quantum mechanics of harmonic oscillators; 15. Quantizing Maxwell's equations; 16. Manifolds; 17. Vector bundles; 18. Connections; 19. Curvature; 20. Maxwell via connections and curvature; 21. The Lagrangian machine, Yang–Mills, and other forces.

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Book SynopsisQuantum information theory is a branch of science at the frontier of physics, mathematics, and information science, and offers a variety of solutions that are impossible using classical theory. This book provides a detailed introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. The second edition contains new sections and entirely new chapters: the hot topic of multipartite entanglement; in-depth discussion of the discrete structures in finite dimensional Hilbert space, including unitary operator bases, mutually unbiased bases, symmetric informationally complete generalized measurements, discrete Wigner function, and unitary designs; the Gleason and KochenSpecker theorems; the proof of the Lieb conjecture; the measure concentration phenomenon; and the Hastings'' non-additivity theorem. This richly-illustrated book will be useful to a broad audience of graduates and researchers interTrade Review'True story: A few years ago my daughter took a break from her usual question, 'Dad, what is your favourite colour?' and asked instead, 'What is your favourite shape?' I was floored! 'What a wonderful question; my favourite shape is Hilbert space!' 'What does it look like?' she asked. My answer: 'I don't know! But every day when I go to work, that's what I think about.' What I was speaking of, of course, is the geometry of quantum-state space. It is as much a mystery today as it was those years ago, and maybe more so as we learn to focus on its most key and mysterious features. This book, the worn first-edition of which I've had on my shelf for 11 years, is the indispensable companion for anyone's journey into that exotic terrain. Beyond all else, I am thrilled about the inclusion of two new chapters in the new edition, one of which I believe goes to the very heart of the meaning of quantum theory.' Christopher A. Fuchs, University of Massachusetts, Boston'The quantum world is full of surprises as is the mathematical theory that describes it. Bengtsson and Życzkowski prove to be expert guides to the deep mathematical structure that underpins quantum information science. Key concepts such as multipartite entanglement and quantum contextuality are discussed with extraordinary clarity. A particular feature of this new edition is the treatment of SIC generalised measurements and the curious bridge they make between quantum physics and number theory.' Gerard J. Milburn, University of QueenslandPraise for the first edition: 'Geometry of Quantum States can be considered an indispensable item on a bookshelf of everyone interest in quantum information theory and its mathematical background.' Miłosz Michalski, editor of Open Systems and Information DynamicsPraise for the first edition: 'Bengtsson's and Zyczkowski's book is an artful presentation of the geometry that lies behind quantum theory … the authors collect, and artfully explain, many important results scattered throughout the literature on mathematical physics. The careful explication of statistical distinguishability metrics (Fubini-Study and Bures) is the best I have read.' Gerard Milburn, University of QueenslandTable of ContentsPreface; 1. Convexity, colours, and statistics; 2. Geometry of probability distributions; 3. Much ado about spheres; 4. Complex projective spaces; 5. Outline of quantum mechanics; 6. Coherent states and group actions; 7. The stellar representation; 8. The space of density matrices; 9. Purification of mixed quantum states; 10. Quantum operations; 11. Duality: maps versus states; 12. Discrete structures in Hilbert space; 13. Density matrices and entropies; 14. Distinguishability measures; 15. Monotone metrics and measures; 16. Quantum entanglement; 17. Multipartite entanglement; Appendix 1. Basic notions of differential geometry; Appendix 2. Basic notions of group theory; Appendix 3. Geometry – do it yourself; Appendix 4. Hints and answers to the exercises; Bibliography; Index.

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Book SynopsisThe continued and dramatic rise in the size of data sets has meant that new methods are required to model and analyze them. This timely account introduces topological data analysis (TDA), a method for modeling data by geometric objects, namely graphs and their higher-dimensional versions: simplicial complexes. The authors outline the necessary background material on topology and data philosophy for newcomers, while more complex concepts are highlighted for advanced learners. The book covers all the main TDA techniques, including persistent homology, cohomology, and Mapper. The final section focuses on the diverse applications of TDA, examining a number of case studies drawn from monitoring the progression of infectious diseases to the study of motion capture data. Mathematicians moving into data science, as well as data scientists or computer scientists seeking to understand this new area, will appreciate this self-contained resource which explains the underlying technology and how it can be used.Table of ContentsPart I. Background: 1. Introduction; 2. Data; Part II. Theory: 3. Topology; 4. Shape of data; 5. Structures on spaces of barcodes; Part III. Practice: 6. Case studies; References; Index.

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Book SynopsisThe Shape of Space, Third Edition maintains the standard of excellence set by the previous editions. This lighthearted textbook covers the basic geometry and topology of two- and three-dimensional spacesâstretching studentsâ minds as they learn to visualize new possibilities for the shape of our universe.Written by a master expositor, leading researcher in the field, and MacArthur Fellow, its informal exposition and engaging exercises appeal to an exceptionally broad audience, from liberal arts students to math undergraduate and graduate students looking for a clear intuitive understanding to supplement more formal texts, and even to laypeople seeking an entertaining self-study book to expand their understanding of space.Features of the Third Edition: Full-color figures throughout Picture proofs have replaced algebraic proofs Simpler handles-and-crosscaps approach to surfaces Updated discussiTable of ContentsPart I Surfaces and Three-Manifolds Flatland Gluing Vocabulary Orientability Classification of Surfaces Products Flat Manifolds Orientability vs. Two-Sidedness Part II Geometries on Surfaces The Sphere The Hyperbolic Plane Geometries on Surfaces Gauss-Bonnet Formula and Euler Number Part III Geometries on Three-Manifolds Four-Dimensional Space The Hypersphere Hyperbolic Space Geometries on Three-Manifolds I Bundles Geometries on Three-Manifolds II Part IV The Universe The Universe The History of Space Appendix A: Answers Appendix B: Bibliography Appendix C: Conway’s ZIP Proof

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Book SynopsisThis book is about the interplay between algebraic topology and the theory of infinite discrete groups. It is a hugely important contribution to the field of topological and geometric group theory, and is bound to become a standard reference in the field.Trade ReviewFrom the reviews:"The author of this book has done a great service to the geometric group theory community by writing a very useful and well-written book on many topics in geometric group theory that every neophyte and researcher in the field should know. … This book is suitable as a textbook for a graduate course, with many good examples and exercises. The reviewer highly recommends this book as a basic reference book for topological methods in group theory." (John G. Ratcliffe, Mathematical Reviews, Issue 2008 j)"This is an interesting book on the interplay between algebraic topology and the theory of infinite discrete groups written for graduate students and group theorists who need to learn more in geometric and homological group theory. … It is a beautiful text in algebraic topology, with modern topics and which points the reader towards new research directions." (Corina Mohorianu, Zentralblatt MATH, Vol. 1142, 2008)“This book is an invaluable resource for anyone wanting a deep understanding of topics related to the ends of groups. … there is a good deal of material in this book that does not appear anywhere else in the literature. … Geoghegan’s book provides a well-presented, concrete development of geometric group theory focused on a topological approach.” (John Meier, Bulletin of the American Mathematical Society, July, 2012)Table of ContentsAlgebraic Topology for Group Theory.- CW Complexes and Homotopy.- Cellular Homology.- Fundamental Group and Tietze Transformation.- Some Techniques in Homotopy Theory.- Elementary Geometric Topology.- Finiteness Properties of Groups.- The Borel Construction and Bass-Serre Theory.- Topological Finiteness Properties and Dimension of Groups.- Homological Finiteness Properties of Groups.- Finiteness Properties of Some Important Groups.- Locally Finite Algebraic Topology for Group Theory.- Locally Finite CW Complexes and Proper Homotopy.- Locally Finite Homology.- Cohomology of CW Complexes.- Topics in the Cohomology of Infinite Groups.- Cohomology of Groups and Ends of Covering Spaces.- Filtered Ends of Pairs of Groups.- Poincaré Duality in Manifolds and Groups.- Homotopical Group Theory.- The Fundamental Group At Infinity.- Higher homotopy theory of groups.- Three Essays.- Three Essays.

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Book SynopsisIndeed, if the order in which the results presented here does not strictly correspond to that in which they were discovered, it nevertheless does correspond to an order in which they might have been discovered had those of us who were working in the area been a little more perspicacious.Table of ContentsI Introductory Notions.- 1. The Fundamental Problems: Extension, Homotopy, and Classification.- 2. Standard Notations and Conventions.- 3. Maps of the n-sphere into Itself.- 4. Compactly Generated Spaces.- 5. NDR-pairs.- 6. Filtered Spaces.- 7. Fibrations.- II CW-complexes.- 1. Construction of CW-complexes.- 2. Homology Theory of CW-complexes.- 3. Compression Theorems.- 4. Cellular Maps.- 5. Local Calculations.- 6. Regular Cell Complexes.- 7. Products and the Cohomology Ring.- III Generalities on Homotopy Classes of Mappings.- 1. Homotopy and the Fundamental Group.- 2. Spaces with Base Points.- 3. Groups of Homotopy Classes.- 4. H-spaces.- 5. H’-spaces.- 6. Exact Sequences of Mapping Functors.- 7. Homology Properties of H-spaces and H’-spaces.- 8. Hopf Algebras.- IV Homotopy Groups.- 1. Relative Homotopy Groups.- 2. The Homotopy Sequence.- 3. The Operations of the Fundamental Group on the Homotopy Sequence.- 4. The Hurewicz Map.- 5. The Eilenberg and Blakers Homology Groups.- 6. The Homotopy Addition Theorem.- 7. The Hurewicz Theorems.- 8. Homotopy Relations in Fibre Spaces.- 9. Fibrations in Which the Base or Fibre is a Sphere.- 10. Elementary Homotopy Theory of Lie Groups and Their Coset Spaces.- V Homotopy Theory of CW-complexes.- 1. The Effect on the Homotopy Groups of a Cellular Extension.- 2. Spaces with Prescribed Homotopy Groups.- 3. Weak Homotopy Equivalence and CW-approximation.- 4. Aspherical Spaces.- 5. Obstruction Theory.- 6. Homotopy Extension and Classification Theorems.- 7. Eilenberg-Mac Lane Spaces.- 8. Cohomology Operations.- VI Homology with Local Coefficients.- 1. Bundles of Groups.- 2. Homology with Local Coefficients.- 3. Computations and Examples.- 4. Local Coefficients in CW-complexes.- 5. Obstruction Theory in Fibre Spaces.- 6. The Primary Obstruction to a Lifting.- 7. Characteristic Classes of Vector Bundles.- VII Homology of Fibre Spaces: Elementary Theory.- 1. Fibrations over a Suspension.- 2. The James Reduced Products.- 3. Further Properties of the Wang Sequence.- 4. Homology of the Classical Groups.- 5. Fibrations Having a Sphere as Fibre.- 6. The Homology Sequence of a Fibration.- 7. The Blakers-Massey Homotopy Excision Theorem.- VIII The Homology Suspension.- 1. The Homology Suspension.- 2. Proof of the Suspension Theorem.- 3. Applications.- 4. Cohomology Operations.- 5. Stable Operations.- 6. The mod 2 Steenrod Algebra.- 7. The Cartan Product Formula.- 8. Some Relations among the Steenrod Squares.- The Action of the Steenrod Algebra on the Cohomology of Some Compact Lie Groups.- IX Postnikov Systems.- 1. Connective Fibrations.- 2. The Postnikov Invariants of a Space.- 3. Amplifying a Space by a Cohomology Class.- 4. Reconstruction of a Space from its Postnikov System.- 5. Some Examples.- 6. Relative Postnikov Systems.- 7. Postnikov Systems and Obstruction Theory.- X On Mappings into Group-like Spaces.- 1. The Category of a Space.- 2. H0-spaces.- 3. Nilpotency of [X, G].- 4. The Case X = X1 × · · · × Xk.- 5. The Samelson Product.- 6. Commutators and Homology.- 7. The Whitehead Product.- 8. Operations in Homotopy Groups.- XI Homotopy Operations.- 1. Homotopy Operations.- 2. The Hopf Invariant.- 3. The Functional Cup Product.- 4. The Hopf Construction.- 5. Geometrical Interpretation of the Hopf Invariant.- 6. The Hilton-Milnor Theorem.- 7. Proof of the Hilton-Milnor Theorem.- 8. The Hopf-Hilton Invariants.- XII Stable Homotopy and Homology.- 1. Homotopy Properties of the James Imbedding.- 2. Suspension and Whitehead Products.- 3. The Suspension Category.- 4. Group Extensions and Homology.- 5. Stable Homotopy as a Homology Theory.- 6. Comparison with the Eilenberg-Steenrod Axioms.- 7. Cohomology Theories.- XIII Homology of Fibre Spaces.- 1. The Homology of a Filtered Space.- 2. Exact Couples.- 3. The Exact Couples of a Filtered Space.- 4. The Spectral Sequence of a Fibration.- 5. Proofs of Theorems (4.7) and 4.8).- 6. The Atiyah-Hirzebruch Spectral Sequence.- 7. The Leray-Serre Spectral Sequence.- 8. Multiplicative Properties of the Leray-Serre Spectral Sequence.- 9. Further Applications of the Leray-Serre Spectral Sequence.- Appendix A.- Compact Lie Groups.- 1. Subgroups, Coset Spaces, Maximal Tori.- 2. Classifying Spaces.- 3. The Spinor Groups.- 6. The Exceptional Jordan Algebra I.- Appendix B.- Additive Relations.- 1. Direct Sums and Products.- 2. Additive Relations.

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Book Synopsis0 Introduction and Foundations.- 0.1 The Fundamental Concepts and Problems of Topology.- 0.2 Simplicial Complexes.- 0.3 The Jordan Curve Theorem.- 0.4 Algorithms.- 0.5 Combinatorial Group Theory.- 1 Complex Analysis and Surface Topology.- 1.1 Riemann Surfaces.- 1.2 Nonorientable Surfaces.- 1.3 The Classification Theorem for Surfaces.- 1.4 Covering Surfaces.- 2 Graphs and Free Groups.- 2.1 Realization of Free Groups by Graphs.- 2.2 Realization of Subgroups.- 3 Foundations for the Fundamental Group.- 3.1 The Fundamental Group.- 3.2 The Fundamental Group of the Circle.- 3.3 Deformation Retracts.- 3.4 The SeifertVan Kampen Theorem.- 3.5 Direct Products.- 4 Fundamental Groups of Complexes.- 4.1 Poincaré's Method for Computing Presentations.- 4.2 Examples.- 4.3 Surface Complexes and Subgroup Theorems.- 5 Homology Theory and Abelianization.- 5.1 Homology Theory.- 5.2 The Structure Theorem for Finitely Generated Abelian Groups.- 5.3 Abelianization.- 6 Curves on Surfaces.- 6.1 Dehn's Algorithm.Table of Contents0 Introduction and Foundations.- 0.1 The Fundamental Concepts and Problems of Topology.- 0.2 Simplicial Complexes.- 0.3 The Jordan Curve Theorem.- 0.4 Algorithms.- 0.5 Combinatorial Group Theory.- 1 Complex Analysis and Surface Topology.- 1.1 Riemann Surfaces.- 1.2 Nonorientable Surfaces.- 1.3 The Classification Theorem for Surfaces.- 1.4 Covering Surfaces.- 2 Graphs and Free Groups.- 2.1 Realization of Free Groups by Graphs.- 2.2 Realization of Subgroups.- 3 Foundations for the Fundamental Group.- 3.1 The Fundamental Group.- 3.2 The Fundamental Group of the Circle.- 3.3 Deformation Retracts.- 3.4 The Seifert—Van Kampen Theorem.- 3.5 Direct Products.- 4 Fundamental Groups of Complexes.- 4.1 Poincaré’s Method for Computing Presentations.- 4.2 Examples.- 4.3 Surface Complexes and Subgroup Theorems.- 5 Homology Theory and Abelianization.- 5.1 Homology Theory.- 5.2 The Structure Theorem for Finitely Generated Abelian Groups.- 5.3 Abelianization.- 6 Curves on Surfaces.- 6.1 Dehn’s Algorithm.- 6.2 Simple Curves on Surfaces.- 6.3 Simplification of Simple Curves by Homeomorphisms.- 6.4 The Mapping Class Group of the Torus.- 7 Knots and Braids.- 7.1 Dehn and Schreier’s Analysis of the Torus Knot Groups.- 7.2 Cyclic Coverings.- 7.3 Braids.- 8 Three-Dimensional Manifolds.- 8.1 Open Problems in Three-Dimensional Topology.- 8.2 Polyhedral Schemata.- 8.3 Heegaard Splittings.- 8.4 Surgery.- 8.5 Branched Coverings.- 9 Unsolvable Problems.- 9.1 Computation.- 9.2 HNN Extensions.- 9.3 Unsolvable Problems in Group Theory.- 9.4 The Homeomorphism Problem.- Bibliography and Chronology.

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Book Synopsis Graphs on Surfaces: Dualities, Polynomials, and Knots also provides a self-contained introduction to graphs on surfaces, generalized duals, topological graph polynomials, and knot polynomials that is accessible both to graph theorists and to knot theorists.Trade ReviewFrom the reviews:“Here, the venerable knot-theoretic and graph-theoretic themes find a host of unifying common generalizations. Undergraduates will appreciate the patient and visual development of the foundations, particularly the dualities (paired representations of a single structure). Summing Up: Recommended. Upper-division undergraduates through researchers/faculty.” (D. V. Feldman, Choice, Vol. 51 (7), March, 2014)“This monograph is aimed at researchers both in graph theory and in knot theory. It should be accessible to a graduate student with a grounding in both subjects. There are (colour) diagrams throughout. … The monograph gives a unified treatment of various ideas that have been studied and used previously, generalising many of them in the process.” (Jessica Banks, zbMATH, Vol. 1283, 2014)“The authors have composed a very interesting and valuable work. … For properly prepared readers … the book under review is the occasion for all sorts of fun including the inner life of ribbon groups, Tait graphs, Penrose polynomials, Tutte polynomials, and of course Jones polynomials and HOMFLY polynomials. This is fascinating mathematics, presented in a clear and accessible way.” (Michael Berg, MAA Reviews, October, 2013)Table of Contents1. Embedded Graphs .- 2. Generalised Dualities .- 3. Twisted duality, cycle family graphs, and embedded graph equivalence .- 4. Interactions with Graph Polynomials .- 5. Applications to Knot Theory .- References .- Index .

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Book SynopsisBasic Objects and Notation.- Linear Algebra Essentials.- Advanced Calculus.- Differential Forms and Tensors.- Riemannian Geometry.- Contact Geometry.- Symplectic Geometry.- References.- Index.Trade ReviewFrom the book reviews:“This books presents an alternative route, aiming to provide the student with an introduction not only to Riemannian geometry, but also to contact and symplectic geometry. … the book is leavened with an excellent collection of illustrative examples, and a wealth of exercises on which students can hone their skills. Each chapter also includes a short guide to further reading on the topic with a helpful brief commentary on the suggestions.” (Robert J. Low, Mathematical Reviews, May, 2014)“This book is a distinctive and ambitious effort to bring modern notions of differential geometry to undergraduates. … Mclnerney’s writing is well constructed and very clear … . Summing Up: Recommended. Upper-division undergraduates and graduate students.” (S. J. Colley, Choice, Vol. 51 (8), April, 2014)“The author does make a considerable effort to keep things as accessible as possible, with fairly detailed explanations, extensive motivational discussions and homework problems … . this book provides a different way of looking at the subject of differential geometry, one that is more modern and sophisticated than is provided by many of the standard undergraduate texts and which will certainly do a good job of preparing the student for additional work in this area down the road.” (Mark Hunacek, MAA Reviews, January, 2014)“This text provides an early and broad view of geometry to mathematical students … . Altogether, this book is easy to read because there are plenty of figures, examples and exercises which make it intuitive and perfect for undergraduate students.” (Teresa Arias-Marco, zbMATH, Vol. 1283, 2014)Table of ContentsBasic Objects and Notation.- Linear Algebra Essentials.- Advanced Calculus.- Differential Forms and Tensors.- Riemannian Geometry.- Contact Geometry.- Symplectic Geometry.- References.- Index.

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Book SynopsisTwo central aspects of Cartan's approach to differential geometry are the theory of exterior differential systems (EDS) and the method of moving frames. This book presents thorough and modern treatments of both subjects, including their applications to both classic and contemporary problems in geometry.Trade Review[T]his book, like the first edition, is an excellent source for graduate students and professional mathematicians who want to learn about moving frames and G-structures in trying to understand differential geometry." - Thomas Garrity, Mathematical Reviews"All the material is carefully developed, many examples supporting the understanding. The reviewer warmly recommends this volume to mathematical university libraries." - Gabriel Eduard Vilcu, Zentralblatt MATHTable of Contents Moving frames and exterior differential systems Euclidean geometry Riemannian geometry Projective geometry I: Basic definitions and examples Cartan-Kahler I: Linear algebra and constant-coefficient homogeneous systems Cartan-Kahler II: The Cartan algorithm for linear Pfaffian systems Applications to PDE Cartan-Kahler III: The general case Geometric structures and connections Superposition for Darboux-integrable systems Conformal differential geometry Projective geometry II: Moving frames and subvarieties of projective space Linear algebra and representation theory Differential forms Complex structures and complex manifolds Initial value problems and the Cauchy-Kowalevski theorem Hints and answers to selected exercises Bibliography Index.

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Book SynopsisTrade ReviewFractals are beautiful and complex geometric objects. Their study, pioneered by Benoît Mandelbrot, is of interest in mathematics, physics and computer science. Their inherent structure, based on their self-similarity, makes the study of their geometry amenable to dynamical approaches. In this book, a theory along these lines is developed by Hillel Furstenberg, one of the foremost experts in ergodic theory, leading to deep results connecting fractal geometry, multiple recurrence, and Ramsey theory. In particular, the notions of fractal dimension and self-similarity are interpreted in terms of ergodic averages and periodicity of classical dynamics; moreover, the methods have deep implications in combinatorics. The exposition is well-structured and clearly written, suitable for graduate students as well as for young researchers with basic familiarity in analysis and probability theory." - Endre Szemerédi, Rényi Institute of Mathematics, BudapestTable of Contents Introduction to fractals Dimension Trees and fractals Invariant sets Probability trees Galleries Probability trees revisited Elements of ergodic theory Galleries of trees General remarks on Markov systems Markov operator $\mathcal{T}$ and measure preserving transformation $T$ Probability trees and galleries Ergodic theorem and the proof of the main theorem An application: The $k$-lane property Dimension and energy Dimension conservation Ergodic theorem for sequences of functions Dimension conservation for homogeneous fractals: The main steps in the proof Verifying the conditions of the ergodic theorem for sequences of functions Bibliography Index

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Book SynopsisGeometric group theory refers to the study of discrete groups using tools from topology, geometry, dynamics and analysis. The field is evolving very rapidly and this volume provides an introduction to and overview of various topics which have played critical roles in this evolution. The book contains lecture notes from courses given at the Park City Math Institute on Geometric Group Theory.Table of Contents CAT(0) cube complexes and groups by M. Sageev Geometric small cancellation by V. Guirardel Lectures on proper CAT(0) spaces and their isometry groups by P.-E. Caprace Lectures on quasi-isometric rigidity by M. Kapovich Geometry of outer space by M. Bestvina Some arithmetic groups that do not act on the circle by D. W. Morris Lectures on lattices and locally symmetric spaces by T. Gelander Lectures on marked length spectrum rigidity by A. Wilkinson Expander graphs, property t and approximate groups by E. Breuillard Cube complexes, subgroups of mapping class groups, and nilpotent genus by M. R. Bridson

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