Topology Books

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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Book SynopsisTrade ReviewThis book covers a lot of ground. But it does so in a clear and careful manner that would make a terrific read for the prepared undergraduate. It is a study in how an intuitive idea can transport one into some deep waters of mathematics, and that is an important story to tell." - John McCleary, Mathematical Reviews"People who teach university-level mathematics for a living often find themselves reading lots of books on the subject. But even for the book-lovers among us, after you've just read about ten linear algebra texts, all of which look like they were stamped from the same cookie cutter, the process can occasionally wear thin. It's very pleasant, then, to stumble across a book that is genuinely unique, that addresses a topic in a way not found elsewhere, and that teaches you something that you didn't know before. It's even nicer when the book in question does a really good job of it, as is the case with the book under review. ...Roe's writing style is succinct, but clear and quite elegant; I could practically hear a British accent as I read the book. This clarity of writing and the numerous appendices help make the book accessible." - MAA OnlineTable of Contents Prelude: Love, hate, and exponentials Paths and homotopies The winding number Topology of the plane Integrals and the winding number Vector fields and the rotation number The winding number in functional analysis Coverings and the fundamental group Coda: The Bott periodicity theorem Linear algebra Metric spaces Extension and approximation theorems Measure zero Calculus on normed spaces Hilbert space Groups and graphs Bibliography Index

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Book SynopsisProvides an elementary introduction to geometric topology and its applications to chemistry, molecular biology, and cosmology. It does not assume any mathematical or scientific background, sophistication, or even motivation to study mathematics. It is meant to be fun and engaging while drawing students in to learn about fundamental topological and geometric ideas.Trade Review[T]his is a wonderful introduction to geometry and topology and their applications to the sciences. The book contains a unique collection of topics that might entice young readers to continue their academic careers by learning more about the world of mathematics." - Claus Ernst, Zentralblatt MATHTable of Contents Universes: An introduction to the shape of the universe Visualizing four dimensions Geometry and topology of different universes Orientability Flat manifolds Connected sums of spaces Products of spaces Geometries of surfaces Knots: Introduction to knot theory Invariants of knots and links Knot polynomials Molecules: Mirror image symmetry from different viewpoints Techniques to prove topological chirality The topology and geometry of DNA The topology of proteins Index

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Book SynopsisThe curvature discussed in this paper is a far reaching generalization of the Riemannian sectional curvature. The authors give a unified definition of curvature which applies to a wide class of geometric structures whose geodesics arise from optimal control problems, including Riemannian, sub-Riemannian, Finsler and sub-Finsler spaces.Table of Contents Introduction Part 1. Statements of the results: General setting Flag and growth vector of an admissible curve Geodesic cost and its asymptotics Sub-Riemannian geometry Part 2. Technical tools and proofs: Jacobi curves Asymptotics of the Jacobi curve: equiregular case Sub-Laplacian and Jacobi curves Part 3. Appendix: Appendix A. Smoothness of value function (Theorem $2.19$) Appendix B. Convergence of approximating Hamiltonian systems (Proposition 5.15) Appendix C. Invariance of geodesic growth vector by dilations (Lemma $5.20$) Appendix D. Regularity of $C(t,s)$ for the Heisenberg group (Proposition $5.51$) Appendix E. Basics on curves in Grassmannians (Lemma $3.5$ and $6.5$) Appendix F. Normal conditions for the canonical frame Appendix G. Coordinate representation of flat, rank 1 Jacobi curves (Proposition $7.7$) Appendix H. A binomial identity (Lemma $7.8$) Appendix I. A geometrical interpretation of $\dot c_t$ Bibliography Index.

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Book SynopsisOffers an introduction to the method of moving frames as developed by Cartan, at a level suitable for beginning graduate students familiar with the geometry of curves and surfaces in Euclidean space. The main focus is on the use of this method to compute local geometric invariants for curves and surfaces in various 3-dimensional homogeneous spaces.Trade ReviewAn excellent and unique graduate level exposition of the differential geometry of curves, surfaces and higher-dimensional submanifolds of homogeneous spaces based on the powerful and elegant method of moving frames. The treatment is self-contained and illustrated through a large number of examples and exercises, augmented by Maple code to assist in both concrete calculations and plotting. Highly recommended." — Niky Kamran, McGill University"The method of moving frames has seen a tremendous explosion of research activity in recent years, expanding into many new areas of applications, from computer vision to the calculus of variations to geometric partial differential equations to geometric numerical integration schemes to classical invariant theory to integrable systems to infinite-dimensional Lie pseudo-groups and beyond. Cartan theory remains a touchstone in modern differential geometry, and Clelland's book provides a fine new introduction that includes both classic and contemporary geometric developments and is supplemented by Maple symbolic software routines that enable the reader to both tackle the exercises and delve further into this fascinating and important field of contemporary mathematics. Recommended for students and researchers wishing to expand their geometric horizons." — Peter Olver, University of MinnesotaTable of Contents Background material: Assorted notions from differential geometry Differential forms Curves and surfaces in homogeneous spaces via the method of moving frames: Homogeneous spaces Curves and surfaces in Euclidean space Curves and surfaces in Minkowski space Curves and surfaces in equi-affine space Curves and surfaces in projective space Applications of moving frames: Minimal surfaces in $\mathbb{E}^3$ and $\mathbb{A}^3$ Pseudospherical surfaces in Backlund's theorem Two classical theorems Beyond the flat case: Moving frames on Riemannian manifolds: Curves and surfaces in elliptic and hyperbolic spaces The nonhomogeneous case: Moving frames on Riemannian manifolds Bibliography Index.

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Book SynopsisDeals with the connections between topology and ordered groups. It begins with a self-contained introduction to orderable groups and from there explores the interactions between orderability and objects in low-dimensional topology. The book also addresses recent applications of orderability in the studies of codimension-one foliations and Heegaard-Floer homology.Trade ReviewThe book finds a good balance between being a resource for researchers and a graduate textbook." - Sebastian Wolfgang Hensel, Mathematical Reviews"The diligent and disciplined reader of this thin (<150 pp) book will be rewarded by a lot more than knowledge of ordered groups." - Lee P. Neuwirth, Zentralblatt MATH"Given the huge popularity enjoyed by low dimensional topology these days, and all for good reason, it should make a very positive impact. The book is easy to read and deals with very pretty mathematics." - Michael Berg, MAA ReviewsTable of Contents Orderable groups and their algebraic properties Hölder’s theorem, convex subgroups and dynamics Free groups, surface groups and covering spaces Knots Three-dimensional manifolds Foliations Left-orderings of the braid groups Groups of homeomorphisms Conradian left-orderings and local indicability Spaces of orderings Bibliography

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Book SynopsisTopological complexity is a numerical homotopy invariant, defined by Farber in the early twenty-first century as part of a topological approach to the motion planning problem in robotics. This volume contains survey articles and original research papers on topological complexity and its many generalizations and variants, to give a snapshot of contemporary research on this exciting topic.Table of Contents Survey Articles: A. Angel and H. Colman, Equivariant topological complexities J. Carrasquel, Rational methods applied to sectional category and topological complexity D. C. Cohen, Topological complexity of classical configuration spaces and related objects P. Pavesic, A topologist's view of kinematic maps and manipulation complexity Research Articles: D. M. Davis, On the cohomology classes of planar polygon spaces J.-P. Doeraene, M. El Haouari, and C. Ribeiro, Sectional category of a class of maps L. Fernandez Suarez and L. Vandembroucq, Q-topological complexity N. Fieldsteel, Topological complexity of graphic arrangements J. Gonzalez, M. Grant, and L. Vandembroucq, Hopf invariants, topological complexity, and LS-category of the cofiber of the diagonal map for two-cell complexes J. Gonzalez and B. Gutierrez, Topological complexity of collision-free multi-tasking motion planning on orientable surfaces M. Grant and D. Recio-Mitter, Topological complexity of subgroups of Artin's braid groups.

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Book SynopsisThe authors construct cornered Floer homology invariants of 3-manifolds with codimension-2 corners and prove that the bordered Floer homology of a 3-manifold with boundary, split into two pieces with corners, can be recovered as a tensor product of the cornered invariants of the pieces.Table of Contents Introduction Some abstract 2-algebra More 2-algebra: bending and smoothing Some homological algebra of 2-modules The algebras and algebra-modules The cornering module-2-modules The trimodules $\mathsf{T}_{DDD}$ and $\mathsf{T}_{DDA}$ Cornered 2-modules for cornered Heegaard diagrams Gradings Practical computations The nilCoxeter planar algebra Bibliography.

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Book SynopsisTable of Contents An example: A tale of two equivalence relations Basics: Cantor sets and orbit equivalence Bratteli diagrams: Generalizing the example The Bratteli-Vershik model: Generalizing the example The Bratteli-Vershik model: Completeness Etale equivalence relations: Unifying the examples The $D$ invariant The Effros-Handelman-Shen theorem The Bratteli-Elliott-Krieger theorem Strong orbit equivalence The $D_m$ invariant The absorption theorem The classification of AF-equivalence relations The classification of $\mathbb{Z}$-actions Examples Bibliography Index of terminology Index of notation

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Book SynopsisContains the proceedings of the CRM Workshops on Probabilistic Methods in Spectral Geometry and PDE, held in 2016 at the Centre de Recherches Mathematiques. This volume covers developments in active research areas at the interface of Probability, Semiclassical Analysis, Mathematical Physics, Theory of Automorphic Forms and Graph Theory.Table of Contents L. Chen and N. Shu, A geometric treatment of log-correlated Gaussian free fields S. Eswarathasan, Tangent nodal sets for random spherical harmonics J. Friedman, Formal zeta function expansions and the frequency of Ramanujan graphs D. Jakobson, T. Langsetmo, I. Rivin, and L. Turner, Rank and Bollobas-Riordan polynomials: Coefficient measures and zeros V. Konakov, S. Menozzi, and S. Molchanov, The Brownian motion on $\mathrm{Aff}(\mathbb{R})$ and quasi-local theorems N. Laaksonen, Quantum limits of Eisenstein series in $\mathbb{H}^3$ F. Macia and G. Riviere, Observability amd quantum limits for the Schrodinger equation on $\mathbb{S}^d$ M. Rossi, Random nodal lengths and Wiener chaos L. Silberman and A. Venkatesh, Entropy bounds and quantum unique ergodity for Hecke eigenfunctions on division algebras.

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Book SynopsisThe aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in $\mathbb{R}^n$ for any $n\ge 3$.

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Book SynopsisThe authors develop a theory of $THH$ and $TC$ of Waldhausen categories and prove the analogues of Waldhausen's theorems for $K$-theory. They resolve the longstanding confusion about localization sequences in $THH$ and $TC$, and establish a specialized devissage theorem.

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Book SynopsisThe authors develop a degree theory for compact immersed hypersurfaces of prescribed $K$-curvature immersed in a compact, orientable Riemannian manifold, where $K$ is any elliptic curvature function. They apply this theory to count the (algebraic) number of immersed hyperspheres in various cases.

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Book SynopsisTable of Contents Basic definitions and examples McMullen's decomposition theorem Valuations on the line McMullen's description of $(n-1)$-homogeneous valuations The Klain-Schneider characterization of simple valuations Digression on the theory of generalized functions on manifolds The Goodey-Weil imbedding Digression on vector bundles The irreducibility theorem Further developments Bibliography

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Book SynopsisContains the proceedings of the AMS Special Session on Topology of Biopolymers, held April 2018, at Northeastern University. The papers cover recent results on the topology and geometry of DNA and protein knotting using techniques from knot theory, spatial graph theory, differential geometry, molecular simulations, and laboratory experimentation.Table of Contents The topology and geometry of DNA: J. M. Fogg and L. Zechiedrich, Beyond the static DNA model of Watson and Crick P. Liu, R. Polischuk, Y. Diao, and J. Arsuaga, Characterizing the topology of kinetoplast DNA using random knotting A. Kasman and B. LeMesurier, Did sequence dependent geometry influence the evolution of the genetic code? T. Deguchi and E. Uehara, Topological sum rules in the knotting probabilities of DNA D. Buck and D. O'Donnol, Knotting of replication intermediates is narrowly restricted A. H. Moore and M. Vazquez, Recent advances on the non-coherent band surgery model for site-specific recombination The topology and geometry of proteins: S. E. Jackson, Why are there knots in proteins? A. Nunes and P. F. N. Faisca, Knotted proteins: Tie etiquette in structural biology D. Goundaroulis, J. Dorier, and A. Stasiak, Knotoids and protein structure K. C. Millett, Topological linking and entanglement in proteins E. Panagiotou and K. W. Plaxco, A topological study of protein folding kinetics.

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Book SynopsisConstructs a Goodwillie tower of categories which interpolates between the category of pointed spaces and the category of spectra. This tower of categories refines the Goodwillie tower of the identity functor in a precise sense.

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Book SynopsisBrings together three approaches to constructing the “virtual” fundamental cycle for the moduli space of pseudo-holomorphic curves. All approaches are based on the idea of local Kuranishi charts for the moduli space. The book offers a comprehensive understanding of the details of these constructions and the assumptions under which they can be made.

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Book SynopsisContains the proceedings of the ICM 2018 satellite school and workshop $K$-theory conference in Argentina. The volume showcases current developments in $K$-theory and related areas, including motives, homological algebra, index theory, operator algebras, and their applications and connections.Table of Contents P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan, Comparison of spaces associated to DGLA via higher holonomy U. Bunke, A. Engel, D. Kasprowski, and C. Winges, Equivariant coarse homotopy theory and coarse algebraic K-homology A. Buss, S. Echterhoff, and R. Willett, Injectivity, crossed products, and amenable group actions L. Hesselholt and T. Nikolaus, Algebraic $K$-theory of planar cuspidal curves D. Juan-Pineda, On $NK_0$ of the group of quaternions M. Karoubi and C. Weibel, The Witt group of real surfaces A. Krishna and H. P. Sarwar, Negative $K$-theory and Chow group of monoid algebras S. T. Melo, The principal-symbol index map for an algebra of pseudodifferential operators R. Meyer, A more general method to classify up to equivariant KK-equivalence II: Computing obstruction classes A. Neeman, Grothendieck duality made simple J. Park, Calculus of absolute Kahler forms and Milnor $K$-theory G. Tabuada, Noncommutative counterparts of celebrated conjectures R. Willett, Bott periodicity and almost commuting matrices.

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Book SynopsisContains the proceedings of the conference Dynamics: Topology and Numbers, held in July 2018. The papers cover diverse fields of mathematics with a unifying theme of relation to dynamical systems. These include arithmetic geometry, flat geometry, complex dynamics, graph theory, relations to number theory, and topological dynamics.Table of Contents L. Snoha, The life and mathematics of Sergii Kolyada I. Kolyada, A. Blokh, and L. Snoha, Recollections about Sergii Kolyada P. Moree, Sergiy and the MPIM Y. I. Manin and M. Marcolli, Homotopy types and geometries below ${\rm Spec}(\mathbb{Z})$ A. Fel'shtyn and M. Zietek, Dynamical zeta functions of Riedemeister type and representations spaces O. Jenkinson and M. Pollicott, Rigorous dimension estimates for Cantor sets arising in Zaremba theory P. Colognese and M. Pollicott, Volume growth for infinite graphs and translation surfaces J. Byszewski, G. Cornelissen, M. Houben, and L. Van Der Meijden, Dynamically affine maps in positive characteristic S. Kolyada, M. Misiurewicz, and L. Snoha, Special $\alpha$-limit sets E. Shi and X. Ye, Equicontinuity of minimal sets for amenable group actions on dendrites E. Akin, E. Glasner, and B. Weiss, On weak rigidity and weakly mixing enveloping semigroups A. Ganguly and A. Ghosh, The inhomogenous Sprindzhuk conjecture over a local field of positive characteristic A. Blokh, L. Oversteegen, and V. Timorin, Dynamical generation of parameter laminations P. Oprocha, T. Yu, and Guohua Zhang, Multi-sensitivity, multi-transitivity and $\delta$-transitivity R. Sharp, Convergence of zeta functions for amenable group extensions of shifts S. Bezuglyi and O. Karpel, Invariant measures for Cantor dynamical systems M. Kapovich, Periods of abelian differentials and dynamics J. Riedl and D. Schleicher, Crossed renormalization of quadratic polynomials.

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Book SynopsisDemonstrates that complex numbers and geometry can be blended together beautifully. This results in easy proofs and natural generalizations of many theorems in plane geometry, such as the Napoleon theorem, the Ptolemy-Euler theorem, the Simson theorem, and the Morley theorem.Trade ReviewProvides a self-contained introduction to complex numbers and college geometry written in an informal style with an emphasis on the motivation behind the ideas ... The author engages the reader with a casual style, motivational questions, interesting problems and historical notes."" - Mathematical Reviews

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Book SynopsisShow that blow-ups or reverse flips (in the sense of the minimal model program) of rational symplectic manifolds with point centers create Floer-non-trivial Lagrangian tori. These results are part of a conjectural decomposition of the Fukaya category of a compact symplectic manifold with a singularity-free running of the minimal model program.

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Book SynopsisCollects and organises hundreds of beautiful and surprising results about four-sided figures. The book contains hundreds of challenging four-sided problems. Instructors of number theory, combinatorics, analysis, and geometry will find examples and problems to enrich their courses.Table of Contents Simple quadrilaterals Quadrilaterals and their circles Diagonals of quadrilaterals Properties of trapezoids Applications of trapezoids Garfield trapezoids and rectangles Parallelograms Rectangles Squares Special quadrilaterals Quadrilateral numbers Solutions to the Challenges A quadrilateral glossary Credits and permissions Bibliography Index.

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Book SynopsisThe theory of persistence modules originated in topological data analysis and became an active area of research in algebraic topology. This book provides a concise and self-contained introduction to persistence modules and focuses on their interactions with pure mathematics, bringing the reader to the cutting edge of current research.Table of Contents A primer of persistence modules: Definition and first examples Barcodes Proof of the isometry theorem What can we read from a barcode? Applications to metric geometry and function theory: Applications of Rips complexes Topological function theory Persistent homology in symplectic geometry: A concise introduction to symplectic geometry Hamiltonian persistence modules Symplectic persistence modules Bibliography Notation index Subject index Name index.

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Book SynopsisView the abstract.

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Book SynopsisView the abstract.

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Book SynopsisOffers a collection of articles written in memory of Boris Dubrovin (1950-2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher.Table of Contents Part I. Integrable Systems: K. Aleshkin and K. Saito, Primitive forms without higher residue structure and integrable hierarchies (I) M. Alkadhem, G. Antoniou, and M, Feigin, Solutions of $BC_n$ type of WDVV equations P. P. Boalch, Topology of the Stokes phenomenon G. Cotti and A. Varchenko, Equivariant quantum differential equation and qKZ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and $\mathcyr{B}$-theorem L. David and C. Hertling, Meromorphic connections over F-manifolds G. Jorjadze and S. Theisen, Canonical maps and integrability in $T\bar{T}$ deformed 2d CFTs I. Krichever and A. Varchenko, Incarnations of XXX $\widehat{\mathfrak{sl}_N}$ Bethe ansatz equations and integrable hierarchies F. Magri, The Kowalewski separability conditions O. Mokhov and N. A. Strizhova, On the Liouville integrable reduction of the associativity equations in the case of three primary fields S. M. Natanzon and A. Yu. Orlov, Hurwitz numbers from matrix integrals over Guassian measure N. Reshetikhin, Spin Calogero-Moser models on symmetric spaces M. Semenov-Tian-Shansky, Quantum toda lattice: A challenge for representation theory A. O. Smirnov, M. V. Pavlov, V. B. Matveev, and V. S. Gerdjikov, Finite-gap solutions of the Mikhalev equation I. A. B. Strachan, Flat coordinates on orbit spaces: From Novikov algebras to cyclic quotient singularities K. Takasaki, Cubic Hodge integrals and integrable hierarchies of Volterra type G. Tian and G. Xu, Gauged Witten equation and adiabatic limit Part II. Quantum Theories and Algebraic Geometry: T. Bridgeland, Geometry from Donaldson-Thomas invariants V. M. Buchstaber and A. P. Veselov, Fricke identities, Frobenius $K$-characters and Markov equation I. Cherednik, On Harish-Chandra theory of global nonsymmetric functions N. C. Combe and Y. I. Manin, Symmetries of genus zero modular operad R. Coquereaux and J.-B. Zuber, On Schur problem and Kostka numbers P. Etingof, E. Frenkel, and D. Kazhdan, An analytic version of the Langlands correspondence for complex curves D. Gaiotto, T. Johnson-Freyd, and E. Witten, A note on some minimally supersymmetric models in two dimensions O. Garcia-Prada and D. Salamon, A moment map interpretation of the Ricci form, Kahler-Einstein structures, and Teichmuller spaces E. Getzler and S. W. Pohorence, Global gauge conditions in the Batalin-Vilkovisky formalism V. Golyshev and D. Zagier, Interpolated Apery numbers, quasiperiods of modular forms, and motivic gamma functions Y.-P. Lee, H.-W. Lin, and C.-L. Wang, Quantum flips I: Local model M. Marcolli, Aspects of $p$-adic geometry related to entanglement entropy S. Merkulov, Grothendieck-Teichmuller group, operads and graph complexes: A survey O. Ogievetsky and S. Shlosman, Platonic compounds of cylinders.

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Book SynopsisOffers a collection of articles written in memory of Boris Dubrovin (1950-2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher.Table of Contents K. Aleshkin and K. Saito, Primitive forms without higher residue structure and integrable hierarchies (I) M. Alkadhem, G. Antoniou, and M, Feigin, Solutions of $BC_n$ type of WDVV equations P. P. Boalch, Topology of the Stokes phenomenon G. Cotti and A. Varchenko, Equivariant quantum differential equation and qKZ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and $\mathcyr{B}$-theorem L. David and C. Hertling, Meromorphic connections over F-manifolds G. Jorjadze and S. Theisen, Canonical maps and integrability in $T\bar{T}$ deformed 2d CFTs I. Krichever and A. Varchenko, Incarnations of XXX $\widehat{\mathfrak{sl}_N}$ Bethe ansatz equations and integrable hierarchies F. Magri, The Kowalewski separability conditions O. Mokhov and N. A. Strizhova, On the Liouville integrable reduction of the associativity equations in the case of three primary fields S. M. Natanzon and A. Yu. Orlov, Hurwitz numbers from matrix integrals over Guassian measure N. Reshetikhin, Spin Calogero-Moser models on symmetric spaces M. Semenov-Tian-Shansky, Quantum toda lattice: A challenge for representation theory A. O. Smirnov, M. V. Pavlov, V. B. Matveev, and V. S. Gerdjikov, Finite-gap solutions of the Mikhalev equation I. A. B. Strachan, Flat coordinates on orbit spaces: From Novikov algebras to cyclic quotient singularities K. Takasaki, Cubic Hodge integrals and integrable hierarchies of Volterra type G. Tian and G. Xu, Gauged Witten equation and adiabatic limit.

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Book SynopsisOffers a collection of articles written in memory of Boris Dubrovin (1950-2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher.Table of Contents T. Bridgeland, Geometry from Donaldson-Thomas invariants V. M. Buchstaber and A. P. Veselov, Fricke identities, Frobenius $K$-characters and Markov equation I. Cherednik, On Harish-Chandra theory of global nonsymmetric functions N. C. Combe and Y. I. Manin, Symmetries of genus zero modular operad R. Coquereaux and J.-B. Zuber, On Schur problem and Kostka numbers P. Etingof, E. Frenkel, and D. Kazhdan, An analytic version of the Langlands correspondence for complex curves D. Gaiotto, T. Johnson-Freyd, and E. Witten, A note on some minimally supersymmetric models in two dimensions O. Garcia-Prada and D. Salamon, A moment map interpretation of the Ricci form, Kahler-Einstein structures, and Teichmuller spaces E. Getzler and S. W. Pohorence, Global gauge conditions in the Batalin-Vilkovisky formalism V. Golyshev and D. Zagier, Interpolated Apery numbers, quasiperiods of modular forms, and motivic gamma functions Y.-P. Lee, H.-W. Lin, and C.-L. Wang, Quantum flips I: Local model M. Marcolli, Aspects of $p$-adic geometry related to entanglement entropy S. Merkulov, Grothendieck-Teichmuller group, operads and graph complexes: A survey O. Ogievetsky and S. Shlosman, Platonic compounds of cylinders.

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Book SynopsisIntroduces advanced undergraduates to Riemannian geometry and mathematical general relativity. The overall strategy of the book is to explain the concept of curvature via the Jacobi equation which, through discussion of tidal forces, further helps motivate the Einstein field equations.Table of Contents Introduction to Riemannian geometry Differential calculus with tensors Curvature General relativity Introduction to geometry analysis Bibliography Index

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Book SynopsisOffers a comprehensive introduction to point-set, algebraic, and geometric topology, designed to support inquiry-based learning (IBL) courses for upper-division undergraduate or beginning graduate students. The book presents an enormous amount of topology, allowing an instructor to choose which topics to treat.Table of Contents Introduction: The enchanting world of topology Point-set topology: Cardinality: To infinity and beyond Topological spaces: Fundamentals Bases, subspaces, products: Creating new spaces Separation properties: Separating this from that Countable features of spaces: Size restrictions Compactness: The next best thing to being finite Continuity: When nearby points stay together Connectedness: When things don't fall into pieces Metric spaces: Getting some distance Algebraic and geometric topology: Transition from point-set topology to algebraic and geometric topology: Similar strategies, different domains Classification of 2-manifolds: Organizing surfaces Fundamental group: Capturing holes Covering spaces: Layering it on Manifolds, simpleces, complexes, and triangulability: Building blocks Simplicial $\mathbb{Z}_2$-homology: Physical algebra Applications of $\mathbb{Z}_2$-homology: A topological superhero Simplicial $\mathbb{Z}$-homology: Getting oriented Singular homology: Abstracting objects to maps The end: A beginning--reflections on topology and learning Appendix: Group theory background Index

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Book SynopsisThis remarkable book has endured as a masterpiece of mathematical exposition. There are few mathematics books that are still so widely read and continue to have so much to offer-even after more than half a century. The book is overflowing with mathematical ideas, which are explained clearly and elegantly, and above all, with penetrating insight.Trade ReviewThis book is a masterpiece -- a delightful classic that should never go out of print. -- MAA Reviews [This] superb introduction to modern geometry was co-authored by David Hilbert, one of the greatest mathematicians of the 20th century. -- Steven StrogatzTable of Contents The simplest curves and surfaces Regular systems of points Projective configurations Differential geometry Kinematics Topology Index.

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Book SynopsisProvides an extensive introduction to a few of the most prominent extrinsic flows, namely the curve shortening flow, the mean curvature flow, the Gauss curvature flow, the inverse-mean curvature flow, and fully nonlinear flows of mean curvature and inverse-mean curvature type.Trade ReviewThis textbook, written by four experts in the field, offers an authoritative introduction and overview to the topic of extrinsic geometric flows. It will serve well as a primary text for a graduate student who already has background knowledge of differential geometry and (some) partial differential equations. It will also serve as a useful reference for experts in the field."" - John Ross, Southwestern UniversityTable of Contents The heat equation Introduction to curve shortening The Gage-Hamilton-Grayson theorem Self-similar and ancient solutions Hypersurfaces in Euclidean space Introduction to mean curvature flow Mean curvature flow of entire graphs Huisken's theorem Mean convex mean curvature flow Monotonicity formulae Singularity analysis Noncollapsing Self-similar solutions Ancient solutions Gauss curvature flows The affine normal flow Flows by superaffine powers of the Gauss curvature Fully nonlinear curvature flows Flows of mean curvature type Flows of inverse-mean curvature type Bibliography Index

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Book SynopsisWe use homotopical algebra (or abstract homotopical methods) to study smooth homotopical problems of infinite-dimensional C$C^{\infty }$-manifolds in convenient calculus. More precisely, we discuss the smoothing of maps, sections, principal bundles, and gauge transformations.

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Book SynopsisWe define enumerative invariants associated to a hybrid Gauged Linear Sigma Model. We prove that in the relevant special cases these invariants recover both the Gromov-Witten type invariants defined by Chang-Li and Fan-Jarvis-Ruan using cosection localization as well as the FJRW type invariants constructed by Polishchuk-Vaintrob.

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Book SynopsisTrade Review“This book is a very welcome, untraditional, thorough and well-organized introduction to a young and quickly developing discipline on the crossroads between mathematics, computer science, and engineering.” —DMV NewsletterTable of Contents Computational geometric topology: Graphs Surfaces Complexes Computational algebraic topology: Homology Duality Morse functions Computational persistent topology: Persistence Stability Applications References Index

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Book Synopsis

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Book Synopsis

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