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The Yang-Baxter equation first appeared in theoretical physics, in a paper by the Nobel laureate C.N. Yang and in the work of R.J. Baxter in the field of Statistical Mechanics. At the 1990 International Mathematics Congress, Vladimir Drinfeld, Vaughan F. R. Jones, and Edward Witten were awarded Fields Medals for their work related to the Yang-Baxter equation. It turned out that this equation is one of the basic equations in mathematical physics; more precisely, it is used for introducing the theory of quantum groups. It also plays a crucial role in: knot theory, braided categories, the analysis of integrable systems, non-commutative descent theory, quantum computing, non-commutative geometry, etc. Many scientists have used the axioms of various algebraic structures (quasi-triangular Hopf algebras, Yetter-Drinfeld categories, quandles, group actions, Lie (super)algebras, brace structures, (co)algebra structures, Jordan triples, Boolean algebras, relations on sets, etc.) or computer calculations (and Grobner bases) in order to produce solutions for the Yang-Baxter equation. However, the full classification of its solutions remains an open problem. At present, the study of solutions of the Yang-Baxter equation attracts the attention of a broad circle of scientists. The current volume highlights various aspects of the Yang-Baxter equation, related algebraic structures, and applications.

braided category --- quasitriangular structure --- quantum projective space --- Hopf algebra --- quantum integrability --- duality --- six-vertex model --- Quantum Group --- Yang-Baxter equation --- star-triangle relation --- R-matrix --- Lie algebra --- bundle --- braid group

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On Knots is a journey through the theory of knots, starting from the simplest combinatorial ideas--ideas arising from the representation of weaving patterns. From this beginning, topological invariants are constructed directly: first linking numbers, then the Conway polynomial and skein theory. This paves the way for later discussion of the recently discovered Jones and generalized polynomials. The central chapter, Chapter Six, is a miscellany of topics and recreations. Here the reader will find the quaternions and the belt trick, a devilish rope trick, Alhambra mosaics, Fibonacci trees, the topology of DNA, and the author's geometric interpretation of the generalized Jones Polynomial.Then come branched covering spaces, the Alexander polynomial, signature theorems, the work of Casson and Gordon on slice knots, and a chapter on knots and algebraic singularities.The book concludes with an appendix about generalized polynomials.

Knot theory --- Knots (Topology) --- Low-dimensional topology --- Knot theory. --- Algebraic topology --- 3-sphere. --- Addition theorem. --- Addition. --- Alexander polynomial. --- Algebraic variety. --- Algorithm. --- Ambient isotopy. --- Arf invariant. --- Basepoint. --- Bijection. --- Bilinear form. --- Borromean rings. --- Bracket polynomial. --- Braid group. --- Branched covering. --- Chiral knot. --- Chromatic polynomial. --- Cobordism. --- Codimension. --- Combination. --- Combinatorics. --- Complex analysis. --- Concentric. --- Conjecture. --- Connected sum. --- Conway polynomial (finite fields). --- Counting. --- Covering space. --- Cyclic group. --- Dense set. --- Determinant. --- Diagram (category theory). --- Diffeomorphism. --- Dimension. --- Disjoint union. --- Disk (mathematics). --- Dual graph. --- Elementary algebra. --- Embedding. --- Enumeration. --- Existential quantification. --- Exotic sphere. --- Fibration. --- Formal power series. --- Fundamental group. --- Geometric topology. --- Geometry and topology. --- Geometry. --- Group action. --- Homotopy. --- Integer. --- Intersection form (4-manifold). --- Isolated singularity. --- Jones polynomial. --- Knot complement. --- Knot group. --- Laws of Form. --- Lens space. --- Linking number. --- Manifold. --- Module (mathematics). --- Morwen Thistlethwaite. --- Normal bundle. --- Notation. --- Obstruction theory. --- Operator algebra. --- Pairing. --- Parity (mathematics). --- Partition function (mathematics). --- Planar graph. --- Point at infinity. --- Polynomial ring. --- Polynomial. --- Quantity. --- Rectangle. --- Reidemeister move. --- Remainder. --- Root of unity. --- Saddle point. --- Seifert surface. --- Singularity theory. --- Slice knot. --- Special case. --- Statistical mechanics. --- Substructure. --- Summation. --- Symmetry. --- Theorem. --- Three-dimensional space (mathematics). --- Topological space. --- Torus knot. --- Trefoil knot. --- Tubular neighborhood. --- Underpinning. --- Unknot. --- Variable (mathematics). --- Whitehead link. --- Wild knot. --- Writhe. --- Variétés topologiques --- Topologie combinatoire --- Theorie des noeuds

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"The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students.The book begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichm©oller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification"--Provided by publisher.

Mappings (Mathematics) --- Class groups (Mathematics) --- Groups, Class (Mathematics) --- Algebraic number theory --- Commutative rings --- Ideals (Algebra) --- Maps (Mathematics) --- Functions --- Functions, Continuous --- Topology --- Transformations (Mathematics) --- 3-manifold theory. --- Alexander method. --- Birman exact sequence. --- BirmanЈilden theorem. --- Dehn twists. --- DehnЌickorish theorem. --- DehnЎielsenЂaer theorem. --- Dennis Johnson. --- Euler class. --- FenchelЎielsen coordinates. --- Gervais presentation. --- Grtzsch's problem. --- Johnson homomorphism. --- Markov partitions. --- Meyer signature cocycle. --- Mod(S). --- Nielsen realization theorem. --- NielsenДhurston classification theorem. --- NielsenДhurston classification. --- Riemann surface. --- Teichmller mapping. --- Teichmller metric. --- Teichmller space. --- Thurston compactification. --- Torelli group. --- Wajnryb presentation. --- algebraic integers. --- algebraic intersection number. --- algebraic relations. --- algebraic structure. --- annulus. --- aspherical manifold. --- bigon criterion. --- braid group. --- branched cover. --- capping homomorphism. --- classifying space. --- closed surface. --- collar lemma. --- compactness criterion. --- complex of curves. --- configuration space. --- conjugacy class. --- coordinates principle. --- cutting homomorphism. --- cyclic subgroup. --- diffeomorphism. --- disk. --- existence theorem. --- extended mapping class group. --- finite index. --- finite subgroup. --- finite-order homeomorphism. --- finite-order mapping class. --- first homology group. --- geodesic laminations. --- geometric classification. --- geometric group theory. --- geometric intersection number. --- geometric operation. --- geometry. --- harmonic maps. --- holomorphic quadratic differential. --- homeomorphism. --- homological criterion. --- homotopy. --- hyperbolic geometry. --- hyperbolic plane. --- hyperbolic structure. --- hyperbolic surface. --- inclusion homomorphism. --- infinity. --- intersection number. --- isotopy. --- lantern relation. --- low-dimensional homology. --- mapping class group. --- mapping torus. --- measured foliation space. --- measured foliations. --- metric geometry. --- moduli space. --- orbifold. --- orbit. --- outer automorphism group. --- pseudo-Anosov homeomorphism. --- punctured disk. --- quasi-isometry. --- quasiconformal map. --- second homology group. --- simple closed curve. --- simplicial complex. --- stretch factors. --- surface bundles. --- surface homeomorphism. --- surface. --- symplectic representation. --- topology. --- torsion. --- torus. --- train track.

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Geometric group theory is the study of the interplay between groups and the spaces they act on, and has its roots in the works of Henri Poincaré, Felix Klein, J.H.C. Whitehead, and Max Dehn. Office Hours with a Geometric Group Theorist brings together leading experts who provide one-on-one instruction on key topics in this exciting and relatively new field of mathematics. It's like having office hours with your most trusted math professors.An essential primer for undergraduates making the leap to graduate work, the book begins with free groups-actions of free groups on trees, algorithmic questions about free groups, the ping-pong lemma, and automorphisms of free groups. It goes on to cover several large-scale geometric invariants of groups, including quasi-isometry groups, Dehn functions, Gromov hyperbolicity, and asymptotic dimension. It also delves into important examples of groups, such as Coxeter groups, Thompson's groups, right-angled Artin groups, lamplighter groups, mapping class groups, and braid groups. The tone is conversational throughout, and the instruction is driven by examples.Accessible to students who have taken a first course in abstract algebra, Office Hours with a Geometric Group Theorist also features numerous exercises and in-depth projects designed to engage readers and provide jumping-off points for research projects.

Geometric group theory. --- "ient. --- 4-valent tree. --- Cantor set. --- Cayley 2-complex. --- Cayley graph. --- Coxeter group. --- DSV method. --- Dehn function. --- Dehn twist. --- Euclidean space. --- Farey complex. --- Farey graph. --- Farey tree. --- Gromov hyperbolicity. --- Klein's criterion. --- Milnor-Schwarz lemma. --- Möbius transformation. --- Nielsen-Schreier Subgroup theorem. --- Perron-Frobenius theorem. --- Riemannian manifold. --- Schottky lemma. --- Thompson's group. --- asymptotic dimension. --- automorphism group. --- automorphism. --- bi-Lipschitz equivalence. --- braid group. --- braids. --- coarse isometry. --- combinatorics. --- compact orientable surface. --- cone type. --- configuration space. --- context-free grammar. --- curvature. --- dead end. --- distortion. --- endomorphism. --- finite group. --- folding. --- formal language. --- free abelian group. --- free action. --- free expansion. --- free group. --- free nonabelian group. --- free reduction. --- generators. --- geometric group theory. --- geometric object. --- geometric space. --- graph. --- group action. --- group element. --- group ends. --- group growth. --- group presentation. --- group theory. --- group. --- homeomorphism. --- homomorphism. --- hyperbolic geometry. --- hyperbolic group. --- hyperbolic space. --- hyperbolicity. --- hyperplane arrangements. --- index. --- infinite graph. --- infinite group. --- integers. --- isoperimetric problem. --- isoperimetry. --- jigsaw puzzle. --- knot theory. --- lamplighter group. --- manifold. --- mapping class group. --- mathematics. --- membership problem. --- metric space. --- non-free action. --- normal subgroup. --- path metric. --- ping-pong lemma. --- ping-pong. --- polynomial growth theorem. --- product. --- punctured disks. --- quasi-isometric equivalence. --- quasi-isometric rigidity. --- quasi-isometry group. --- quasi-isometry invariant. --- quasi-isometry. --- reflection group. --- reflection. --- relators. --- residual finiteness. --- right-angled Artin group. --- robotics. --- semidirect product. --- space. --- surface group. --- surface. --- symmetric group. --- symmetry. --- topological model. --- topology. --- train track. --- tree. --- word length. --- word metric. --- word problem.

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Geometric group theory is the study of the interplay between groups and the spaces they act on, and has its roots in the works of Henri Poincaré, Felix Klein, J.H.C. Whitehead, and Max Dehn. Office Hours with a Geometric Group Theorist brings together leading experts who provide one-on-one instruction on key topics in this exciting and relatively new field of mathematics. It's like having office hours with your most trusted math professors.An essential primer for undergraduates making the leap to graduate work, the book begins with free groups-actions of free groups on trees, algorithmic questions about free groups, the ping-pong lemma, and automorphisms of free groups. It goes on to cover several large-scale geometric invariants of groups, including quasi-isometry groups, Dehn functions, Gromov hyperbolicity, and asymptotic dimension. It also delves into important examples of groups, such as Coxeter groups, Thompson's groups, right-angled Artin groups, lamplighter groups, mapping class groups, and braid groups. The tone is conversational throughout, and the instruction is driven by examples.Accessible to students who have taken a first course in abstract algebra, Office Hours with a Geometric Group Theorist also features numerous exercises and in-depth projects designed to engage readers and provide jumping-off points for research projects.

Geometric group theory. --- "ient. --- 4-valent tree. --- Cantor set. --- Cayley 2-complex. --- Cayley graph. --- Coxeter group. --- DSV method. --- Dehn function. --- Dehn twist. --- Euclidean space. --- Farey complex. --- Farey graph. --- Farey tree. --- Gromov hyperbolicity. --- Klein's criterion. --- Milnor-Schwarz lemma. --- Möbius transformation. --- Nielsen-Schreier Subgroup theorem. --- Perron-Frobenius theorem. --- Riemannian manifold. --- Schottky lemma. --- Thompson's group. --- asymptotic dimension. --- automorphism group. --- automorphism. --- bi-Lipschitz equivalence. --- braid group. --- braids. --- coarse isometry. --- combinatorics. --- compact orientable surface. --- cone type. --- configuration space. --- context-free grammar. --- curvature. --- dead end. --- distortion. --- endomorphism. --- finite group. --- folding. --- formal language. --- free abelian group. --- free action. --- free expansion. --- free group. --- free nonabelian group. --- free reduction. --- generators. --- geometric group theory. --- geometric object. --- geometric space. --- graph. --- group action. --- group element. --- group ends. --- group growth. --- group presentation. --- group theory. --- group. --- homeomorphism. --- homomorphism. --- hyperbolic geometry. --- hyperbolic group. --- hyperbolic space. --- hyperbolicity. --- hyperplane arrangements. --- index. --- infinite graph. --- infinite group. --- integers. --- isoperimetric problem. --- isoperimetry. --- jigsaw puzzle. --- knot theory. --- lamplighter group. --- manifold. --- mapping class group. --- mathematics. --- membership problem. --- metric space. --- non-free action. --- normal subgroup. --- path metric. --- ping-pong lemma. --- ping-pong. --- polynomial growth theorem. --- product. --- punctured disks. --- quasi-isometric equivalence. --- quasi-isometric rigidity. --- quasi-isometry group. --- quasi-isometry invariant. --- quasi-isometry. --- reflection group. --- reflection. --- relators. --- residual finiteness. --- right-angled Artin group. --- robotics. --- semidirect product. --- space. --- surface group. --- surface. --- symmetric group. --- symmetry. --- topological model. --- topology. --- train track. --- tree. --- word length. --- word metric. --- word problem.