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The description for this book, Knot Groups. Annals of Mathematics Studies. (AM-56), Volume 56, will be forthcoming.

Topology --- 512 --- Algebra --- 512 Algebra --- Knot theory. --- Knots (Topology) --- Low-dimensional topology --- Abelian group. --- Alexander duality. --- Alexander polynomial. --- Algebraic theory. --- Algorithm. --- Analytic continuation. --- Associative property. --- Automorphism. --- Axiom. --- Bijection. --- Binary relation. --- Calculation. --- Central series. --- Characterization (mathematics). --- Cobordism. --- Coefficient. --- Cohomology. --- Combinatorics. --- Commutator subgroup. --- Complete theory. --- Computation. --- Conjugacy class. --- Conjugate element (field theory). --- Connected space. --- Connectedness. --- Coprime integers. --- Coset. --- Covering space. --- Curve. --- Cyclic group. --- Dehn's lemma. --- Determinant. --- Diagonalization. --- Diagram (category theory). --- Dimension. --- Direct product. --- Equivalence class. --- Equivalence relation. --- Euclidean space. --- Euler characteristic. --- Existential quantification. --- Fiber bundle. --- Finite group. --- Finitely generated module. --- Frattini subgroup. --- Free abelian group. --- Fundamental group. --- Geometry. --- Group ring. --- Group theory. --- Group with operators. --- Hausdorff space. --- Homeomorphism. --- Homology (mathematics). --- Homomorphism. --- Homotopy group. --- Homotopy. --- Identity matrix. --- Inner automorphism. --- Interior (topology). --- Intersection number (graph theory). --- Knot group. --- Linear combination. --- Manifold. --- Mathematical induction. --- Monomorphism. --- Morphism. --- Morse theory. --- Natural transformation. --- Non-abelian group. --- Normal subgroup. --- Orientability. --- Permutation. --- Polynomial. --- Presentation of a group. --- Principal ideal domain. --- Principal ideal. --- Root of unity. --- Semigroup. --- Simplicial complex. --- Simply connected space. --- Special case. --- Square matrix. --- Subgroup. --- Subset. --- Summation. --- Theorem. --- Three-dimensional space (mathematics). --- Topological space. --- Topology. --- Torus knot. --- Transfinite number. --- Trefoil knot. --- Trichotomy (mathematics). --- Trivial group. --- Triviality (mathematics). --- Two-dimensional space. --- Unit vector. --- Wreath product.

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Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories. The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are "ed without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.

Algebraic geometry --- Differential geometry. Global analysis --- 512.7 --- Algebraic geometry. Commutative rings and algebras --- Toric varieties. --- 512.7 Algebraic geometry. Commutative rings and algebras --- Toric varieties --- Embeddings, Torus --- Torus embeddings --- Varieties, Toric --- Algebraic varieties --- Addition. --- Affine plane. --- Affine space. --- Affine variety. --- Alexander Grothendieck. --- Alexander duality. --- Algebraic curve. --- Algebraic group. --- Atiyah–Singer index theorem. --- Automorphism. --- Betti number. --- Big O notation. --- Characteristic class. --- Chern class. --- Chow group. --- Codimension. --- Cohomology. --- Combinatorics. --- Commutative property. --- Complete intersection. --- Convex polytope. --- Convex set. --- Coprime integers. --- Cotangent space. --- Dedekind sum. --- Dimension (vector space). --- Dimension. --- Direct proof. --- Discrete valuation ring. --- Discrete valuation. --- Disjoint union. --- Divisor (algebraic geometry). --- Divisor. --- Dual basis. --- Dual space. --- Equation. --- Equivalence class. --- Equivariant K-theory. --- Euler characteristic. --- Exact sequence. --- Explicit formula. --- Facet (geometry). --- Fundamental group. --- Graded ring. --- Grassmannian. --- H-vector. --- Hirzebruch surface. --- Hodge theory. --- Homogeneous coordinates. --- Homomorphism. --- Hypersurface. --- Intersection theory. --- Invertible matrix. --- Invertible sheaf. --- Isoperimetric inequality. --- Lattice (group). --- Leray spectral sequence. --- Limit point. --- Line bundle. --- Line segment. --- Linear subspace. --- Local ring. --- Mathematical induction. --- Mixed volume. --- Moduli space. --- Moment map. --- Monotonic function. --- Natural number. --- Newton polygon. --- Open set. --- Picard group. --- Pick's theorem. --- Polytope. --- Projective space. --- Quadric. --- Quotient space (topology). --- Regular sequence. --- Relative interior. --- Resolution of singularities. --- Restriction (mathematics). --- Resultant. --- Riemann–Roch theorem. --- Serre duality. --- Sign (mathematics). --- Simplex. --- Simplicial complex. --- Simultaneous equations. --- Spectral sequence. --- Subgroup. --- Subset. --- Summation. --- Surjective function. --- Tangent bundle. --- Theorem. --- Topology. --- Toric variety. --- Unit disk. --- Vector space. --- Weil conjecture. --- Zariski topology.

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Nilpotence and Periodicity in Stable Homotopy Theory describes some major advances made in algebraic topology in recent years, centering on the nilpotence and periodicity theorems, which were conjectured by the author in 1977 and proved by Devinatz, Hopkins, and Smith in 1985. During the last ten years a number of significant advances have been made in homotopy theory, and this book fills a real need for an up-to-date text on that topic. Ravenel's first few chapters are written with a general mathematical audience in mind. They survey both the ideas that lead up to the theorems and their applications to homotopy theory. The book begins with some elementary concepts of homotopy theory that are needed to state the problem. This includes such notions as homotopy, homotopy equivalence, CW-complex, and suspension. Next the machinery of complex cobordism, Morava K-theory, and formal group laws in characteristic p are introduced. The latter portion of the book provides specialists with a coherent and rigorous account of the proofs. It includes hitherto unpublished material on the smash product and chromatic convergence theorems and on modular representations of the symmetric group.

Homotopie --- Homotopy theory --- Homotopy theory. --- Deformations, Continuous --- Topology --- Abelian category. --- Abelian group. --- Adams spectral sequence. --- Additive category. --- Affine space. --- Algebra homomorphism. --- Algebraic closure. --- Algebraic structure. --- Algebraic topology (object). --- Algebraic topology. --- Algebraic variety. --- Algebraically closed field. --- Atiyah–Hirzebruch spectral sequence. --- Automorphism. --- Boolean algebra (structure). --- CW complex. --- Canonical map. --- Cantor set. --- Category of topological spaces. --- Category theory. --- Classification theorem. --- Classifying space. --- Cohomology operation. --- Cohomology. --- Cokernel. --- Commutative algebra. --- Commutative ring. --- Complex projective space. --- Complex vector bundle. --- Computation. --- Conjecture. --- Conjugacy class. --- Continuous function. --- Contractible space. --- Coproduct. --- Differentiable manifold. --- Disjoint union. --- Division algebra. --- Equation. --- Explicit formulae (L-function). --- Functor. --- G-module. --- Groupoid. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy group. --- Homotopy. --- Hopf algebra. --- Hurewicz theorem. --- Inclusion map. --- Infinite product. --- Integer. --- Inverse limit. --- Irreducible representation. --- Isomorphism class. --- K-theory. --- Loop space. --- Mapping cone (homological algebra). --- Mathematical induction. --- Modular representation theory. --- Module (mathematics). --- Monomorphism. --- Moore space. --- Morava K-theory. --- Morphism. --- N-sphere. --- Noetherian ring. --- Noetherian. --- Noncommutative ring. --- Number theory. --- P-adic number. --- Piecewise linear manifold. --- Polynomial ring. --- Polynomial. --- Power series. --- Prime number. --- Principal ideal domain. --- Profinite group. --- Reduced homology. --- Ring (mathematics). --- Ring homomorphism. --- Ring spectrum. --- Simplicial complex. --- Simply connected space. --- Smash product. --- Special case. --- Spectral sequence. --- Steenrod algebra. --- Sub"ient. --- Subalgebra. --- Subcategory. --- Subring. --- Symmetric group. --- Tensor product. --- Theorem. --- Topological space. --- Topology. --- Vector bundle. --- Zariski topology.

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This book contains accounts of talks held at a symposium in honorof John C. Moore in October 1983 at Princeton University, The workincludes papers in classical homotopy theory, homological algebra,rational homotopy theory, algebraic K-theory of spaces, and othersubjects.

Algebraic topology --- K-theory --- 512.73 --- 515.14 --- 512.73 Cohomology theory of algebraic varieties and schemes --- Cohomology theory of algebraic varieties and schemes --- 515.14 Algebraic topology --- Moore, John C. --- Abelian group. --- Adams spectral sequence. --- Adjoint functors. --- Adjunction (field theory). --- Algebraic K-theory. --- Algebraic closure. --- Algebraic geometry. --- Algebraic group. --- Algebraic number field. --- Algebraic space. --- Algebraic topology. --- Algebraically closed field. --- Associative algebra. --- Boundary (topology). --- CW complex. --- Classification theorem. --- Closure (mathematics). --- Coalgebra. --- Cofibration. --- Cohomology. --- Commutative diagram. --- Commutative property. --- Coproduct. --- Deformation theory. --- Degenerate bilinear form. --- Diagram (category theory). --- Differentiable manifold. --- Dimension (vector space). --- Division algebra. --- Eilenberg–Moore spectral sequence. --- Epimorphism. --- Exterior (topology). --- Formal power series. --- Free Lie algebra. --- Free algebra. --- Freudenthal suspension theorem. --- Function (mathematics). --- Function space. --- Functor. --- G-module. --- Galois extension. --- Global dimension. --- Group cohomology. --- Group homomorphism. --- H-space. --- Hilbert's Theorem 90. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy group. --- Homotopy. --- Hopf algebra. --- Hopf invariant. --- Hurewicz theorem. --- Inclusion map. --- Inequality (mathematics). --- Integral domain. --- Isometry. --- Isomorphism class. --- K-theory. --- Lie algebra. --- Lie group. --- Limit (category theory). --- Loop space. --- Mathematician. --- Mathematics. --- Noetherian ring. --- Order topology. --- P-adic number. --- Polynomial ring. --- Polynomial. --- Prime number. --- Principal bundle. --- Principal ideal domain. --- Projective module. --- Projective plane. --- Pullback (category theory). --- Pushout (category theory). --- Ring of integers. --- Series (mathematics). --- Sheaf (mathematics). --- Simplicial category. --- Simplicial complex. --- Simplicial set. --- Special case. --- Spectral sequence. --- Square (algebra). --- Stable homotopy theory. --- Steenrod algebra. --- Superalgebra. --- Theorem. --- Topological K-theory. --- Topological space. --- Topology. --- Triviality (mathematics). --- Uniqueness theorem. --- Universal enveloping algebra. --- Vector bundle. --- Weak equivalence (homotopy theory). --- William Browder (mathematician). --- Géométrie algébrique --- K-théorie

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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.