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Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics. The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

Algebraic geometry --- Topology --- Toposes --- Categories (Mathematics) --- Categories (Mathematics). --- Toposes. --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Category theory (Mathematics) --- Topoi (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Functor theory --- Adjoint functors. --- Associative property. --- Base change map. --- Base change. --- CW complex. --- Canonical map. --- Cartesian product. --- Category of sets. --- Category theory. --- Coequalizer. --- Cofinality. --- Coherence theorem. --- Cohomology. --- Cokernel. --- Commutative property. --- Continuous function (set theory). --- Contractible space. --- Coproduct. --- Corollary. --- Derived category. --- Diagonal functor. --- Diagram (category theory). --- Dimension theory (algebra). --- Dimension theory. --- Dimension. --- Enriched category. --- Epimorphism. --- Equivalence class. --- Equivalence relation. --- Existence theorem. --- Existential quantification. --- Factorization system. --- Functor category. --- Functor. --- Fundamental group. --- Grothendieck topology. --- Grothendieck universe. --- Group homomorphism. --- Groupoid. --- Heyting algebra. --- Higher Topos Theory. --- Higher category theory. --- Homotopy category. --- Homotopy colimit. --- Homotopy group. --- Homotopy. --- I0. --- Inclusion map. --- Inductive dimension. --- Initial and terminal objects. --- Inverse limit. --- Isomorphism class. --- Kan extension. --- Limit (category theory). --- Localization of a category. --- Maximal element. --- Metric space. --- Model category. --- Monoidal category. --- Monoidal functor. --- Monomorphism. --- Monotonic function. --- Morphism. --- Natural transformation. --- Nisnevich topology. --- Noetherian topological space. --- Noetherian. --- O-minimal theory. --- Open set. --- Power series. --- Presheaf (category theory). --- Prime number. --- Pullback (category theory). --- Pushout (category theory). --- Quillen adjunction. --- Quotient by an equivalence relation. --- Regular cardinal. --- Retract. --- Right inverse. --- Sheaf (mathematics). --- Sheaf cohomology. --- Simplicial category. --- Simplicial set. --- Special case. --- Subcategory. --- Subset. --- Surjective function. --- Tensor product. --- Theorem. --- Topological space. --- Topology. --- Topos. --- Total order. --- Transitive relation. --- Universal property. --- Upper and lower bounds. --- Weak equivalence (homotopy theory). --- Yoneda lemma. --- Zariski topology. --- Zorn's lemma.

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Part exposition and part presentation of new results, this monograph deals with that area of mathematics which has both combinatorial group theory and mathematical logic in common. Its main topics are the word problem for groups, the conjugacy problem for groups, and the isomorphism problem for groups. The presentation depends on previous results of J. L. Britton, which, with other factual background, are treated in detail.

Group theory --- 510.6 --- Mathematical logic --- 510.6 Mathematical logic --- Group theory. --- Logic, Symbolic and mathematical. --- Groupes, Théorie des --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Algebra of logic --- Logic, Universal --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Abelian group. --- Betti number. --- Characteristic function (probability theory). --- Characterization (mathematics). --- Combinatorial group theory. --- Conjecture. --- Conjugacy class. --- Conjugacy problem. --- Contradiction. --- Corollary. --- Cyclic permutation. --- Decision problem. --- Diffeomorphism. --- Direct product. --- Direct proof. --- Effective method. --- Elementary class. --- Embedding. --- Enumeration. --- Epimorphism. --- Equation. --- Equivalence relation. --- Exact sequence. --- Existential quantification. --- Finite group. --- Finite set. --- Finitely generated group. --- Finitely presented. --- Free group. --- Free product. --- Fundamental group. --- Fundamental theorem. --- Group (mathematics). --- Gödel numbering. --- Homomorphism. --- Homotopy. --- Inner automorphism. --- Markov property. --- Mathematical logic. --- Mathematical proof. --- Mathematics. --- Monograph. --- Natural number. --- Nilpotent group. --- Normal subgroup. --- Notation. --- Permutation. --- Polycyclic group. --- Presentation of a group. --- Quotient group. --- Recursive set. --- Requirement. --- Residually finite group. --- Semigroup. --- Simple set. --- Simplicial complex. --- Solvable group. --- Statistical hypothesis testing. --- Subgroup. --- Theorem. --- Theory. --- Topology. --- Transitive relation. --- Triviality (mathematics). --- Truth table. --- Turing degree. --- Turing machine. --- Without loss of generality. --- Word problem (mathematics). --- Groupes, Théorie des --- Décidabilité (logique mathématique)

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In The Structure of Affine Buildings, Richard Weiss gives a detailed presentation of the complete proof of the classification of Bruhat-Tits buildings first completed by Jacques Tits in 1986. The book includes numerous results about automorphisms, completions, and residues of these buildings. It also includes tables correlating the results in the locally finite case with the results of Tits's classification of absolutely simple algebraic groups defined over a local field. A companion to Weiss's The Structure of Spherical Buildings, The Structure of Affine Buildings is organized around the classification of spherical buildings and their root data as it is carried out in Tits and Weiss's Moufang Polygons.

Buildings (Group theory) --- Moufang loops --- Automorphisms --- Affine algebraic groups --- Moufang loops. --- Automorphisms. --- Affine algebraic groups. --- Algebraic groups, Affine --- Loops, Moufang --- Theory of buildings (Group theory) --- Tits's theory of buildings (Group theory) --- Group schemes (Mathematics) --- Group theory --- Symmetry (Mathematics) --- Loops (Group theory) --- Linear algebraic groups --- Buildings (Group theory). --- Addition. --- Additive group. --- Additive inverse. --- Algebraic group. --- Algebraic structure. --- Ambient space. --- Associative property. --- Automorphism. --- Big O notation. --- Bijection. --- Bilinear form. --- Bounded set (topological vector space). --- Bounded set. --- Calculation. --- Cardinality. --- Cauchy sequence. --- Commutative property. --- Complete graph. --- Complete metric space. --- Composition algebra. --- Connected component (graph theory). --- Consistency. --- Continuous function. --- Coordinate system. --- Corollary. --- Coxeter group. --- Coxeter–Dynkin diagram. --- Diagram (category theory). --- Diameter. --- Dimension. --- Discrete valuation. --- Division algebra. --- Dot product. --- Dynkin diagram. --- E6 (mathematics). --- E7 (mathematics). --- E8 (mathematics). --- Empty set. --- Equipollence (geometry). --- Equivalence class. --- Equivalence relation. --- Euclidean geometry. --- Euclidean space. --- Existential quantification. --- Free monoid. --- Fundamental domain. --- Hyperplane. --- Infimum and supremum. --- Jacques Tits. --- K0. --- Linear combination. --- Mathematical induction. --- Metric space. --- Multiple edges. --- Multiplicative inverse. --- Number theory. --- Octonion. --- Parameter. --- Permutation group. --- Permutation. --- Pointwise. --- Polygon. --- Projective line. --- Quadratic form. --- Quaternion. --- Remainder. --- Root datum. --- Root system. --- Scientific notation. --- Sphere. --- Subgroup. --- Subring. --- Subset. --- Substructure. --- Theorem. --- Topology of uniform convergence. --- Topology. --- Torus. --- Tree (data structure). --- Tree structure. --- Two-dimensional space. --- Uniform continuity. --- Valuation (algebra). --- Vector space. --- Without loss of generality.

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Fibre bundles, now an integral part of differential geometry, are also of great importance in modern physics--such as in gauge theory. This book, a succinct introduction to the subject by renown mathematician Norman Steenrod, was the first to present the subject systematically. It begins with a general introduction to bundles, including such topics as differentiable manifolds and covering spaces. The author then provides brief surveys of advanced topics, such as homotopy theory and cohomology theory, before using them to study further properties of fibre bundles. The result is a classic and timeless work of great utility that will appeal to serious mathematicians and theoretical physicists alike.

#WWIS:d.d. Prof. L. Bouckaert/ALTO --- 515.1 --- 515.1 Topology --- Topology --- Topology. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Algebraic topology. --- Associated bundle. --- Associative algebra. --- Associative property. --- Atlas (topology). --- Automorphism. --- Axiomatic system. --- Barycentric subdivision. --- Bilinear map. --- Bundle map. --- Classification theorem. --- Coefficient. --- Cohomology ring. --- Cohomology. --- Conjugacy class. --- Connected component (graph theory). --- Connected space. --- Coordinate system. --- Coset. --- Cup product. --- Cyclic group. --- Determinant. --- Differentiable manifold. --- Differential structure. --- Dimension (vector space). --- Direct product. --- Division algebra. --- Equivalence class. --- Equivalence relation. --- Euler number. --- Existence theorem. --- Existential quantification. --- Factorization. --- Fiber bundle. --- Frenet–Serret formulas. --- Gram–Schmidt process. --- Group theory. --- Homeomorphism. --- Homology (mathematics). --- Homomorphism. --- Homotopy group. --- Homotopy. --- Hopf theorem. --- Hurewicz theorem. --- Identity element. --- Inclusion map. --- Inner automorphism. --- Invariant subspace. --- Invertible matrix. --- Jacobian matrix and determinant. --- Klein bottle. --- Lattice of subgroups. --- Lie group. --- Line element. --- Line segment. --- Linear map. --- Linear space (geometry). --- Linear subspace. --- Manifold. --- Mapping cylinder. --- Metric tensor. --- N-sphere. --- Natural topology. --- Octonion. --- Open set. --- Orientability. --- Orthogonal group. --- Orthogonalization. --- Permutation. --- Principal bundle. --- Product topology. --- Quadratic form. --- Quaternion. --- Retract. --- Separable space. --- Set theory. --- Simplicial complex. --- Special case. --- Stiefel manifold. --- Subalgebra. --- Subbase. --- Subgroup. --- Subset. --- Symmetric tensor. --- Tangent bundle. --- Tangent space. --- Tangent vector. --- Tensor field. --- Tensor. --- Theorem. --- Tietze extension theorem. --- Topological group. --- Topological space. --- Transitive relation. --- Transpose. --- Union (set theory). --- Unit sphere. --- Universal bundle. --- Vector field.

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Based on a series of lectures given by Harish-Chandra at the Institute for Advanced Study in 1971-1973, this book provides an introduction to the theory of harmonic analysis on reductive p-adic groups.Originally published in 1979.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

512.74 --- p-adic groups --- Banach algebras --- Groups, p-adic --- Algebraic groups. Abelian varieties --- p-adic groups. --- 512.74 Algebraic groups. Abelian varieties --- P-adic groups. --- Harmonic analysis. Fourier analysis --- Harmonic analysis --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Group theory --- Harmonic analysis. --- Adjoint representation. --- Admissible representation. --- Algebra homomorphism. --- Algebraic group. --- Analytic continuation. --- Analytic function. --- Associative property. --- Automorphic form. --- Automorphism. --- Banach space. --- Bijection. --- Bilinear form. --- Borel subgroup. --- Cartan subgroup. --- Central simple algebra. --- Characteristic function (probability theory). --- Characterization (mathematics). --- Class function (algebra). --- Commutative property. --- Compact space. --- Composition series. --- Conjugacy class. --- Corollary. --- Dimension (vector space). --- Discrete series representation. --- Division algebra. --- Double coset. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Epimorphism. --- Equivalence class. --- Equivalence relation. --- Existential quantification. --- Factorization. --- Fourier series. --- Function (mathematics). --- Functional equation. --- Fundamental domain. --- Fundamental lemma (Langlands program). --- G-module. --- Group isomorphism. --- Haar measure. --- Hecke algebra. --- Holomorphic function. --- Identity element. --- Induced representation. --- Inner automorphism. --- Lebesgue measure. --- Levi decomposition. --- Lie algebra. --- Locally constant function. --- Locally integrable function. --- Mathematical induction. --- Matrix coefficient. --- Maximal compact subgroup. --- Meromorphic function. --- Module (mathematics). --- Module homomorphism. --- Open set. --- Order of integration (calculus). --- Orthogonal complement. --- P-adic number. --- Pole (complex analysis). --- Product measure. --- Projection (linear algebra). --- Quotient module. --- Quotient space (topology). --- Radon measure. --- Reductive group. --- Representation of a Lie group. --- Representation theorem. --- Representation theory. --- Ring homomorphism. --- Schwartz space. --- Semisimple algebra. --- Separable extension. --- Sesquilinear form. --- Set (mathematics). --- Sign (mathematics). --- Square-integrable function. --- Sub"ient. --- Subalgebra. --- Subgroup. --- Subset. --- Summation. --- Support (mathematics). --- Surjective function. --- Tempered representation. --- Tensor product. --- Theorem. --- Topological group. --- Topological space. --- Topology. --- Trace (linear algebra). --- Transitive relation. --- Unitary representation. --- Universal enveloping algebra. --- Variable (mathematics). --- Vector space. --- Analyse harmonique (mathématiques)