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Written and revised by D. B. A. Epstein.

Category theory. Homological algebra --- 515.14 --- Algebraic topology --- Homology theory. --- 515.14 Algebraic topology --- Cohomology theory --- Contrahomology theory --- Algebra homomorphism. --- Algebra over a field. --- Algebraic structure. --- Approximation. --- Axiom. --- Basis (linear algebra). --- CW complex. --- Cartesian product. --- Classical group. --- Coefficient. --- Cohomology operation. --- Cohomology ring. --- Cohomology. --- Commutative property. --- Complex number. --- Computation. --- Continuous function. --- Cup product. --- Cyclic group. --- Diagram (category theory). --- Dimension. --- Direct limit. --- Embedding. --- Existence theorem. --- Fibration. --- Homomorphism. --- Hopf algebra. --- Hopf invariant. --- Ideal (ring theory). --- Integer. --- Inverse limit. --- Manifold. --- Mathematics. --- Monomial. --- N-skeleton. --- Natural transformation. --- Permutation. --- Quaternion. --- Ring (mathematics). --- Scalar (physics). --- Special unitary group. --- Steenrod algebra. --- Stiefel manifold. --- Subgroup. --- Subset. --- Summation. --- Symmetric group. --- Symplectic group. --- Theorem. --- Uniqueness theorem. --- Upper and lower bounds. --- Vector field. --- Vector space. --- W0.

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One of the most exciting new subjects in Algebraic Number Theory and Arithmetic Algebraic Geometry is the theory of Euler systems. Euler systems are special collections of cohomology classes attached to p-adic Galois representations. Introduced by Victor Kolyvagin in the late 1980's in order to bound Selmer groups attached to p-adic representations, Euler systems have since been used to solve several key problems. These include certain cases of the Birch and Swinnerton-Dyer Conjecture and the Main Conjecture of Iwasawa Theory. Because Selmer groups play a central role in Arithmetic Algebraic Geometry, Euler systems should be a powerful tool in the future development of the field. Here, in the first book to appear on the subject, Karl Rubin presents a self-contained development of the theory of Euler systems. Rubin first reviews and develops the necessary facts from Galois cohomology. He then introduces Euler systems, states the main theorems, and develops examples and applications. The remainder of the book is devoted to the proofs of the main theorems as well as some further speculations. The book assumes a solid background in algebraic Number Theory, and is suitable as an advanced graduate text. As a research monograph it will also prove useful to number theorists and researchers in Arithmetic Algebraic Geometry.

Algebraic number theory. --- p-adic numbers. --- Numbers, p-adic --- Number theory --- p-adic analysis --- Galois cohomology --- Cohomologie galoisienne. --- Algebraic number theory --- p-adic numbers --- Abelian extension. --- Abelian variety. --- Absolute Galois group. --- Algebraic closure. --- Barry Mazur. --- Big O notation. --- Birch and Swinnerton-Dyer conjecture. --- Cardinality. --- Class field theory. --- Coefficient. --- Cohomology. --- Complex multiplication. --- Conjecture. --- Corollary. --- Cyclotomic field. --- Dimension (vector space). --- Divisibility rule. --- Eigenvalues and eigenvectors. --- Elliptic curve. --- Error term. --- Euler product. --- Euler system. --- Exact sequence. --- Existential quantification. --- Field of fractions. --- Finite set. --- Functional equation. --- Galois cohomology. --- Galois group. --- Galois module. --- Gauss sum. --- Global field. --- Heegner point. --- Ideal class group. --- Integer. --- Inverse limit. --- Inverse system. --- Karl Rubin. --- Local field. --- Mathematical induction. --- Maximal ideal. --- Modular curve. --- Modular elliptic curve. --- Natural number. --- Orthogonality. --- P-adic number. --- Pairing. --- Principal ideal. --- R-factor (crystallography). --- Ralph Greenberg. --- Remainder. --- Residue field. --- Ring of integers. --- Scientific notation. --- Selmer group. --- Subgroup. --- Tate module. --- Taylor series. --- Tensor product. --- Theorem. --- Upper and lower bounds. --- Victor Kolyvagin. --- Courbes elliptiques --- Nombres, Théorie des

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The description for this book, Convergence and Uniformity in Topology. (AM-2), Volume 2, will be forthcoming.

Topology. --- Absolute value. --- Abstract algebra. --- Algebraic topology. --- Axiom of choice. --- Binary relation. --- Cardinal number. --- Characteristic function (probability theory). --- Closed set. --- Closure operator. --- Combinatorial topology. --- Compact space. --- Complete lattice. --- Complete metric space. --- Continuous function (set theory). --- Continuous function. --- Countable set. --- Counterexample. --- Dimension theory (algebra). --- Dimension theory. --- Discrete space. --- Domain of a function. --- Empty set. --- Enumeration. --- Equivalence class. --- Equivalence relation. --- Existential quantification. --- Family of sets. --- Finite set. --- General topology. --- Geometry. --- Hahn–Banach theorem. --- Hausdorff space. --- Homeomorphism. --- Infimum and supremum. --- Integer. --- Interval (mathematics). --- Lebesgue constant (interpolation). --- Limit point. --- Linear space (geometry). --- Mathematician. --- Mathematics. --- Maximal element. --- Metric space. --- Monotonic function. --- Mutual exclusivity. --- Natural number. --- Negation. --- Normal space. --- Open set. --- Ordinal number. --- Real number. --- Regular space. --- Requirement. --- Scientific notation. --- Separation axiom. --- Set (mathematics). --- Set theory. --- Special case. --- Subsequence. --- Subset. --- Suggestion. --- Summation. --- Superspace. --- Theorem. --- Theory. --- Topological algebra. --- Total order. --- Transfinite induction. --- Transfinite number. --- Transfinite. --- Transitive relation. --- Tychonoff space. --- Ultrafilter. --- Uncountable set. --- Uniform continuity. --- Union (set theory). --- Upper and lower bounds. --- Zorn's lemma.

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The description for this book, The Calculi of Lambda Conversion. (AM-6), Volume 6, will be forthcoming.

Logic, Symbolic and mathematical. --- Recursive functions. --- 2H. --- A-normal form. --- Addition. --- Alphabetical order. --- Ambiguity. --- Argument of a function. --- Axiom. --- Bibliography. --- Big O notation. --- Calculation. --- Characteristic function (probability theory). --- Combination. --- Complex number. --- Computability. --- Computation. --- Consistency. --- Corollary. --- Definition. --- Denotation. --- Determination. --- Differential calculus. --- Enumeration. --- Equation. --- Exc. --- Existential quantification. --- Exponentiation. --- Finitary. --- Finite set. --- Formal system. --- Frege (programming language). --- Function (mathematics). --- Gödel numbering. --- Identity function. --- In the process of. --- Integer. --- Iteration. --- Limit (mathematics). --- Logic. --- Logical conjunction. --- Logical disjunction. --- Mathematical induction. --- Mathematical logic. --- Mathematics. --- Metamathematics. --- Natural number. --- Negation. --- Notation. --- Null set. --- Number theory. --- Ordinal number. --- Pairing. --- Paul Bernays. --- Primitive recursive function. --- Principia Mathematica. --- Propositional function. --- Quantifier (logic). --- Real number. --- Recursion (computer science). --- Recursion. --- Reduction of order. --- Requirement. --- Resultant. --- Rule of inference. --- Scientific notation. --- Sequence. --- Set theory. --- Special case. --- Successor function. --- Theorem. --- Theory. --- Transfinite number. --- Transfinite. --- Truth value. --- Uncertainty. --- Universal quantification. --- Upper and lower bounds. --- Variable (mathematics). --- Well-formed formula. --- Without loss of generality.

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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.