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Algebraic cycles

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Noncommutative geometry combines themes from algebra, analysis and geometry and has significant applications to physics. This book focuses on cyclic theory, and is based upon the lecture courses by Daniel G. Quillen at the University of Oxford from 1988-92, which developed his own approach to the subject. The basic definitions, examples and exercises provided here allow non-specialists and students with a background in elementary functional analysis, commutative algebra and differential geometry to get to grips with the subject. Quillen's development of cyclic theory emphasizes analogies between commutative and noncommutative theories, in which he reinterpreted classical results of Hamiltonian mechanics, operator algebras and differential graded algebras into a new formalism. In this book, cyclic theory is developed from motivating examples and background towards general results. Themes covered are relevant to current research, including homomorphisms modulo powers of ideals, traces on noncommutative differential forms, quasi-free algebras and Chern characters on connections.

Algebraic cycles.

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This is a collection of lecture notes from the Summer School 'Cycles Algébriques; Aspects Transcendents, Grenoble 2001'. The topics range from introductory lectures on algebraic cycles to more advanced material. The advanced lectures are grouped under three headings: Lawson (co)homology, motives and motivic cohomology and Hodge theoretic invariants of cycles. Among the topics treated are: cycle spaces, Chow topology, morphic cohomology, Grothendieck motives, Chow-Künneth decompositions of the diagonal, motivic cohomology via higher Chow groups, the Hodge conjecture for certain fourfolds, an effective version of Nori's connectivity theorem, Beilinson's Hodge and Tate conjecture for open complete intersections. As the lectures were intended for non-specialists many examples have been included to illustrate the theory. As such this book will be ideal for graduate students or researchers seeking a modern introduction to the state-of-the-art theory in this subject.

Algebraic cycles --- Cycles, Algebraic --- Geometry, Algebraic

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Algebraic geometry is a central subfield of mathematics in which the study of cycles is an important theme. Alexander Grothendieck taught that algebraic cycles should be considered from a motivic point of view and in recent years this topic has spurred a lot of activity. This 2007 book is one of two volumes that provide a self-contained account of the subject. Together, the two books contain twenty-two contributions from leading figures in the field which survey the key research strands and present interesting new results. Topics discussed include: the study of algebraic cycles using Abel-Jacobi/regulator maps and normal functions; motives (Voevodsky's triangulated category of mixed motives, finite-dimensional motives); the conjectures of Bloch-Beilinson and Murre on filtrations on Chow groups and Bloch's conjecture. Researchers and students in complex algebraic geometry and arithmetic geometry will find much of interest here.

Algebraic cycles --- Motives (Mathematics) --- Theory of motives (Mathematics) --- Algebraic varieties --- Homology theory --- Cycles, Algebraic --- Geometry, Algebraic

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The behaviour of vanishing cycles is the cornerstone for understanding the geometry and topology of families of hypersurfaces, usually regarded as singular fibrations. This self-contained tract proposes a systematic geometro-topological approach to vanishing cycles, especially those appearing in non-proper fibrations, such as the fibration defined by a polynomial function. Topics which have been the object of active research over the past 15 years, such as holomorphic germs, polynomial functions, and Lefschetz pencils on quasi-projective spaces, are here shown in a new light: conceived as aspects of a single theory with vanishing cycles at its core. Throughout the book the author presents the current state of the art. Transparent proofs are provided so that non-specialists can use this book as an introduction, but all researchers and graduate students working in differential and algebraic topology, algebraic geometry, and singularity theory will find this book of great use.

Polynomials --- Algebraic cycles --- Vanishing theorems --- Hypersurfaces --- SINGULARITIES (Mathematics) --- Polynomials. --- Mathematics. --- Math --- Science --- Algebra --- Algebraic cycles. --- Vanishing theorems. --- Hypersurfaces. --- Singularities (Mathematics) --- Geometry, Algebraic --- Hyperspace --- Surfaces --- Theorems, Vanishing --- Complex manifolds --- Fiber bundles (Mathematics) --- Homology theory --- Vector bundles --- Cycles, Algebraic