Mixed Motives
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Mixed Motives
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Author : Marc Levine
language : en
Publisher: American Mathematical Soc.
Release Date : 1998
Mixed Motives written by Marc Levine and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998 with Mathematics categories.
This book combines foundational constructions in the theory of motives and results relating motivic cohomology to more explicit constructions. Prerequisite for understanding the work is a basic background in algebraic geometry. The author constructs and describes a triangulated category of mixed motives over an arbitrary base scheme. Most of the classical constructions of cohomology are described in the motivic setting, including Chern classes from higher $K$-theory, push-forward for proper maps, Riemann-Roch, duality, as well as an associated motivic homology, Borel-Moore homology and cohomology with compact supports.
Motives
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Author : Uwe Jannsen
language : en
Publisher: American Mathematical Soc.
Release Date : 1994
Motives written by Uwe Jannsen and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994 with Mathematics categories.
Motives were introduced in the mid-1960s by Grothendieck to explain the analogies among the various cohomology theories for algebraic varieties, to play the role of the missing rational cohomology, and to provide a blueprint for proving Weil's conjectures about the zeta function of a variety over a finite field. Over the last ten years or so, researchers in various areas--Hodge theory, algebraic $K$-theory, polylogarithms, automorphic forms, $L$-functions, $\ell$-adic representations, trigonometric sums, and algebraic cycles--have discovered that an enlarged (and in part conjectural) theory of ``mixed'' motives indicates and explains phenomena appearing in each area. Thus the theory holds the potential of enriching and unifying these areas. This is one of two volumes containing the revised texts of nearly all the lectures presented at the AMS-IMS-SIAM Joint Summer Research Conference on Motives, held in Seattle, in 1991. A number of related works are also included, making for a total of forty-seven papers, from general introductions to specialized surveys to research papers.
Triangulated Categories Of Mixed Motives
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Author : Denis-Charles Cisinski
language : en
Publisher: Springer Nature
Release Date : 2019-11-09
Triangulated Categories Of Mixed Motives written by Denis-Charles Cisinski and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-11-09 with Mathematics categories.
The primary aim of this monograph is to achieve part of Beilinson’s program on mixed motives using Voevodsky’s theories of A1-homotopy and motivic complexes. Historically, this book is the first to give a complete construction of a triangulated category of mixed motives with rational coefficients satisfying the full Grothendieck six functors formalism as well as fulfilling Beilinson’s program, in particular the interpretation of rational higher Chow groups as extension groups. Apart from Voevodsky’s entire work and Grothendieck’s SGA4, our main sources are Gabber’s work on étale cohomology and Ayoub’s solution to Voevodsky’s cross functors theory. We also thoroughly develop the theory of motivic complexes with integral coefficients over general bases, along the lines of Suslin and Voevodsky. Besides this achievement, this volume provides a complete toolkit for the study of systems of coefficients satisfying Grothendieck’ six functors formalism, including Grothendieck-Verdier duality. It gives a systematic account of cohomological descent theory with an emphasis on h-descent. It formalizes morphisms of coefficient systems with a view towards realization functors and comparison results. The latter allows to understand the polymorphic nature of rational mixed motives. They can be characterized by one of the following properties: existence of transfers, universality of rational algebraic K-theory, h-descent, étale descent, orientation theory. This monograph is a longstanding research work of the two authors. The first three parts are written in a self-contained manner and could be accessible to graduate students with a background in algebraic geometry and homotopy theory. It is designed to be a reference work and could also be useful outside motivic homotopy theory. The last part, containing the most innovative results, assumes some knowledge of motivic homotopy theory, although precise statements and references are given.
Mixed Motives And Their Realization In Derived Categories
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Author : Annette Huber
language : en
Publisher: Springer
Release Date : 2006-11-17
Mixed Motives And Their Realization In Derived Categories written by Annette Huber and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-17 with Mathematics categories.
The conjectural theory of mixed motives would be a universal cohomology theory in arithmetic algebraic geometry. The monograph describes the approach to motives via their well-defined realizations. This includes a review of several known cohomology theories. A new absolute cohomology is introduced and studied. The book assumes knowledge of the standard cohomological techniques in algebraic geometry as well as K-theory. So the monograph is primarily intended for researchers. Advanced graduate students can use it as a guide to the literature.
Mixed Motives And Algebraic K Theory
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Author : Uwe Jannsen
language : en
Publisher: Springer
Release Date : 2006-11-14
Mixed Motives And Algebraic K Theory written by Uwe Jannsen and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-14 with Mathematics categories.
The relations that could or should exist between algebraic cycles, algebraic K-theory, and the cohomology of - possibly singular - varieties, are the topic of investigation of this book. The author proceeds in an axiomatic way, combining the concepts of twisted Poincaré duality theories, weights, and tensor categories. One thus arrives at generalizations to arbitrary varieties of the Hodge and Tate conjectures to explicit conjectures on l-adic Chern characters for global fields and to certain counterexamples for more general fields. It is to be hoped that these relations ions will in due course be explained by a suitable tensor category of mixed motives. An approximation to this is constructed in the setting of absolute Hodge cycles, by extending this theory to arbitrary varieties. The book can serve both as a guide for the researcher, and as an introduction to these ideas for the non-expert, provided (s)he knows or is willing to learn about K-theory and the standard cohomology theories of algebraic varieties.
Mixed Motives
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Author : W. H. Watson
language : en
Publisher: Athena PressPub Company
Release Date : 2006-01
Mixed Motives written by W. H. Watson and has been published by Athena PressPub Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-01 with Fiction categories.
The author has created a set of appealing short stories by cleverly exploring the theme of human relationships played out in a variety of ways. The characters face some tough challenges in the stories about aid work in Africa and they must learn to pull together as a team. In 'The Rifle', a retired doctor recalls how he left his family behind to join a medical relief group and help the desperate Biafrans. He and his support staff find the lack of clean water is a mere inconvenience compared with the danger of being accused of gun-running! In complete contrast, 'A Hero' is a touching story of a little boy, Daniel, who falls hopelessly in love with his piano teacher. The teacher is unaware of the emotional turmoil she causes, especially as her attention is centred on her fiance. Daniel faces a real dilemma: how can he prove himself and get noticed? Each story in the collection is complete in itself and introduces new characters and new situations. Something for everyone! Born in 1921, WH Watson was wounded and captured during the commando raid on St Nazaire in 1942. After years in family practice he and his wife, Wyn, also a doctor, worked for charities over short periods in Somalia, Sierra Leone, Nepal, Afghanistan and India, and for two years in Malawi."
Noncommutative Geometry Quantum Fields And Motives
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Author : Alain Connes
language : en
Publisher: American Mathematical Soc.
Release Date : 2019-03-13
Noncommutative Geometry Quantum Fields And Motives written by Alain Connes and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-03-13 with categories.
The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role: space-time, where the guiding principle is the problem of developing a quantum theory of gravity, and the space of primes, where one can regard the Riemann Hypothesis as a long-standing problem motivating the development of new geometric tools. The book stresses the relevance of noncommutative geometry in dealing with these two spaces. The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative geometry. The main result is a complete derivation of the full Standard Model Lagrangian from a very simple mathematical input. Other topics covered in the first part of the book are a noncommutative geometry model of dimensional regularization and its role in anomaly computations, and a brief introduction to motives and their conjectural relation to quantum field theory. The second part of the book gives an interpretation of the Weil explicit formula as a trace formula and a spectral realization of the zeros of the Riemann zeta function. This is based on the noncommutative geometry of the adèle class space, which is also described as the space of commensurability classes of Q-lattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the space of Q-lattices, in one and two dimensions, exhibits spontaneous symmetry breaking. In the low-temperature regime, the equilibrium states of the corresponding systems are related to points of classical moduli spaces and the symmetries to the class field theory of the field of rational numbers and of imaginary quadratic fields, as well as to the automorphisms of the field of modular functions. The book ends with a set of analogies between the noncommutative geometries underlying the mathematical formulation of the Standard Model minimally coupled to gravity and the moduli spaces of Q-lattices used in the study of the zeta function.
Feynman Motives
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Author : Matilde Marcolli
language : en
Publisher: World Scientific
Release Date : 2010
Feynman Motives written by Matilde Marcolli and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010 with Science categories.
This book presents recent and ongoing research work aimed at understanding the mysterious relation between the computations of Feynman integrals in perturbative quantum field theory and the theory of motives of algebraic varieties and their periods. The main question is whether residues of Feynman integrals always evaluate to periods of mixed Tate motives, as appears to be the case from extensive computations of Feynman integrals carried out by Broadhurst and Kreimer. Two different approaches to the subject are described. The first, a "bottom-up" approach, constructs explicit algebraic varieties and periods from Feynman graphs and parametric Feynman integrals. This approach grew out of work of Bloch–Esnault–Kreimer and suggests that, while the algebraic varieties associated to the Feynman graphs can be arbitrarily complicated as motives, the part that is involved in the Feynman integral computation might still be of the special mixed Tate kind. A second, "top-down" approach to the problem, developed in the work of Connes and the author, consists of comparing a Tannakian category constructed out of the data of renormalization with those formed by mixed Tate motives. The book draws connections between these two approaches and gives an overview of various ongoing directions of research in the field. The text is aimed at researchers in mathematical physics, high energy physics, number theory and algebraic geometry. Based on lecture notes for a graduate course given by the author at Caltech in the fall of 2008, it cal also be used by graduate students interested in working in this area.
Noncommutative Motives
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Author : Gonçalo Tabuada
language : en
Publisher: American Mathematical Soc.
Release Date : 2015-09-21
Noncommutative Motives written by Gonçalo Tabuada and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-09-21 with Algebraic varieties categories.
The theory of motives began in the early 1960s when Grothendieck envisioned the existence of a "universal cohomology theory of algebraic varieties". The theory of noncommutative motives is more recent. It began in the 1980s when the Moscow school (Beilinson, Bondal, Kapranov, Manin, and others) began the study of algebraic varieties via their derived categories of coherent sheaves, and continued in the 2000s when Kontsevich conjectured the existence of a "universal invariant of noncommutative algebraic varieties". This book, prefaced by Yuri I. Manin, gives a rigorous overview of some of the main advances in the theory of noncommutative motives. It is divided into three main parts. The first part, which is of independent interest, is devoted to the study of DG categories from a homotopical viewpoint. The second part, written with an emphasis on examples and applications, covers the theory of noncommutative pure motives, noncommutative standard conjectures, noncommutative motivic Galois groups, and also the relations between these notions and their commutative counterparts. The last part is devoted to the theory of noncommutative mixed motives. The rigorous formalization of this latter theory requires the language of Grothendieck derivators, which, for the reader's convenience, is revised in a brief appendix.
Periods And Nori Motives
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Author : Annette Huber
language : en
Publisher: Springer
Release Date : 2017-03-08
Periods And Nori Motives written by Annette Huber and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-03-08 with Mathematics categories.
This book casts the theory of periods of algebraic varieties in the natural setting of Madhav Nori’s abelian category of mixed motives. It develops Nori’s approach to mixed motives from scratch, thereby filling an important gap in the literature, and then explains the connection of mixed motives to periods, including a detailed account of the theory of period numbers in the sense of Kontsevich-Zagier and their structural properties. Period numbers are central to number theory and algebraic geometry, and also play an important role in other fields such as mathematical physics. There are long-standing conjectures about their transcendence properties, best understood in the language of cohomology of algebraic varieties or, more generally, motives. Readers of this book will discover that Nori’s unconditional construction of an abelian category of motives (over fields embeddable into the complex numbers) is particularly well suited for this purpose. Notably, Kontsevich's formal period algebra represents a torsor under the motivic Galois group in Nori's sense, and the period conjecture of Kontsevich and Zagier can be recast in this setting. Periods and Nori Motives is highly informative and will appeal to graduate students interested in algebraic geometry and number theory as well as researchers working in related fields. Containing relevant background material on topics such as singular cohomology, algebraic de Rham cohomology, diagram categories and rigid tensor categories, as well as many interesting examples, the overall presentation of this book is self-contained.