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Introduction To Abelian Model Structures And Gorenstein Homological Dimensions

Author: Marco A. P. Bullones
Publisher: CRC Press
ISBN: 149872535X
Size: 77.74 MB
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Introduction to Abelian Model Structures and Gorenstein Homological Dimensions provides a starting point to study the relationship between homological and homotopical algebra, a very active branch of mathematics. The book shows how to obtain new model structures in homological algebra by constructing a pair of compatible complete cotorsion pairs related to a specific homological dimension and then applying the Hovey Correspondence to generate an abelian model structure. The first part of the book introduces the definitions and notations of the universal constructions most often used in category theory. The next part presents a proof of the Eklof and Trlifaj theorem in Grothedieck categories and covers M. Hovey’s work that connects the theories of cotorsion pairs and model categories. The final two parts study the relationship between model structures and classical and Gorenstein homological dimensions and explore special types of Grothendieck categories known as Gorenstein categories. As self-contained as possible, this book presents new results in relative homological algebra and model category theory. The author also re-proves some established results using different arguments or from a pedagogical point of view. In addition, he proves folklore results that are difficult to locate in the literature.

Mathematical Modelling Of Waves In Multi Scale Structured Media

Author: Alexander B. Movchan
Publisher: CRC Press
ISBN: 1498782108
Size: 35.13 MB
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Mathematical Modelling of Waves in Multi-Scale Structured Media presents novel analytical and numerical models of waves in structured elastic media, with emphasis on the asymptotic analysis of phenomena such as dynamic anisotropy, localisation, filtering and polarisation as well as on the modelling of photonic, phononic, and platonic crystals.

Delay Differential Evolutions Subjected To Nonlocal Initial Conditions

Author: Monica-Dana Burlică
Publisher: CRC Press
ISBN: 1498746462
Size: 62.34 MB
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Filling a gap in the literature, Delay Differential Evolutions Subjected to Nonlocal Initial Conditions reveals important results on ordinary differential equations (ODEs) and partial differential equations (PDEs). It presents very recent results relating to the existence, boundedness, regularity, and asymptotic behavior of global solutions for differential equations and inclusions, with or without delay, subjected to nonlocal implicit initial conditions. After preliminaries on nonlinear evolution equations governed by dissipative operators, the book gives a thorough study of the existence, uniqueness, and asymptotic behavior of global bounded solutions for differential equations with delay and local initial conditions. It then focuses on two important nonlocal cases: autonomous and quasi-autonomous. The authors next discuss sufficient conditions for the existence of almost periodic solutions, describe evolution systems with delay and nonlocal initial conditions, examine delay evolution inclusions, and extend some results to the multivalued case of reaction-diffusion systems. The book concludes with results on viability for nonlocal evolution inclusions.

Monomial Algebras Second Edition

Author: Rafael Villarreal
Publisher: CRC Press
ISBN: 148223470X
Size: 15.43 MB
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Monomial Algebras, Second Edition presents algebraic, combinatorial, and computational methods for studying monomial algebras and their ideals, including Stanley–Reisner rings, monomial subrings, Ehrhart rings, and blowup algebras. It emphasizes square-free monomials and the corresponding graphs, clutters, or hypergraphs. New to the Second Edition Four new chapters that focus on the algebraic properties of blowup algebras in combinatorial optimization problems of clutters and hypergraphs Two new chapters that explore the algebraic and combinatorial properties of the edge ideal of clutters and hypergraphs Full revisions of existing chapters to provide an up-to-date account of the subject Bringing together several areas of pure and applied mathematics, this book shows how monomial algebras are related to polyhedral geometry, combinatorial optimization, and combinatorics of hypergraphs. It directly links the algebraic properties of monomial algebras to combinatorial structures (such as simplicial complexes, posets, digraphs, graphs, and clutters) and linear optimization problems.

Cohen Macaulay Representations

Author: Graham J. Leuschke
Publisher: American Mathematical Soc.
ISBN: 0821875817
Size: 16.46 MB
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This book is a comprehensive treatment of the representation theory of maximal Cohen-Macaulay (MCM) modules over local rings. This topic is at the intersection of commutative algebra, singularity theory, and representations of groups and algebras. Two introductory chapters treat the Krull-Remak-Schmidt Theorem on uniqueness of direct-sum decompositions and its failure for modules over local rings. Chapters 3-10 study the central problem of classifying the rings with only finitely many indecomposable MCM modules up to isomorphism, i.e., rings of finite CM type. The fundamental material--ADE/simple singularities, the double branched cover, Auslander-Reiten theory, and the Brauer-Thrall conjectures--is covered clearly and completely. Much of the content has never before appeared in book form. Examples include the representation theory of Artinian pairs and Burban-Drozd's related construction in dimension two, an introduction to the McKay correspondence from the point of view of maximal Cohen-Macaulay modules, Auslander-Buchweitz's MCM approximation theory, and a careful treatment of nonzero characteristic. The remaining seven chapters present results on bounded and countable CM type and on the representation theory of totally reflexive modules.

Fundamental Algebraic Geometry

Author: Barbara Fantechi
Publisher: American Mathematical Soc.
ISBN: 0821842455
Size: 78.45 MB
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Alexander Grothendieck introduced many concepts into algebraic geometry; they turned out to be astoundingly powerful and productive and truly revolutionized the subject. Grothendieck sketched his new theories in a series of talks at the Seminaire Bourbaki between 1957 and 1962 and collected his write-ups in a volume entitled ``Fondements de la Geometrie Algebrique,'' known as FGA. Much of FGA is now common knowledge; however, some of FGA is less well known, and its full scope is familiar to few. The present book resulted from the 2003 ``Advanced School in Basic Algebraic Geometry'' at the ICTP in Trieste, Italy. The book aims to fill in Grothendieck's brief sketches. There are four themes: descent theory, Hilbert and Quot schemes, the formal existence theorem, and the Picard scheme. Most results are proved in full detail; furthermore, newer ideas are introduced to promote understanding, and many connections are drawn to newer developments. The main prerequisite is a thorough acquaintance with basic scheme theory. Thus this book is a valuable resource for anyone doing algebraic geometry.

Abelian Groups Rings Modules And Homological Algebra

Author: Pat Goeters
Publisher: CRC Press
ISBN: 9781420010763
Size: 30.16 MB
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About the book... In honor of Edgar Enochs and his venerable contributions to a broad range of topics in Algebra, top researchers from around the world gathered at Auburn University to report on their latest work and exchange ideas on some of today's foremost research topics. This carefully edited volume presents the refereed papers of the participants of these talks along with contributions from other veteran researchers who were unable to attend. These papers reflect many of the current topics in Abelian Groups, Commutative Algebra, Commutative Rings, Group Theory, Homological Algebra, Lie Algebras, and Module Theory. Accessible even to beginning mathematicians, many of these articles suggest problems and programs for future study. This volume is an outstanding addition to the literature and a valuable handbook for beginning as well as seasoned researchers in Algebra. about the editors... H. PAT GOETERS completed his undergraduate studies in mathematics and computer science at Southern Connecticut State University and received his Ph.D. in 1984 from the University of Connecticut under the supervision of William J. Wickless. After spending one year in a post-doctoral position in Wesleyan University under the tutelage of James D. Reid, Goeters was invited for a tenure track position in Auburn University by Ulrich F. Albrecht. Soon afterwards, William Ullery and Overtoun Jenda were hired, and so began a lively Algebra group. OVERTOUN M. G. JENDA received his bachelor's degree in Mathematics from Chancellor College, the University of Malawi. He moved to the U.S. 1977 to pursue graduate studies at University of Kentucky, earning his Ph.D. in 1981 under the supervision of Professor Edgar Enochs. He then returned to Chancellor College, where he was a lecturer (assistant professor) for three years. He moved to the University of Botswana for another three-year stint as a lecturer before moving back to the University of Kentucky as a visiting assistant professor in 1987. In 1988, he joined the Algebra research group at Auburn University.

Gorenstein Dimensions

Author: Lars W. Christensen
Publisher: Springer
ISBN: 3540400087
Size: 12.11 MB
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This book is intended as a reference for mathematicians working with homological dimensions in commutative algebra and as an introduction to Gorenstein dimensions for graduate students with an interest in the same. Any admirer of classics like the Auslander-Buchsbaum-Serre characterization of regular rings, and the Bass and Auslander-Buchsbaum formulas for injective and projective dimension of f.g. modules will be intrigued by this book's content. Readers should be well-versed in commutative algebra and standard applications of homological methods. The framework is that of complexes, but all major results are restated for modules in traditional notation, and an appendix makes the proofs accessible for even the casual user of hyperhomological methods.

Dirichlet Branes And Mirror Symmetry

Publisher: American Mathematical Soc.
ISBN: 0821838482
Size: 77.89 MB
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Research in string theory over the last several decades has yielded a rich interaction with algebraic geometry. In 1985, the introduction of Calabi-Yau manifolds into physics as a way to compactify ten-dimensional space-time has led to exciting cross-fertilization between physics and mathematics, especially with the discovery of mirror symmetry in 1989. A new string revolution in the mid-1990s brought the notion of branes to the forefront. As foreseen by Kontsevich, these turned out to have mathematical counterparts in the derived category of coherent sheaves on an algebraic variety and the Fukaya category of a symplectic manifold. This has led to exciting new work, including the Strominger-Yau-Zaslow conjecture, which used the theory of branes to propose a geometric basis for mirror symmetry, the theory of stability conditions on triangulated categories, and a physical basis for the McKay correspondence. These developments have led to a great deal of new mathematical work. One difficulty in understanding all aspects of this work is that it requires being able to speak two different languages, the language of string theory and the language of algebraic geometry. The 2002 Clay School on Geometry and String Theory set out to bridge this gap, and this monograph builds on the expository lectures given there to provide an up-to-date discussion including subsequent developments. A natural sequel to the first Clay monograph on Mirror Symmetry, it presents the new ideas coming out of the interactions of string theory and algebraic geometry in a coherent logical context. We hope it will allow students and researchers who are familiar with the language of one of the two fields to gain acquaintance with the language of the other. The book first introduces the notion of Dirichlet brane in the context of topological quantum field theories, and then reviews the basics of string theory. After showing how notions of branes arose in string theory, it turns to an introduction to the algebraic geometry, sheaf theory, and homological algebra needed to define and work with derived categories. The physical existence conditions for branes are then discussed and compared in the context of mirror symmetry, culminating in Bridgeland's definition of stability structures, and its applications to the McKay correspondence and quantum geometry. The book continues with detailed treatments of the Strominger-Yau-Zaslow conjecture, Calabi-Yau metrics and homological mirror symmetry, and discusses more recent physical developments. This book is suitable for graduate students and researchers with either a physics or mathematics background, who are interested in the interface between string theory and algebraic geometry.

Interactions Between Homotopy Theory And Algebra

Author: Luchezar L. Avramov
Publisher: American Mathematical Soc.
ISBN: 0821838148
Size: 63.38 MB
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This book is based on talks presented at the Summer School on Interactions between Homotopy theory and Algebra held at the University of Chicago in the summer of 2004. The goal of this book is to create a resource for background and for current directions of research related to deep connections between homotopy theory and algebra, including algebraic geometry, commutative algebra, and representation theory. The articles in this book are aimed at the audience of beginning researchers with varied mathematical backgrounds and have been written with both the quality of exposition and the accessibility to novices in mind.