Syzygies and Homotopy Theory

Author: F.E.A. Johnson

Publisher: Springer Science & Business Media

ISBN: 9781447122944

Category: Mathematics

Page: 296

View: 5570

The most important invariant of a topological space is its fundamental group. When this is trivial, the resulting homotopy theory is well researched and familiar. In the general case, however, homotopy theory over nontrivial fundamental groups is much more problematic and far less well understood. Syzygies and Homotopy Theory explores the problem of nonsimply connected homotopy in the first nontrivial cases and presents, for the first time, a systematic rehabilitation of Hilbert's method of syzygies in the context of non-simply connected homotopy theory. The first part of the book is theoretical, formulated to allow a general finitely presented group as a fundamental group. The innovation here is to regard syzygies as stable modules rather than minimal modules. Inevitably this forces a reconsideration of the problems of noncancellation; these are confronted in the second, practical, part of the book. In particular, the second part of the book considers how the theory works out in detail for the specific examples Fn ́F where Fn is a free group of rank n and F is finite. Another innovation is to parametrize the first syzygy in terms of the more familiar class of stably free modules. Furthermore, detailed description of these stably free modules is effected by a suitable modification of the method of Milnor squares. The theory developed within this book has potential applications in various branches of algebra, including homological algebra, ring theory and K-theory. Syzygies and Homotopy Theory will be of interest to researchers and also to graduate students with a background in algebra and algebraic topology.

Advances in Two-Dimensional Homotopy and Combinatorial Group Theory

Author: Wolfgang Metzler,Stephan Rosebrock

Publisher: Cambridge University Press

ISBN: 1108640869

Category: Mathematics

Page: 179

View: 2807

This volume presents the current state of knowledge in all aspects of two-dimensional homotopy theory. Building on the foundations laid a quarter of a century ago in the volume Two-dimensional Homotopy and Combinatorial Group Theory (LMS 197), the editors here bring together much remarkable progress that has been obtained in the intervening years. And while the fundamental open questions, such as the Andrews–Curtis Conjecture and the Whitehead asphericity problem remain to be (fully) solved, this book will provide both students and experts with an overview of the state of the art and work in progress. Ample references are included to the LMS 197 volume, as well as a comprehensive bibliography bringing matters entirely up to date.

Interactions Between Homotopy Theory and Algebra

Summer School on Interactions Between Homotopy Theory and Algebra, University of Chicago, July 26-August 6, 2004, Chicago, Illinois

Author: Luchezar L. Avramov

Publisher: American Mathematical Soc.

ISBN: 0821838148

Category: Mathematics

Page: 334

View: 5909

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.

The Geometry of Syzygies

A Second Course in Algebraic Geometry and Commutative Algebra

Author: David Eisenbud

Publisher: Springer Science & Business Media

ISBN: 0387264566

Category: Mathematics

Page: 246

View: 5314

First textbook-level account of basic examples and techniques in this area. Suitable for self-study by a reader who knows a little commutative algebra and algebraic geometry already. David Eisenbud is a well-known mathematician and current president of the American Mathematical Society, as well as a successful Springer author.

Two-Dimensional Homotopy and Combinatorial Group Theory

Author: Cynthia Hog-Angeloni,Wolfgang Metzler,Allan J. Sieradski

Publisher: Cambridge University Press

ISBN: 9780521447003

Category: Mathematics

Page: 412

View: 4688

Basic work on two-dimensional homotopy theory dates back to K. Reidemeister and J. H. C. Whitehead. Much work in this area has been done since then, and this book considers the current state of knowledge in all the aspects of the subject. The editors start with introductory chapters on low-dimensional topology, covering both the geometric and algebraic sides of the subject, the latter including crossed modules, Reidemeister-Peiffer identities, and a concrete and modern discussion of Whitehead's algebraic classification of 2-dimensional homotopy types. Further chapters have been skilfully selected and woven together to form a coherent picture. The latest algebraic results and their applications to 3- and 4-dimensional manifolds are dealt with. The geometric nature of the subject is illustrated to the full by over 100 diagrams. Final chapters summarize and contribute to the present status of the conjectures of Zeeman, Whitehead, and Andrews-Curtis. No other book covers all these topics. Some of the material here has been used in courses, making this book valuable for anyone with an interest in two-dimensional homotopy theory, from graduate students to research workers.

Graded Syzygies

Author: Irena Peeva

Publisher: Springer Science & Business Media

ISBN: 9780857291776

Category: Mathematics

Page: 304

View: 3623

The study of free resolutions is a core and beautiful area in Commutative Algebra. The main goal of this book is to inspire the readers and develop their intuition about syzygies and Hilbert functions. Many examples are given in order to illustrate ideas and key concepts. A valuable feature of the book is the inclusion of open problems and conjectures; these provide a glimpse of exciting, and often challenging, research directions in the field. Three types of problems are presented: Conjectures, Problems, and Open-Ended Problems. The latter do not describe specific problems but point to interesting directions for exploration. The first part of the monograph contains basic background material on graded free resolutions. Further coverage of topics includes syzygies over a polynomial ring, resolutions over quotient rings, lex ideals and Hilbert functions, compression, resolutions of monomial ideals, and syzygies of toric ideals. With a clear and self-contained exposition this text is intended for advanced graduate students and postdoctorates; it will be also of interest to senior mathematicians.

Mathematics Mechanization and Applications

Author: Xiao-Shan Gao

Publisher: N.A

ISBN: 9780127347608

Category: Computers

Page: 551

View: 5470

Mathematics Mechanization and Applications provides a uniform presentation of major developments, carried out mostly in Wu's extended Chinese group, on algorithms and software tools for mechanizing algebraic equations solving and geometric theorem proving together with their applications to problems in science and engineering. It is distinguished by its uniform presentation with all-Chinese contributors and a 40-page list of references. There are 20 chapters written by experienced researchers. The book is divided into four parts: polynomial system solving, automated geometric reasoning, algebraic computation, and implementations and applications. Each chapter is devoted to surveying and expounding the main results achieved from one selected subject. The book contains surveys for diverse applications of the theories and methods to real world problems, ranging from the analysis of robotics and mechanisms to nonlinear programming and chemical equilibrium computation. Part of the theoretical and practical work reviewed in the book has been either unpublished or published only in Chinese journals or even only in the Chinese language. This book therefore provides Western readers working in symbolic and algebraic computation, geometric reasoning and modeling, algorithmic mathematics, robotics, CAGD, and other relevant areas with an easily accessible source of references for what the Chinese researchers have been doing under the banner of mathematics mechanization. * Addresses the frontiers of research with original ideas and results * Includes sophisticated, successful applications to scientific and engineering problems * Covers polynomial system solving, geometric reasoning, computer algebra, and mathematical software * Is comprehensive and focused * Contains an extensive bibliography--of high reference value--particularly for western readers

An Introduction to Homological Algebra

Author: Charles A. Weibel

Publisher: Cambridge University Press

ISBN: 113964307X

Category: Mathematics

Page: N.A

View: 2459

The landscape of homological algebra has evolved over the last half-century into a fundamental tool for the working mathematician. This book provides a unified account of homological algebra as it exists today. The historical connection with topology, regular local rings, and semi-simple Lie algebras are also described. This book is suitable for second or third year graduate students. The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences. Homology of group and Lie algebras illustrate these topics. Intermingled are less canonical topics, such as the derived inverse limit functor lim1, local cohomology, Galois cohomology, and affine Lie algebras. The last part of the book covers less traditional topics that are a vital part of the modern homological toolkit: simplicial methods, Hochschild and cyclic homology, derived categories and total derived functors. By making these tools more accessible, the book helps to break down the technological barrier between experts and casual users of homological algebra.

Commutative Algebra and Algebraic Geometry

Joint International Meeting of the American Mathematical Society and the Indian Mathematical Society on Commutative Algebra and Algebraic Geometry, Bangalore, India, December 17-20, 2003

Author: Sudhir Ghorpade,Hema Srinivasan,Jugal Verma

Publisher: American Mathematical Soc.

ISBN: 0821836293

Category: Mathematics

Page: 173

View: 509

The first Joint AMS-India Mathematics Meeting was held in Bangalore (India). This book presents articles written by speakers from a special session on commutative algebra and algebraic geometry. Included are contributions from some leading researchers around the world in this subject area. The volume contains new and original research papers and survey articles suitable for graduate students and researchers interested in commutative algebra and algebraic geometry.

A Course in Homological Algebra

Author: P.J. Hilton,Urs Stammbach

Publisher: Springer Science & Business Media

ISBN: 146849936X

Category: Mathematics

Page: 340

View: 1397

In this chapter we are largely influenced in our choice of material by the demands of the rest of the book. However, we take the view that this is an opportunity for the student to grasp basic categorical notions which permeate so much of mathematics today, including, of course, algebraic topology, so that we do not allow ourselves to be rigidly restricted by our immediate objectives. A reader totally unfamiliar with category theory may find it easiest to restrict his first reading of Chapter II to Sections 1 to 6; large parts of the book are understandable with the material presented in these sections. Another reader, who had already met many examples of categorical formulations and concepts might, in fact, prefer to look at Chapter II before reading Chapter I. Of course the reader thoroughly familiar with category theory could, in principal, omit Chapter II, except perhaps to familiarize himself with the notations employed. In Chapter III we begin the proper study of homological algebra by looking in particular at the group ExtA(A, B), where A and Bare A-modules. It is shown how this group can be calculated by means of a projective presentation of A, or an injective presentation of B; and how it may also be identified with the group of equivalence classes of extensions of the quotient module A by the submodule B.

Geometry and Complexity Theory

Author: J. M. Landsberg

Publisher: Cambridge University Press

ISBN: 110819141X

Category: Computers

Page: N.A

View: 1172

Two central problems in computer science are P vs NP and the complexity of matrix multiplication. The first is also a leading candidate for the greatest unsolved problem in mathematics. The second is of enormous practical and theoretical importance. Algebraic geometry and representation theory provide fertile ground for advancing work on these problems and others in complexity. This introduction to algebraic complexity theory for graduate students and researchers in computer science and mathematics features concrete examples that demonstrate the application of geometric techniques to real world problems. Written by a noted expert in the field, it offers numerous open questions to motivate future research. Complexity theory has rejuvenated classical geometric questions and brought different areas of mathematics together in new ways. This book will show the beautiful, interesting, and important questions that have arisen as a result.

Syzygies and Hilbert Functions

Author: Irena Peeva

Publisher: CRC Press

ISBN: 9781420050912

Category: Mathematics

Page: 304

View: 1355

Hilbert functions and resolutions are both central objects in commutative algebra and fruitful tools in the fields of algebraic geometry, combinatorics, commutative algebra, and computational algebra. Spurred by recent research in this area, Syzygies and Hilbert Functions explores fresh developments in the field as well as fundamental concepts. Written by international mathematics authorities, the book first examines the invariant of Castelnuovo-Mumford regularity, blowup algebras, and bigraded rings. It then outlines the current status of two challenging conjectures: the lex-plus-power (LPP) conjecture and the multiplicity conjecture. After reviewing results of the geometry of Hilbert functions, the book considers minimal free resolutions of integral subschemes and of equidimensional Cohen-Macaulay subschemes of small degree. It also discusses relations to subspace arrangements and the properties of the infinite graded minimal free resolution of the ground field over a projective toric ring. The volume closes with an introduction to multigraded Hilbert functions, mixed multiplicities, and joint reductions. By surveying exciting topics of vibrant current research, Syzygies and Hilbert Functions stimulates further study in this hot area of mathematical activity.

Category Theory in Context

Author: Emily Riehl

Publisher: Courier Dover Publications

ISBN: 0486820807

Category: Mathematics

Page: 272

View: 1045

Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.

Tensors

Geometry and Applications

Author: J. M. Landsberg

Publisher: American Mathematical Soc.

ISBN: 0821869078

Category: Mathematics

Page: 439

View: 9243

Tensors are ubiquitous in the sciences. The geometry of tensors is both a powerful tool for extracting information from data sets, and a beautiful subject in its own right. This book has three intended uses: a classroom textbook, a reference work for researchers in the sciences, and an account of classical and modern results in (aspects of) the theory that will be of interest to researchers in geometry. For classroom use, there is a modern introduction to multilinear algebra and to the geometry and representation theory needed to study tensors, including a large number of exercises. For researchers in the sciences, there is information on tensors in table format for easy reference and a summary of the state of the art in elementary language. This is the first book containing many classical results regarding tensors. Particular applications treated in the book include the complexity of matrix multiplication, P versus NP, signal processing, phylogenetics, and algebraic statistics. For geometers, there is material on secant varieties, G-varieties, spaces with finitely many orbits and how these objects arise in applications, discussions of numerous open questions in geometry arising in applications, and expositions of advanced topics such as the proof of the Alexander-Hirschowitz theorem and of the Weyman-Kempf method for computing syzygies.

Colored Operads

Author: Donald Yau

Publisher: American Mathematical Soc.

ISBN: 1470427230

Category: Algebra, Homological

Page: 428

View: 1672

The subject of this book is the theory of operads and colored operads, sometimes called symmetric multicategories. A (colored) operad is an abstract object which encodes operations with multiple inputs and one output and relations between such operations. The theory originated in the early 1970s in homotopy theory and quickly became very important in algebraic topology, algebra, algebraic geometry, and even theoretical physics (string theory). Topics covered include basic graph theory, basic category theory, colored operads, and algebras over colored operads. Free colored operads are discussed in complete detail and in full generality. The intended audience of this book includes students and researchers in mathematics and other sciences where operads and colored operads are used. The prerequisite for this book is minimal. Every major concept is thoroughly motivated. There are many graphical illustrations and about 150 exercises. This book can be used in a graduate course and for independent study.

The Transformation of the English Novel, 1890-1930

Studies in Hardy, Conrad, Joyce, Lawrence, Forster and Woolf

Author: D. Schwarz

Publisher: Springer

ISBN: 0230379338

Category: Literary Criticism

Page: 336

View: 7269

In an exciting and important book... The theoretical chapters are a model of elegantly styled accommodation; yet they brook no fudging of the issues, no comfortable ambiguities - Modern Fiction Studies The Transformation of the English Novel, 1890-1930: Studies in Hardy, Conrad, Joyce, Lawrence, Forster and Woolf is a provocative exploration of a crucial period in the development of the English novel, integrating critical theory, historical background and sophisticated close reading. Divided into two major sections, the first shows how historical and contextual material is essential for developing powerful readings. The second section is theoretical and speaks of the transformation in the way that we read and think about authors, readers, characters and form in the light of recent theory, offering an alternative to the deconstructive and Marxist trends in literary studies.

Computations in Algebraic Geometry with Macaulay 2

Author: David Eisenbud,Daniel R. Grayson,Mike Stillman,Bernd Sturmfels

Publisher: Springer Science & Business Media

ISBN: 3662048515

Category: Mathematics

Page: 329

View: 7470

This book presents algorithmic tools for algebraic geometry, with experimental applications. It also introduces Macaulay 2, a computer algebra system supporting research in algebraic geometry, commutative algebra, and their applications. The algorithmic tools presented here are designed to serve readers wishing to bring such tools to bear on their own problems. The first part of the book covers Macaulay 2 using concrete applications; the second emphasizes details of the mathematics.

Numerical Polynomial Algebra

Author: Hans J. Stetter

Publisher: SIAM

ISBN: 0898715571

Category: Mathematics

Page: 472

View: 1168

This book is the first comprehensive treatment of numerical polynomial algebra, an area which so far has received little attention.