A User's Guide to Spectral Sequences

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Author: John McCleary

Publisher: Cambridge University Press

ISBN: 9780521567596

Category: Mathematics

Page: 561

View: 6909

Spectral sequences are among the most elegant and powerful methods of computation in mathematics. This book describes some of the most important examples of spectral sequences and some of their most spectacular applications. The first part treats the algebraic foundations for this sort of homological algebra, starting from informal calculations. The heart of the text is an exposition of the classical examples from homotopy theory, with chapters on the Leray-Serre spectral sequence, the Eilenberg-Moore spectral sequence, the Adams spectral sequence, and, in this new edition, the Bockstein spectral sequence. The last part of the book treats applications throughout mathematics, including the theory of knots and links, algebraic geometry, differential geometry and algebra. This is an excellent reference for students and researchers in geometry, topology, and algebra.

Cohen-Macaulay Rings

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Author: Winfried Bruns,H. Jürgen Herzog

Publisher: Cambridge University Press

ISBN: 9780521566742

Category: Mathematics

Page: 453

View: 1487

In the last two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules. A separate chapter is devoted to Hilbert functions (including Macaulay's theorem) and numerical invariants derived from them. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. So the general theory is applied to Stanley-Reisner rings, semigroup rings, determinantal rings, and rings of invariants. Their connections with combinatorics are highlighted, e.g. Stanley's upper bound theorem or Ehrhart's reciprocity law for rational polytopes. The final chapters are devoted to Hochster's theorem on big Cohen-Macaulay modules and its applications, including Peskine-Szpiro's intersection theorem, the Evans-Griffith syzygy theorem, bounds for Bass numbers, and tight closure. Throughout each chapter the authors have supplied many examples and exercises which, combined with the expository style, will make the book very useful for graduate courses in algebra. As the only modern, broad account of the subject it will be essential reading for researchers in commutative algebra.

Cohomology of Groups

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Author: Kenneth S. Brown

Publisher: Springer Science & Business Media

ISBN: 1468493272

Category: Mathematics

Page: 306

View: 8525

Aimed at second year graduate students, this text introduces them to cohomology theory (involving a rich interplay between algebra and topology) with a minimum of prerequisites. No homological algebra is assumed beyond what is normally learned in a first course in algebraic topology, and the basics of the subject, as well as exercises, are given prior to discussion of more specialized topics.

Introduction to Homotopy Theory

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Author: Martin Arkowitz

Publisher: Springer Science & Business Media

ISBN: 9781441973290

Category: Mathematics

Page: 344

View: 7511

This is a book in pure mathematics dealing with homotopy theory, one of the main branches of algebraic topology. The principal topics are as follows: Basic Homotopy; H-spaces and co-H-spaces; fibrations and cofibrations; exact sequences of homotopy sets, actions, and coactions; homotopy pushouts and pullbacks; classical theorems, including those of Serre, Hurewicz, Blakers-Massey, and Whitehead; homotopy Sets; homotopy and homology decompositions of spaces and maps; and obstruction theory. The underlying theme of the entire book is the Eckmann-Hilton duality theory. The book can be used as a text for the second semester of an advanced ungraduate or graduate algebraic topology course.

Topological Modular Forms

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Author: Christopher L. Douglas, John Francis, André G. Henriques, Michael A. Hill

Publisher: American Mathematical Soc.

ISBN: 1470418843

Category: Mathematics

Page: 318

View: 3641

The theory of topological modular forms is an intricate blend of classical algebraic modular forms and stable homotopy groups of spheres. The construction of this theory combines an algebro-geometric perspective on elliptic curves over finite fields with techniques from algebraic topology, particularly stable homotopy theory. It has applications to and connections with manifold topology, number theory, and string theory. This book provides a careful, accessible introduction to topological modular forms. After a brief history and an extended overview of the subject, the book proper commences with an exposition of classical aspects of elliptic cohomology, including background material on elliptic curves and modular forms, a description of the moduli stack of elliptic curves, an explanation of the exact functor theorem for constructing cohomology theories, and an exploration of sheaves in stable homotopy theory. There follows a treatment of more specialized topics, including localization of spectra, the deformation theory of formal groups, and Goerss-Hopkins obstruction theory for multiplicative structures on spectra. The book then proceeds to more advanced material, including discussions of the string orientation, the sheaf of spectra on the moduli stack of elliptic curves, the homotopy of topological modular forms, and an extensive account of the construction of the spectrum of topological modular forms. The book concludes with the three original, pioneering and enormously influential manuscripts on the subject, by Hopkins, Miller, and Mahowald.

Cubical Homotopy Theory

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Author: Brian A. Munson,Ismar Volić

Publisher: Cambridge University Press

ISBN: 1316351939

Category: Mathematics

Page: N.A

View: 9678

Graduate students and researchers alike will benefit from this treatment of classical and modern topics in homotopy theory of topological spaces with an emphasis on cubical diagrams. The book contains 300 examples and provides detailed explanations of many fundamental results. Part I focuses on foundational material on homotopy theory, viewed through the lens of cubical diagrams: fibrations and cofibrations, homotopy pullbacks and pushouts, and the Blakers–Massey Theorem. Part II includes a brief example-driven introduction to categories, limits and colimits, an accessible account of homotopy limits and colimits of diagrams of spaces, and a treatment of cosimplicial spaces. The book finishes with applications to some exciting new topics that use cubical diagrams: an overview of two versions of calculus of functors and an account of recent developments in the study of the topology of spaces of knots.

An Introduction to Invariants and Moduli

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Author: Shigeru Mukai,W. M. Oxbury

Publisher: Cambridge University Press

ISBN: 9780521809061

Category: Mathematics

Page: 503

View: 8810

Incorporated in this volume are the first two books in Mukai's series on Moduli Theory. The notion of a moduli space is central to geometry. However, it's influence is not confined there; for example the theory of moduli spaces is a crucial ingredient in the proof of Fermat's last theorem. An accurate account of Mukai's influential Japanese texts, this tranlation will be a valuable resource for researchers and graduate students working in a range of areas.

Generalized Cohomology

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Author: Akira Kōno,Dai Tamaki

Publisher: American Mathematical Soc.

ISBN: 9780821835142

Category:

Page: N.A

View: 7832

Hyperplane Arrangements

An Introduction

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Author: Alexandru Dimca

Publisher: Springer

ISBN: 3319562215

Category: Mathematics

Page: 200

View: 2731

This textbook provides an accessible introduction to the rich and beautiful area of hyperplane arrangement theory, where discrete mathematics, in the form of combinatorics and arithmetic, meets continuous mathematics, in the form of the topology and Hodge theory of complex algebraic varieties. The topics discussed in this book range from elementary combinatorics and discrete geometry to more advanced material on mixed Hodge structures, logarithmic connections and Milnor fibrations. The author covers a lot of ground in a relatively short amount of space, with a focus on defining concepts carefully and giving proofs of theorems in detail where needed. Including a number of surprising results and tantalizing open problems, this timely book also serves to acquaint the reader with the rapidly expanding literature on the subject. Hyperplane Arrangements will be particularly useful to graduate students and researchers who are interested in algebraic geometry or algebraic topology. The book contains numerous exercises at the end of each chapter, making it suitable for courses as well as self-study.

Toric Varieties

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Author: David A. Cox,John B. Little,Henry K. Schenck

Publisher: American Mathematical Soc.

ISBN: 0821848194

Category: Mathematics

Page: 841

View: 7734

Toric varieties form a beautiful and accessible part of modern algebraic geometry. This book covers the standard topics in toric geometry; a novel feature is that each of the first nine chapters contains an introductory section on the necessary background material in algebraic geometry. Other topics covered include quotient constructions, vanishing theorems, equivariant cohomology, GIT quotients, the secondary fan, and the minimal model program for toric varieties. The subject lends itself to rich examples reflected in the 134 illustrations included in the text. The book also explores connections with commutative algebra and polyhedral geometry, treating both polytopes and their unbounded cousins, polyhedra. There are appendices on the history of toric varieties and the computational tools available to investigate nontrivial examples in toric geometry. Readers of this book should be familiar with the material covered in basic graduate courses in algebra and topology, and to a somewhat lesser degree, complex analysis. In addition, the authors assume that the reader has had some previous experience with algebraic geometry at an advanced undergraduate level. The book will be a useful reference for graduate students and researchers who are interested in algebraic geometry, polyhedral geometry, and toric varieties.

An Introduction to Invariants and Moduli

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Author: Shigeru Mukai,W. M. Oxbury

Publisher: Cambridge University Press

ISBN: 9780521809061

Category: Mathematics

Page: 503

View: 750

Incorporated in this volume are the first two books in Mukai's series on Moduli Theory. The notion of a moduli space is central to geometry. However, it's influence is not confined there; for example the theory of moduli spaces is a crucial ingredient in the proof of Fermat's last theorem. An accurate account of Mukai's influential Japanese texts, this tranlation will be a valuable resource for researchers and graduate students working in a range of areas.

Simplicial Homotopy Theory

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Author: Paul G. Goerss,John F. Jardine

Publisher: Birkhäuser

ISBN: 3034887078

Category: Mathematics

Page: 510

View: 1035

Elliptic Curves. (MN-40)

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Author: Anthony W. Knapp

Publisher: Princeton University Press

ISBN: 0691186901

Category: Mathematics

Page: N.A

View: 7102

An Introduction to Homological Algebra

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Author: Charles A. Weibel

Publisher: Cambridge University Press

ISBN: 113964307X

Category: Mathematics

Page: N.A

View: 4362

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.

Connective Real K-theory of Finite Groups

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Author: Robert Ray Bruner,John Patrick Campbell Greenlees

Publisher: American Mathematical Soc.

ISBN: 0821851896

Category: Mathematics

Page: 318

View: 4239

This book is about equivariant real and complex topological $K$-theory for finite groups. Its main focus is on the study of real connective $K$-theory including $ko^*(BG)$ as a ring and $ko_*(BG)$ as a module over it. In the course of their study the authors define equivariant versions of connective $KO$-theory and connective $K$-theory with reality, in the sense of Atiyah, which give well-behaved, Noetherian, uncompleted versions of the theory. They prove local cohomology and completion theorems for these theories, giving a means of calculation as well as establishing their formal credentials. In passing from the complex to the real theories in the connective case, the authors describe the known failure of descent and explain how the $\eta$-Bockstein spectral sequence provides an effective substitute. This formal framework allows the authors to give a systematic calculation scheme to quantify the expectation that $ko^*(BG)$ should be a mixture of representation theory and group cohomology. It is characteristic that this starts with $ku^*(BG)$ and then uses the local cohomology theorem and the Bockstein spectral sequence to calculate $ku_*(BG)$, $ko^*(BG)$, and $ko_*(BG)$. To give the skeleton of the answer, the authors provide a theory of $ko$-characteristic classes for representations, with the Pontrjagin classes of quaternionic representations being the most important. Building on the general results, and their previous calculations, the authors spend the bulk of the book giving a large number of detailed calculations for specific groups (cyclic, quaternion, dihedral, $A_4$, and elementary abelian 2-groups). The calculations illustrate the richness of the theory and suggest many further lines of investigation. They have been applied in the verification of the Gromov-Lawson-Rosenberg conjecture for several new classes of finite groups.|This book is about equivariant real and complex topological $K$-theory for finite groups. Its main focus is on the study of real connective $K$-theory including $ko^*(BG)$ as a ring and $ko_*(BG)$ as a module over it. In the course of their study the authors define equivariant versions of connective $KO$-theory and connective $K$-theory with reality, in the sense of Atiyah, which give well-behaved, Noetherian, uncompleted versions of the theory. They prove local cohomology and completion theorems for these theories, giving a means of calculation as well as establishing their formal credentials. In passing from the complex to the real theories in the connective case, the authors describe the known failure of descent and explain how the $\eta$-Bockstein spectral sequence provides an effective substitute. This formal framework allows the authors to give a systematic calculation scheme to quantify the expectation that $ko^*(BG)$ should be a mixture of representation theory and group cohomology. It is characteristic that this starts with $ku^*(BG)$ and then uses the local cohomology theorem and the Bockstein spectral sequence to calculate $ku_*(BG)$, $ko^*(BG)$, and $ko_*(BG)$. To give the skeleton of the answer, the authors provide a theory of $ko$-characteristic classes for representations, with the Pontrjagin classes of quaternionic representations being the most important. Building on the general results, and their previous calculations, the authors spend the bulk of the book giving a large number of detailed calculations for specific groups (cyclic, quaternion, dihedral, $A_4$, and elementary abelian 2-groups). The calculations illustrate the richness of the theory and suggest many further lines of investigation. They have been applied in the verification of the Gromov-Lawson-Rosenberg conjecture for several new classes of finite groups.

Combinatorial Algebraic Topology

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Author: Dimitry Kozlov

Publisher: Springer Science & Business Media

ISBN: 3540719628

Category: Mathematics

Page: 390

View: 6470

This volume is the first comprehensive treatment of combinatorial algebraic topology in book form. The first part of the book constitutes a swift walk through the main tools of algebraic topology. Readers - graduate students and working mathematicians alike - will probably find particularly useful the second part, which contains an in-depth discussion of the major research techniques of combinatorial algebraic topology. Although applications are sprinkled throughout the second part, they are principal focus of the third part, which is entirely devoted to developing the topological structure theory for graph homomorphisms.

Differential Geometrical Theory of Statistics

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Author: Frédéric Barbaresco,Frank Nielsen

Publisher: MDPI

ISBN: 3038424242

Category: Computers

Page: 472

View: 6061

This book is a printed edition of the Special Issue "Differential Geometrical Theory of Statistics" that was published in Entropy

An Introduction to Algebraic Geometry and Algebraic Groups

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Author: Meinolf Geck

Publisher: Oxford University Press

ISBN: 019967616X

Category: Mathematics

Page: 320

View: 7816

An accessible text introducing algebraic groups at advanced undergraduate and early graduate level, this book covers the conjugacy of Borel subgroups and maximal tori, the theory of algebraic groups with a BN-pair, Frobenius maps on affine varieties and algebraic groups, zeta functions and Lefschetz numbers for varieties over finite fields.

Introduction to Foliations and Lie Groupoids

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Author: I. Moerdijk,J. Mrcun

Publisher: Cambridge University Press

ISBN: 9781139438988

Category: Mathematics

Page: N.A

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This book gives a quick introduction to the theory of foliations, Lie groupoids and Lie algebroids. An important feature is the emphasis on the interplay between these concepts: Lie groupoids form an indispensable tool to study the transverse structure of foliations as well as their noncommutative geometry, while the theory of foliations has immediate applications to the Lie theory of groupoids and their infinitesimal algebroids. The book starts with a detailed presentation of the main classical theorems in the theory of foliations then proceeds to Molino's theory, Lie groupoids, constructing the holonomy groupoid of a foliation and finally Lie algebroids. Among other things, the authors discuss to what extent Lie's theory for Lie groups and Lie algebras holds in the more general context of groupoids and algebroids. Based on the authors' extensive teaching experience, this book contains numerous examples and exercises making it ideal for graduate students and their instructors.

Stars and Their Spectra

An Introduction to the Spectral Sequence

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Author: James B. Kaler

Publisher: Cambridge University Press

ISBN: 0521899540

Category: Science

Page: 394

View: 6080

Revised and expanded, the second edition of this popular book provides a thorough introduction to stellar spectra. Each chapter explores a different star type, including new classes L and T. With modern digital spectra and updates from two decades of astronomical discoveries, it is invaluable for amateur astronomers and students.