Topics in Algebraic and Topological K Theory

This volume is an introductory textbook to K-theory, both algebraic and topological, and to various current research topics within the field, including Kasparov's bivariant K-theory, the Baum-Connes conjecture, the comparison between ...

Author: Paul Frank Baum

Publisher: Springer Science & Business Media

ISBN: 3642157076

Category: Mathematics

Page: 308

View: 463

This volume is an introductory textbook to K-theory, both algebraic and topological, and to various current research topics within the field, including Kasparov's bivariant K-theory, the Baum-Connes conjecture, the comparison between algebraic and topological K-theory of topological algebras, the K-theory of schemes, and the theory of dg-categories.



K Theory for Operator Algebras

We will not treat applications in detail; however, we will outline the most striking of the applications to date in a section at the end, as well as mentioning others at suitable points in the text.

Author: Bruce Blackadar

Publisher: Springer

ISBN: 9781461395744

Category: Mathematics

Page: 337

View: 277

K -Theory has revolutionized the study of operator algebras in the last few years. As the primary component of the subject of "noncommutative topol ogy," K -theory has opened vast new vistas within the structure theory of C* algebras, as well as leading to profound and unexpected applications of opera tor algebras to problems in geometry and topology. As a result, many topolo gists and operator algebraists have feverishly begun trying to learn each others' subjects, and it appears certain that these two branches of mathematics have become deeply and permanently intertwined. Despite the fact that the whole subject is only about a decade old, operator K -theory has now reached a state of relative stability. While there will undoubtedly be many more revolutionary developments and applications in the future, it appears the basic theory has more or less reached a "final form." But because of the newness of the theory, there has so far been no comprehensive treatment of the subject. It is the ambitious goal of these notes to fill this gap. We will develop the K -theory of Banach algebras, the theory of extensions of C*-algebras, and the operator K -theory of Kasparov from scratch to its most advanced aspects. We will not treat applications in detail; however, we will outline the most striking of the applications to date in a section at the end, as well as mentioning others at suitable points in the text.

Algebraic K Theory and Its Applications

This text is designed to help graduate students in other areas learn the basics of K-Theory and get a feel for its many applications.

Author: Jonathan Rosenberg

Publisher: Springer Science & Business Media

ISBN: 1461243149

Category: Mathematics

Page: 394

View: 553

Algebraic K-Theory is crucial in many areas of modern mathematics, especially algebraic topology, number theory, algebraic geometry, and operator theory. This text is designed to help graduate students in other areas learn the basics of K-Theory and get a feel for its many applications. Topics include algebraic topology, homological algebra, algebraic number theory, and an introduction to cyclic homology and its interrelationship with K-Theory.

K Theory for Group C Algebras and Semigroup C Algebras

This book gives an account of the necessary background for group algebras and crossed products for actions of a group or a semigroup on a space and reports on some very recently developed techniques with applications to particular examples.

Author: Joachim Cuntz

Publisher: Birkhäuser

ISBN: 9783319599144

Category: Mathematics

Page: 322

View: 844

This book gives an account of the necessary background for group algebras and crossed products for actions of a group or a semigroup on a space and reports on some very recently developed techniques with applications to particular examples. Much of the material is available here for the first time in book form. The topics discussed are among the most classical and intensely studied C*-algebras. They are important for applications in fields as diverse as the theory of unitary group representations, index theory, the topology of manifolds or ergodic theory of group actions. Part of the most basic structural information for such a C*-algebra is contained in its K-theory. The determination of the K-groups of C*-algebras constructed from group or semigroup actions is a particularly challenging problem. Paul Baum and Alain Connes proposed a formula for the K-theory of the reduced crossed product for a group action that would permit, in principle, its computation. By work of many hands, the formula has by now been verified for very large classes of groups and this work has led to the development of a host of new techniques. An important ingredient is Kasparov's bivariant K-theory. More recently, also the C*-algebras generated by the regular representation of a semigroup as well as the crossed products for actions of semigroups by endomorphisms have been studied in more detail. Intriguing examples of actions of such semigroups come from ergodic theory as well as from algebraic number theory. The computation of the K-theory of the corresponding crossed products needs new techniques. In cases of interest the K-theory of the algebras reflects ergodic theoretic or number theoretic properties of the action.

Topology and K Theory

These are notes from a graduate student course on algebraic topology and K-theory given by Daniel Quillen at the Massachusetts Institute of Technology during 1979-1980.

Author: Robert Penner

Publisher: Springer Nature

ISBN: 3030439968

Category: Mathematics

Page: 213

View: 144

These are notes from a graduate student course on algebraic topology and K-theory given by Daniel Quillen at the Massachusetts Institute of Technology during 1979-1980. He had just received the Fields Medal for his work on these topics among others and was funny and playful with a confident humility from the start. These are not meant to be polished lecture notes, rather, things are presented as did Quillen reflected in the hand-written notes, resisting any temptation to change or add notation, details or elaborations. Indeed, the text is faithful to Quillen's own exposition, even respecting the {\sl board-like presentation} of formulae, diagrams and proofs, omitting numbering theorems in favor of names and so on. This is meant to be Quillen on Quillen as it happened forty years ago, an informal text for a second-semester graduate student on topology, category theory and K-theory, a potential preface to studying Quillen's own landmark papers and an informal glimpse of his great mind. The intellectual pace of the lectures, namely fast and lively, is Quillen himself, and part of the point here is to capture some of this intimacy. To be sure, much has happened since then from this categorical perspective started by Grothendieck, and Misha Kapranov has contributed an Afterword in order to make it more useful to current students.


K theory in Algebra Analysis and Topology

This volume contains the proceedings of the ICM 2018 satellite school and workshop K-theory conference in Argentina.

Author: Guillermo Cortiñas

Publisher: American Mathematical Soc.

ISBN: 1470450267

Category: Education

Page: 388

View: 732

This volume contains the proceedings of the ICM 2018 satellite school and workshop K-theory conference in Argentina. The school was held from July 16–20, 2018, in La Plata, Argentina, and the workshop was held from July 23–27, 2018, in Buenos Aires, Argentina. The volume showcases current developments in K-theory and related areas, including motives, homological algebra, index theory, operator algebras, and their applications and connections. Papers cover topics such as K-theory of group rings, Witt groups of real algebraic varieties, coarse homology theories, topological cyclic homology, negative K-groups of monoid algebras, Milnor K-theory and regulators, noncommutative motives, the classification of C∗-algebras via Kasparov's K-theory, the comparison between full and reduced C∗-crossed products, and a proof of Bott periodicity using almost commuting matrices.

Algebraic K Theory and Algebraic Topology

This book is the volume of proceedings for this meeting.

Author: P.G. Goerss

Publisher: Springer Science & Business Media

ISBN: 9401706956

Category: Mathematics

Page: 328

View: 110

A NATO Advanced Study Institute entitled "Algebraic K-theory and Algebraic Topology" was held at Chateau Lake Louise, Lake Louise, Alberta, Canada from December 12 to December 16 of 1991. This book is the volume of proceedings for this meeting. The papers that appear here are representative of most of the lectures that were given at the conference, and therefore present a "snapshot" of the state ofthe K-theoretic art at the end of 1991. The underlying objective of the meeting was to discuss recent work related to the Lichtenbaum-Quillen complex of conjectures, fro~ both the algebraic and topological points of view. The papers in this volume deal with a range of topics, including motivic cohomology theories, cyclic homology, intersection homology, higher class field theory, and the former telescope conjecture. This meeting was jointly funded by grants from NATO and the National Science Foun dation in the United States. I would like to take this opportunity to thank these agencies for their support. I would also like to thank the other members of the organizing com mittee, namely Paul Goerss, Bruno Kahn and Chuck Weibel, for their help in making the conference successful. This was the second NATO Advanced Study Institute to be held in this venue; the first was in 1987. The success of both conferences owes much to the professionalism and helpfulness of the administration and staff of Chateau Lake Louise.

Manifolds and K Theory

This volume contains the proceedings of the conference on Manifolds, -Theory, and Related Topics, held from June 23–27, 2014, in Dubrovnik, Croatia.

Author: Gregory Arone

Publisher: American Mathematical Soc.

ISBN: 1470417006

Category: $K$-theory -- Higher algebraic $K$-theory -- Algebraic $K$-theory of spaces

Page: 259

View: 718

This volume contains the proceedings of the conference on Manifolds, -Theory, and Related Topics, held from June 23–27, 2014, in Dubrovnik, Croatia. The articles contained in this volume are a collection of research papers featuring recent advances in homotopy theory, -theory, and their applications to manifolds. Topics covered include homotopy and manifold calculus, structured spectra, and their applications to group theory and the geometry of manifolds. This volume is a tribute to the influence of Tom Goodwillie in these fields.



An Algebraic Introduction to K Theory

This is an introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra (including Galois theory and modules over a principal ideal domain).

Author: Bruce A. Magurn

Publisher: Cambridge University Press

ISBN: 1107079446

Category: Mathematics

Page:

View: 545

This is an introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra (including Galois theory and modules over a principal ideal domain). The presentation is almost entirely self-contained, and is divided into short sections with exercises to reinforce the ideas and suggest further lines of inquiry. No experience with analysis, geometry, number theory or topology is assumed. Within the context of linear algebra, K-theory organises and clarifies the relations among ideal class groups, group representations, quadratic forms, dimensions of a ring, determinants, quadratic reciprocity and Brauer groups of fields. By including introductions to standard algebra topics (tensor products, localisation, Jacobson radical, chain conditions, Dedekind domains, semi-simple rings, exterior algebras), the author makes algebraic K-theory accessible to first-year graduate students and other mathematically sophisticated readers. Even if your algebra is rusty, you can read this book; the necessary background is here, with proofs.

Algebraic K theory

This book contains proceedings of the research conference on algebraic $K$-theory that took place in Poznan, Poland, in September 1995.

Author: Grzegorz Banaszak

Publisher: American Mathematical Soc.

ISBN: 0821805118

Category: Mathematics

Page: 210

View: 470

This book contains proceedings of the research conference on algebraic $K$-theory that took place in Poznan, Poland, in September 1995. The conference concluded the activity of the algebraic $K$-theory seminar held at the Adam Mickiewicz University in the academic year 1994-1995. Talks at the conference covered a wide range of current research activities in algebraic $K$-theory. In particular, the following topics were covered: $K$-theory of fields and rings of integers; $K$-theory of elliptic and modular curves; theory of motives, motivic cohomology, Beilinson conjectures; and, algebraic $K$-theory of topological spaces, topological Hochschild homology and cyclic homology. With contributions by some leading experts in the field, this book provides a look at the state of current research in algebraic $K$-theory.

An Algebraic Introduction to K Theory

The blend of K - theory with these topics motivates and enhances their exposition . By including these algebra topics with the K - theory , I also hope to reach the mathematically sophisticated reader , who may have heard that K ...

Author: Bruce A. Magurn

Publisher: Cambridge University Press

ISBN: 9780521800785

Category: Mathematics

Page: 676

View: 782

An introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra.


Algebraic K theory and Its Applications

This proceedings volume is divided into two parts.

Author: Hyman Bass

Publisher: World Scientific Publishing Company Incorporated

ISBN: 9789810234911

Category: Mathematics

Page: 607

View: 504

This proceedings volume is divided into two parts. The first part consists of the lectures given during the first two weeks of the School on "Overview of Algebraic K-Theory", including various constructions, examples and illustrations from algebra, number theory, algebraic topology and algebraic geometry; more concentrated topics on "K-theory and cyclic/Maclane (co)homology", K-theory and motivic (co)homology; Chow groups and algebraic cycles; K-theory and values of zeta functions. The second part consists of research papers arising from the lectures during the third week of the School.

Michael Atiyah Collected Works

For the first time, these volumes bring together Atiyah's collected papers--both monographs and collaborative works-- including those dealing with mathematical education and current topics of research such as K-theory and gauge theory.

Author: Michael Atiyah

Publisher: Oxford University Press

ISBN: 9780198532767

Category: Mathematics

Page: 854

View: 109

One of the greatest mathematicians in the world, Michael Atiyah has earned numerous honors, including a Fields Medal, the mathematical equivalent of the Nobel Prize. While the focus of his work has been in the areas of algebraic geometry and topology, he has also participated in research with theoretical physicists. For the first time, these volumes bring together Atiyah's collected papers--both monographs and collaborative works-- including those dealing with mathematical education and current topics of research such as K-theory and gauge theory. The volumes are organized thematically. They will be of great interest to research mathematicians, theoretical physicists, and graduate students in these areas.

Polytopes Rings and K Theory

Inthe second partof Chapter two topics that extend the K-theory of Chapters and directlyarehighlightedbyafulldiscussion:theexistenceofsimplicialprojective toric varieties with “huge” K -groups, and the triviality of ...

Author: Winfried Bruns

Publisher: Springer Science & Business Media

ISBN: 0387763562

Category: Mathematics

Page: 461

View: 868

This book examines interactions of polyhedral discrete geometry and algebra. What makes this book unique is the presentation of several central results in all three areas of the exposition - from discrete geometry, to commutative algebra, and K-theory.