Bousfield Classes And Ohkawa S Theorem

Author : Takeo Ohsawa,Norihiko Minami
Publisher : Springer Nature
Page : 438 pages
File Size : 53,9 Mb
Release : 2020-03-18
Category : Mathematics
ISBN : 9789811515880

Get Book

Bousfield Classes and Ohkawa's Theorem by Takeo Ohsawa,Norihiko Minami Pdf

This volume originated in the workshop held at Nagoya University, August 28–30, 2015, focusing on the surprising and mysterious Ohkawa's theorem: the Bousfield classes in the stable homotopy category SH form a set. An inspiring, extensive mathematical story can be narrated starting with Ohkawa's theorem, evolving naturally with a chain of motivational questions: Ohkawa's theorem states that the Bousfield classes of the stable homotopy category SH surprisingly forms a set, which is still very mysterious. Are there any toy models where analogous Bousfield classes form a set with a clear meaning? The fundamental theorem of Hopkins, Neeman, Thomason, and others states that the analogue of the Bousfield classes in the derived category of quasi-coherent sheaves Dqc(X) form a set with a clear algebro-geometric description. However, Hopkins was actually motivated not by Ohkawa's theorem but by his own theorem with Smith in the triangulated subcategory SHc, consisting of compact objects in SH. Now the following questions naturally occur: (1) Having theorems of Ohkawa and Hopkins-Smith in SH, are there analogues for the Morel-Voevodsky A1-stable homotopy category SH(k), which subsumes SH when k is a subfield of C?, (2) Was it not natural for Hopkins to have considered Dqc(X)c instead of Dqc(X)? However, whereas there is a conceptually simple algebro-geometrical interpretation Dqc(X)c = Dperf(X), it is its close relative Dbcoh(X) that traditionally, ever since Oka and Cartan, has been intensively studied because of its rich geometric and physical information. This book contains developments for the rest of the story and much more, including the chromatics homotopy theory, which the Hopkins–Smith theorem is based upon, and applications of Lurie's higher algebra, all by distinguished contributors.

Author : Jean-Pierre Meyer
Publisher : American Mathematical Soc.
Page : 392 pages
File Size : 51,6 Mb
Release : 1999
Category : Mathematics
ISBN : 9780821810576

Get Book

Homotopy Invariant Algebraic Structures by Jean-Pierre Meyer Pdf

This volume presents the proceedings of the conference held in honor of J. Michael Boardman's 60th birthday. It brings into print his classic work on conditionally convergent spectral sequences. Over the past 30 years, it has become evident that some of the deepest questions in algebra are best understood against the background of homotopy theory. Boardman and Vogt's theory of homotopy-theoretic algebraic structures and the theory of spectra, for example, were two benchmark breakthroughs underlying the development of algebraic $K$-theory and the recent advances in the theory of motives. The volume begins with short notes by Mac Lane, May, Stasheff, and others on the early and recent history of the subject. But the bulk of the volume consists of research papers on topics that have been strongly influenced by Boardman's work. Articles give readers a vivid sense of the current state of the theory of "homotopy-invariant algebraic structures". Also included are two major foundational papers by Goerss and Strickland on applications of methods of algebra (i.e., Dieudonné modules and formal schemes) to problems of topology. Boardman is known for the depth and wit of his ideas. This volume is intended to reflect and to celebrate those fine characteristics.

Author : John Greenlees
Publisher : Springer Science & Business Media
Page : 396 pages
File Size : 52,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789400709485

Get Book

Axiomatic, Enriched and Motivic Homotopy Theory by John Greenlees Pdf

The NATO Advanced Study Institute "Axiomatic, enriched and rna tivic homotopy theory" took place at the Isaac Newton Institute of Mathematical Sciences, Cambridge, England during 9-20 September 2002. The Directors were J.P.C.Greenlees and I.Zhukov; the other or ganizers were P.G.Goerss, F.Morel, J.F.Jardine and V.P.Snaith. The title describes the content well, and both the event and the contents of the present volume reflect recent remarkable successes in model categor ies, structured ring spectra and homotopy theory of algebraic geometry. The ASI took the form of a series of 15 minicourses and a few extra lectures, and was designed to provide background, and to bring the par ticipants up to date with developments. The present volume is based on a number of the lectures given during the workshop. The ASI was the opening workshop of the four month programme "New Contexts for Stable Homotopy Theory" which explored several themes in greater depth. I am grateful to the Isaac Newton Institute for providing such an ideal venue, the NATO Science Committee for their funding, and to all the speakers at the conference, whether or not they were able to contribute to the present volume. All contributions were refereed, and I thank the authors and referees for their efforts to fit in with the tight schedule. Finally, I would like to thank my coorganizers and all the staff at the Institute for making the ASI run so smoothly. J.P.C.GREENLEES.

Author : John Harold Palmieri
Publisher : American Mathematical Soc.
Page : 193 pages
File Size : 49,9 Mb
Release : 2001
Category : Homotopy theory
ISBN : 9780821826683

Get Book

Stable Homotopy over the Steenrod Algebra by John Harold Palmieri Pdf

This title applys the tools of stable homotopy theory to the study of modules over the mod $p$ Steenrod algebra $A DEGREES{*}$. More precisely, let $A$ be the dual of $A DEGREES{*}$; then we study the category $\mathsf{stable}(A)$ of unbounded cochain complexes of injective comodules over $A$, in which the morphisms are cochain homotopy classes of maps. This category is triangulated. Indeed, it is a stable homotopy category, so we can use Brown representability, Bousfield localization, Brown-Comenetz duality, and other homotopy-theoretic tools to study it. One focus of attention is the analogue of the stable homotopy groups of spheres, which in this setting is the cohomology of $A$, $\mathrm{Ext}_A DEGREES{**}(\mathbf{F}_p, \mathbf{F}_p)$. This title also has nilpotence theorems, periodicity theorems, a convergent chromatic tower, and a nu

Author : Jean-Michel Morel,Bernard Teissier
Publisher : Springer Nature
Page : 629 pages
File Size : 47,6 Mb
Release : 2023-06-14
Category : Mathematics
ISBN : 9783031122446

Get Book

Mathematics Going Forward by Jean-Michel Morel,Bernard Teissier Pdf

This volume is an original collection of articles by 44 leading mathematicians on the theme of the future of the discipline. The contributions range from musings on the future of specific fields, to analyses of the history of the discipline, to discussions of open problems and conjectures, including first solutions of unresolved problems. Interestingly, the topics do not cover all of mathematics, but only those deemed most worthy to reflect on for future generations. These topics encompass the most active parts of pure and applied mathematics, including algebraic geometry, probability, logic, optimization, finance, topology, partial differential equations, category theory, number theory, differential geometry, dynamical systems, artificial intelligence, theory of groups, mathematical physics and statistics.

Author : Anonim
Publisher : Unknown
Page : 704 pages
File Size : 45,7 Mb
Release : 2007
Category : Group actions (Mathematics)
ISBN : CORNELL:31924099690822

Get Book

Groups, Geometry, and Dynamics by Anonim Pdf

Publishes research articles that focus on groups or group actions as well as articles in other areas of mathematics in which groups or group actions are used as a main tool. Covers all topics of modern group theory with preference given to geometric, asymptotic and combinatorial group theory, dynamics of group actions, probabilistic and analytical methods, interaction with ergodic theory and operator algebras, and other related fields.

Author : Anonim
Publisher : Unknown
Page : 888 pages
File Size : 40,6 Mb
Release : 2001
Category : Mathematics
ISBN : UVA:X006170284

Get Book

Mathematical Reviews by Anonim Pdf

Author : Thorsten Holm,Peter Jørgensen,Raphaël Rouquier
Publisher : Cambridge University Press
Page : 473 pages
File Size : 54,7 Mb
Release : 2010-06-24
Category : Mathematics
ISBN : 9781139488884

Get Book

Triangulated Categories by Thorsten Holm,Peter Jørgensen,Raphaël Rouquier Pdf

A 2010 collection of survey articles by leading experts covering fundamental aspects of triangulated categories, as well as applications in algebraic geometry, representation theory, commutative algebra, microlocal analysis and algebraic topology. This is a valuable reference for experts and a useful introduction for graduate students entering the field.

Author : Dennis P. Sullivan
Publisher : Springer
Page : 286 pages
File Size : 48,7 Mb
Release : 2009-09-03
Category : Mathematics
ISBN : 9048103509

Get Book

Geometric Topology: Localization, Periodicity and Galois Symmetry by Dennis P. Sullivan Pdf

The seminal ‘MIT notes’ of Dennis Sullivan were issued in June 1970 and were widely circulated at the time. The notes had a - jor in?uence on the development of both algebraic and geometric topology, pioneering the localization and completion of spaces in homotopy theory, including p-local, pro?nite and rational homotopy theory, le- ing to the solution of the Adams conjecture on the relationship between vector bundles and spherical ?brations, the formulation of the ‘Sullivan conjecture’ on the contractibility of the space of maps from the classifying space of a ?nite group to a ?nite dimensional CW complex, theactionoftheGalois groupoverQofthealgebraicclosureQof Q on smooth manifold structures in pro?nite homotopy theory, the K-theory orientation ofPL manifolds and bundles. Some of this material has been already published by Sullivan him- 1 self: in an article in the Proceedings of the 1970 Nice ICM, and in the 1974 Annals of Mathematics papers Genetics of homotopy theory and the Adams conjecture and The transversality character- 2 istic class and linking cycles in surgery theory . Many of the ideas originating in the notes have been the starting point of subsequent 1 reprinted at the end of this volume 2 joint with John Morgan vii viii 3 developments . However, the text itself retains a unique ?avour of its time, and of the range of Sullivan’s ideas.

Author : Mark Hovey,Neil P. Strickland
Publisher : American Mathematical Soc.
Page : 100 pages
File Size : 43,7 Mb
Release : 1999
Category : Mathematics
ISBN : 9780821810798

Get Book

Morava K-Theories and Localisation by Mark Hovey,Neil P. Strickland Pdf

This book is intended for graduate students and research mathematicians working in group theory and generalizations.

Author : Vijay Kumar Thakur,Manju Kumari Thakur
Publisher : Springer
Page : 576 pages
File Size : 50,8 Mb
Release : 2015-07-20
Category : Technology & Engineering
ISBN : 9788132224730

Get Book

Eco-friendly Polymer Nanocomposites by Vijay Kumar Thakur,Manju Kumari Thakur Pdf

This book contains precisely referenced chapters, emphasizing environment-friendly polymer nanocomposites with basic fundamentals, practicality and alternatives to traditional nanocomposites through detailed reviews of different environmental friendly materials procured from different resources, their synthesis and applications using alternative green approaches. The book aims at explaining basics of eco-friendly polymer nanocomposites from different natural resources and their chemistry along with practical applications which present a future direction in the biomedical, pharmaceutical and automotive industry. The book attempts to present emerging economic and environmentally friendly polymer nanocomposites that are free from side effects studied in the traditional nanocomposites. This book is the outcome of contributions by many experts in the field from different disciplines, with various backgrounds and expertises. This book will appeal to researchers as well as students from different disciplines. The content includes industrial applications and will fill the gap between the research works in laboratory to practical applications in related industries.

Author : Ali Chamseddine,Caterina Consani,Nigel Higson,Masoud Khalkhali,Henri Moscovici,Guoliang Yu
Publisher : Springer Nature
Page : 753 pages
File Size : 46,8 Mb
Release : 2020-01-13
Category : Mathematics
ISBN : 9783030295974

Get Book

Advances in Noncommutative Geometry by Ali Chamseddine,Caterina Consani,Nigel Higson,Masoud Khalkhali,Henri Moscovici,Guoliang Yu Pdf

This authoritative volume in honor of Alain Connes, the foremost architect of Noncommutative Geometry, presents the state-of-the art in the subject. The book features an amalgam of invited survey and research papers that will no doubt be accessed, read, and referred to, for several decades to come. The pertinence and potency of new concepts and methods are concretely illustrated in each contribution. Much of the content is a direct outgrowth of the Noncommutative Geometry conference, held March 23–April 7, 2017, in Shanghai, China. The conference covered the latest research and future areas of potential exploration surrounding topology and physics, number theory, as well as index theory and its ramifications in geometry.

Author : Mark Hovey,John Harold Palmieri,Neil P. Strickland
Publisher : American Mathematical Soc.
Page : 114 pages
File Size : 48,9 Mb
Release : 1997
Category : Mathematics
ISBN : 9780821806241

Get Book

Axiomatic Stable Homotopy Theory by Mark Hovey,John Harold Palmieri,Neil P. Strickland Pdf

This book gives an axiomatic presentation of stable homotopy theory. It starts with axioms defining a 'stable homotopy category'; using these axioms, one can make various constructions - cellular towers, Bousfield localization, and Brown representability, to name a few. Much of the book is devoted to these constructions and to the study of the global structure of stable homotopy categories. Next, a number of examples of such categories are presented. Some of these arise in topology (the ordinary stable homotopy category of spectra, categories of equivariant spectra, and Bousfield localizations of these), and others in algebra (coming from the representation theory of groups or of Lie algebras, as well as the derived category of a commutative ring). Hence one can apply many of the tools of stable homotopy theory to these algebraic situations.This work: provides a reference for standard results and constructions in stable homotopy theory; discusses applications of those results to algebraic settings, such as group theory and commutative algebra; provides a unified treatment of several different situations in stable homotopy, including equivariant stable homotopy and localizations of the stable homotopy category; and, also provides a context for nilpotence and thick subcategory theorems, such as the nilpotence theorem of Devinatz-Hopkins-Smith and the thick subcategory theorem of Hopkins-Smith in stable homotopy theory, and the thick subcategory theorem of Benson-Carlson-Rickard in representation theory. This book presents stable homotopy theory as a branch of mathematics in its own right with applications in other fields of mathematics. It is a first step toward making stable homotopy theory a tool useful in many disciplines of mathematics.

Author : M. A. Mandell,J. Peter May
Publisher : American Mathematical Soc.
Page : 108 pages
File Size : 49,6 Mb
Release : 2002
Category : Mathematics
ISBN : 9780821829363

Get Book

Equivariant Orthogonal Spectra and S-Modules by M. A. Mandell,J. Peter May Pdf

The last few years have seen a revolution in our understanding of the foundations of stable homotopy theory. Many symmetric monoidal model categories of spectra whose homotopy categories are equivalent to the stable homotopy category are now known, whereas no such categories were known before 1993. The most well-known examples are the category of $S$-modules and the category of symmetric spectra. We focus on the category of orthogonal spectra, which enjoys some of the best features of $S$-modules and symmetric spectra and which is particularly well-suited to equivariant generalization. We first complete the nonequivariant theory by comparing orthogonal spectra to $S$-modules. We then develop the equivariant theory.For a compact Lie group $G$, we construct a symmetric monoidal model category of orthogonal $G$-spectra whose homotopy category is equivalent to the classical stable homotopy category of $G$-spectra. We also complete the theory of $S_G$-modules and compare the categories of orthogonal $G$-spectra and $S_G$-modules. A key feature is the analysis of change of universe, change of group, fixed point, and orbit functors in these two highly structured categories for the study of equivariant stable homotopy theory.

Author : Scott Balchin,David Barnes,Magdalena Kędziorek,Markus Szymik
Publisher : Cambridge University Press
Page : 357 pages
File Size : 44,9 Mb
Release : 2021-11-18
Category : Mathematics
ISBN : 9781108931946

Get Book

Equivariant Topology and Derived Algebra by Scott Balchin,David Barnes,Magdalena Kędziorek,Markus Szymik Pdf

A collection of research papers, both new and expository, based on the interests of Professor J. P. C. Greenlees.