V. A. Smirnov, Moscow State Pedagogical Institute, Russia

Simplicial and Operad Methods in Algebraic Topology

Expected publication date is March 21, 2001

Description

In recent years, for solving problems of algebraic topology and, in particular, difficult problems of homotopy theory, algebraic structures more complicated than just a topological monoid, an algebra, a coalgebra, etc., have been used more and more often. A convenient language for describing various structures arising naturally on topological spaces and on their cohomology and homotopy groups is the language of operads and algebras over an operad. This language was proposed by J. P. May in the 1970s to describe the structures on various loop spaces.

This book presents a detailed study of the concept of an operad in the categories of topological spaces and of chain complexes. The notions of an algebra and a coalgebra over an operad are introduced, and their properties are investigated. The algebraic structure of the singular chain complex of a topological space is explained, and it is shown how the problem of homotopy classification of topological spaces can be solved using this structure. For algebras and coalgebras over operads, standard constructions are defined, particularly the bar and cobar consturctions. Operad methods are applied to computing the homology of iterated loop spaces, investigating the algebraic structure of generalized cohomology theories, describing cohomology of groups and algebras, computing differential in the Adams spectral sequence for the homotopy groups of the spheres, and some other problems.

Contents

Operads in the category of topological spaces
Simplicial objects and homotopy theory
Algebraic structures on chain complexes
$A_\infty$-structures on chain complexes
Operads and algebras over operads
Homoloty of iterated loop spaces
Homotopy theories and $E_\infty$-structures
Operad methods in cobordism theory
Description of the cohomology of groups and algebras
Homology operations and differentials in the Adams spectral sequence
Bibliography
Index

Details:

Series: Translations of Mathematical Monographs, Volume: 198
Publication Year: 2001
ISBN: 0-8218-2170-9
Paging: 235 pp.
Binding: Hardcover

Edited by: A. V. Kelarev, University of Tasmania R. Gobel, University of Essen
K. M. Rangaswamy,University of Colorado,, P. Schultz, The University of Western Australia, , and C. Vinsonhaler,University of Connecticut, Storrs, CT

Abelian Groups, Rings and Modules

Expected publication date is March 23, 2001

Description

This volume presents the proceedings from the conference on Abelian Groups, Rings, and Modules (AGRAM) held at the University of Western Australia (Perth). Included are articles based on talks given at the conference, as well as a few specially invited papers. The proceedings are dedicated to Professor L?szl? Fuchs. The book includes a tribute and a review of his work by his long-time collaborator, Professor Luigi Salce.

Four surveys from leading experts follow Professor Salce's article. They present recent results from active research areas:

Error correcting codes as ideals in group rings,
Duality in module categories,
Automorphism groups of abelian groups, and
Generalizations of isomorphism in torsion-free abelian groups.

In addition to these surveys, the volume contains 22 research articles in diverse areas connected with the themes of the conference. The areas discussed include abelian groups and their endomorphism rings, modules over various rings, commutative and non-commutative ring theory, varieties of groups, and topological aspects of algebra. The book offers a comprehensive source for recent research in this active area of study.

Contents
Introduction
L. Salce -- L?szl? Fuchs and his "moddom" work
Survey articles
A. V. Kelarev and P. Sol? -- Error-correcting codes as ideals in group rings
B. Olberding -- Homomorphisms and duality for torsion-free modules
K. C. O'Meara and C. Vinsonhaler -- Generalizations of isomorphism in torsion-free abelian groups
P. Schultz -- Automorphism groups of abelian groups
Contributed papers
D. M. Arnold -- Direct sum decompositions of torsion-free abelian groups of finite rank
M. A. Avi?? and P. Schultz -- The endomorphism ring of a bounded abelian $p$-group
E. Blagoveshchenskaya, G. Ivanov, and P. Schultz -- The Baer-Kaplansky theorem for almost completely decomposable groups
A. Blass and J. Irwin -- Maximal pure independent sets
D. Dikranjan and M. Tkachenko -- Characterization of the tori via density of the solution set of linear equations
A. A. Fomin -- Quotient divisible mixed groups
L. Fuchs and S. B. Lee -- Stacked bases over h-local Pr?fer domains
A. J. Giovannitti -- Groups with locally defined heights and products of $\mathfrak{R}^*$ groups
R. G?bel and S. Shelah -- Reflexive subgroups of the Baer-Specker group and Martin's axiom
P. Hill, C. Megibben, and W. Ullery -- $\Sigma$-isotype subgroups of local $k$-groups
G. Ivanov -- Character modules and endomorphism rings of modules over Artinian serial rings
P. Loth -- Topologically pure extensions
N. R. McConnell and T. Stokes -- Rings having simple adjoint semigroup
A. Mader, L. G. Nongxa, and M. A. Ould-Beddi -- Invariants of global crq-groups
V. H. Mikaelian -- On varieties of groups generated by wreath products of abelian groups
O. Mutzbauer -- Existence of rigid indecomposable almost completely decomposable groups
W. K. Nicholson and M. F. Yousif -- C2-rings and the FGF-conjecture
B. L. Osofsky -- Lifting direct sum decompositions of bounded abelian $p$-groups
K. M. Rangaswamy -- On modules and submodules with finite projective dimension
L. Str?ngmann and S. L. Wallutis -- On the torsion groups in cotorsion classes
J. Trlifaj -- Cotorsion theories induced by tilting and cotilting modules
J. Zemlicka -- Steadiness is tested by a single module

Details:

Series: Contemporary Mathematics, Volume: 273
Publication Year: 2001
ISBN: 0-8218-2751-0
Paging: 308 pp.
Binding: Softcover

Edited by: Ken-ichi Maruyama, Chiba University, Japan,
and John W. Rutter, University of Liverpool, England

Groups of Homotopy Self-Equivalences and Related Topics

Expected publication date is March 29, 2001

Description

This volume offers the proceedings from the workshop held at the Gargnano Institute of the University of Milan (Italy) on groups of homotopy self-equivalences and related topics. The book comprises articles of current research on the group of homotopy self-equivalences, the homotopy of function spaces, rational homotopy theory, the classification of homotopy types, and equivariant homotopy theory.

Mathematicians from many areas of the globe attended the workshops to discuss their research and to share ideas. Included are two specially-written articles, by J. W. Rutter, reviewing the work done in the area of homotopy self-equivalences since 1988. Included also is a bibliography of some 122 articles published since 1988 and a list of problems. This book is suitable for both advanced graduate students and researchers.

Contents

J. W. Rutter -- Homotopy self-equivalences 1988-1999
J. W. Rutter -- Bibliography on $\mathcal E(X)$ 1988-1999
M. Arkowitz, G. Lupton, and A. Murillo -- Subgroups of the group of self-homotopy equivalences
S. Bauer, M. Crabb, and M. Spreafico -- The space of free loops on a real projective space
H.-J. Baues and Y. Drozd -- Indecomposable homotopy types with at most two non-trivial homology groups
H.-J. Baues and N. Iwase -- Square rings associated to elements in homotopy groups of spheres
P. I. Booth -- Fibrations with product of Eilenberg-MacLane space fibres I
D. L. Ferrario -- Self homotopy equivalences of equivariant spheres
Y. Felix -- Two examples to illustrate properties of the group of self-equivalences of a finite CW complex $X$
A. Garv?n, A. Murillo, P. Pavesic, and A. Viruel -- Nilpotency and localization of groups of fibre homotopy equivalences
K. A. Hardie and K. H. Kamps -- The homotopy groups of the homotopy fibre of an induced map of function spaces
V. Hauschild -- Fibrations, self homotopy equivalences and negative derivations
K. Ishiguro -- Classifying spaces and a subgroup of the exceptional Lie group $G_2$
D. Kahn and C. Schwartz -- The structure of the Hurewicz homomorphism
H. J. Marcum -- Joins, diagonals and Hopf invariants
K.-i. Maruyama -- A subgroup of self homotopy equivalences which is invariant on genus
K. Morisugi -- Composition structure of the self maps of $SU(3)$ or $Sp(2)$
J. Mukai -- Self-homotopy of a suspension of the real 4-projective space
J. Pan and M. H. Woo -- Phantom elements and its applications
J. W. Rutter -- Homotopy equivalences of lens spaces of one-relator groups
H. Shiga, K. Tsukiyama, and T. Yamaguchi -- Principal $S^1$-bundles and forgetful maps
S. B. Smith -- Rational type of classifying spaces for fibrations
M. Arkowitz -- Problems on self-homotopy equivalences

Details:

Series: Contemporary Mathematics, ISSN: Volume: 274
Publication Year: 2001
ISBN: 0-8218-2683-2
Paging: 315 pp.
Binding: Softcover

Jorgen Jost, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

Bosonic Strings: A Mathematical Treatment

Expected publication date is April 22, 2001

Description

Presented in this book is a mathematical treatment of Bosonic string theory from the point of view of global geometry. As motivation, the author presents the theory of point particles and Feynman path integrals. He considers the theory of strings as a quantization of the classical Plateau problem for minimal surfaces. The conformal variance of the relevant functional, the Polyakov action or (in mathematical terminology) the Dirichlet integral, leads to an anomaly in the process of quantization. The mathematical concepts needed to resolve this anomaly via the Faddeev-Popov method are introduced, specifically the geometry of the Teichmu?ller and moduli spaces of Riemann surfaces and the corresponding function spaces, i.e., Hilbert spaces of Sobolev type and diffeomorphism groups. Other useful tools are the algebraic geometry of Riemann surfaces and infinite-dimensional determinants. Also discussed are the boundary regularity questions. The main result is a presentation of the string partition
function as an integral over a moduli space of Riemann surfaces. Some new physical concepts, such as D-branes, are also discussed.

This volume offers a mathematically rigorous treatment of some aspects of string theory, employs a global geometry approach, systematically treats strings with boundary, and carefully explains all mathematical concepts and tools.

Titles in this series are copublished with International Press, Cambridge, MA.

Contents

Point particles
The Bosonic string
Bibliography
Index

Details:

Series: AMS/IP Studies in Advanced Mathematics, Publication Year: 2001
ISBN: 0-8218-2644-1
Paging: approximately 112 pp.
Binding: Hardcover

Elliott H. Lieb, Princeton University, NJ, and Michael Loss,
Georgia Institute of Technology, Atlanta, GA

Analysis: Second Edition

Expected publication date is April 19, 2001

Description

Significantly revised and expanded, this new Second Edition provides readers at all levels--from beginning students to practicing analysts--with the basic concepts and standard tools necessary to solve problems of analysis, and how to apply these concepts to research in a variety of areas.

Authors Elliott Lieb and Michael Loss take you quickly from basic topics to methods that work successfully in mathematics and its applications. While omitting many usual typical textbook topics, Analysis includes all necessary definitions, proofs, explanations, examples, and exercises to bring the reader to an advanced level of understanding with a minimum of fuss, and, at the same time, doing so in a rigorous and pedagogical way. Many topics that are useful and important, but usually left to advanced monographs, are presented in Analysis, and these give the beginner a sense that the subject is alive and growing.

This new Second Edition incorporates numerous changes since the publication of the original 1997 edition, and includes:

Features:

a new chapter on eigenvalues that covers the min-max principle, semi-classical approximation, coherent states, Lieb-Thirring inequalities, and more extensive additions to chapters covering Sobolev Inequalities, including the Nash and Log Sobolev inequalities new material on Measure and Integration many new exercises and much more ...

The Second Edition continues its no-nonsense approach to the topic that has made it one of the best selling books on the subject. It is an authoritative, straight-forward volume that readers--from the graduate student, to the professional mathematician, to the physicist or engineer using analytical methods--will find useful both as a reference and as a guide to real problem solving.

About the authors: Elliott Lieb is Professor of Mathematics and Physics at Princeton University and is a member of the US, Austrian, and Danish Academies of Science. He is also the recipient of several prizes including the 1988 AMS/SIAM Birkhoff prize. Michael Loss is Professor of Mathematics at the Georgia Institute of Technology.

Contents

Measure and integration
$L^p$-spaces
Rearrangement inequalities
Integral inequalities
The Fourier transform
Distributions
The Sobolev spaces $H^1$ and $H^{1/2}$
Sobolev inequalities
Potential theory and Coulomb energies
Regularity of solutions of Poisson's equation
Introduction to the calculus of variations
More about eigenvalues
References
List of symbols
Index

Details:

Series: Graduate Studies in Mathematics, Volume: 14
Publication Year: 2001
ISBN: 0-8218-2783-9
Paging: approximately 326 pp.
Binding: Hardcover