The connective topological modular forms spectrum, tmf, is in a sense initial among elliptic spectra, and as such is an important link between the homotopy groups of spheres and modular forms. A primary goal of this volume is to give a complete account, with full proofs, of the homotopy of tmf and several tmf-module spectra by means of the classical Adams spectral sequence, thus verifying, correcting, and extending existing approaches. In the process, folklore results are made precise and generalized. Anderson and Brown-Comenetz duality, and the corresponding dualities in homotopy groups, are carefully proved. The volume also includes an account of the homotopy groups of spheres through degree 44, with complete proofs, except that the Adams conjecture is used without proof. Also presented are modern stable proofs of classical results which are hard to extract from the literature.
Tools used in this book include a multiplicative spectral sequence generalizing a construction of Davis and Mahowald, and computer software which computes the cohomology of modules over the Steenrod algebra and products therein. Techniques from commutative algebra are used to make the calculation precise and finite. The H∞ ring structure of the sphere and of tmf are used to determine many differentials and relations.
Graduate students and researchers interested in algebraic topology, specifically in stable homotopy theory.
Mathematical Surveys and Monographs, Volume: 253
2021; 690 pp; Softcover
MSC: Primary 18; 55;
Print ISBN: 978-1-4704-6958-0
Product Code: SURV/253.S
Ultrafilters and ultraproducts provide a useful generalization of the ordinary limit processes which have applications to many areas of mathematics. Typically, this topic is presented to students in specialized courses such as logic, functional analysis, or geometric group theory. In this book, the basic facts about ultrafilters and ultraproducts are presented to readers with no prior knowledge of the subject and then these techniques are applied to a wide variety of topics. The first part of the book deals solely with ultrafilters and presents applications to voting theory, combinatorics, and topology, while also dealing also with foundational issues. The second part presents the classical ultraproduct construction and provides applications to algebra, number theory, and nonstandard analysis. The third part discusses a metric generalization of the ultraproduct construction and gives example applications to geometric group theory and functional analysis. The final section returns to more advanced topics of a more foundational nature.
The book should be of interest to undergraduates, graduate students, and researchers from all areas of mathematics interested in learning how ultrafilters and ultraproducts can be applied to their specialty.
Undergraduate and graduate students and researchers interested in ultrafilters and ultraproducts in geometric group theory, combinatorics, and number theory.
Graduate Studies in Mathematics, Volume: 220
2022; 408 pp; Hardcover
MSC: Primary 03; 54;
Print ISBN: 978-1-4704-6900-9
Product Code: GSM/220
William Thurston's work has had a profound influence on mathematics. He connected whole mathematical subjects in entirely new ways and changed the way mathematicians think about geometry, topology, foliations, group theory, dynamical systems, and the way these areas interact. His emphasis on understanding and imagination in mathematical learning and thinking are integral elements of his distinctive legacy.
This four-part collection brings together in one place Thurston's major writings, many of which are appearing in publication for the first time. Volumes I?III contain commentaries by the Editors. Volume IV includes a preface by Steven P. Kerckhoff.
Volume IV contains Thurston's highly influential, though previously unpublished, 1977?78 Princeton Course Notes on the Geometry and Topology of 3-manifolds. It is an indispensable part of the Thurston collection but can also be used on its own as a textbook or for self-study.
Graduate students and researchers interested in geometric topology, geometric group theory, low-dimensional topology, and dynamical systems of rational maps.
Collected Works, Volume: 27
2021; 316 pp; Hardcover
MSC: Primary 57; 53;
Print ISBN: 978-1-4704-6391-5
Product Code: CWORKS/27.4
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
This book provides the main results and ideas in the theories of completely bounded maps, operator spaces, and operator algebras, along with some of their main applications. It requires only a basic background in functional analysis to read through the book. The descriptions and discussions of the topics are self-explained. It is appropriate for graduate students new to the subject and the field.
The book starts with the basic representation theorems for abstract operator spaces and their mappings, followed by a discussion of tensor products and the analogue of Grothendieck's approximation property. Next, the operator space analogues of the nuclear, integral, and absolutely summing mappings are discussed. In what is perhaps the deepest part of the book, the authors present the remarkable “non-classical” phenomena that occur when one considers local reflexivity and exactness for operator spaces. This is an area of great beauty and depth, and it represents one of the triumphs of the subject. In the final part of the book, the authors consider applications to non-commutative harmonic analysis and non-self-adjoint operator algebra theory.
Operator space theory provides a synthesis of Banach space theory with the non-commuting variables of operator algebra theory, and it has led to exciting new approaches in both disciplines. This book is an indispensable introduction to the theory of operator spaces.
ReadershipGraduate students and researchers interested in functional analysis and operator theory.
AMS Chelsea Publishing, Volume: 386
2022; 358 pp; Softcover
MSC: Primary 46; Secondary 47
Print ISBN: 978-1-4704-6505-6
Product Code: CHEL/386
William Thurston's work has had a profound influence on mathematics. He connected whole mathematical subjects in entirely new ways and changed the way mathematicians think about geometry, topology, foliations, group theory, dynamical systems, and the way these areas interact. His emphasis on understanding and imagination in mathematical learning and thinking are integral elements of his distinctive legacy.
This four-part collection brings together in one place Thurston's major writings, many of which are appearing in publication for the first time. Volumes I?III contain commentaries by the Editors. Volume IV includes a preface by Steven P. Kerckhoff.
Volume I contains William Thurston's papers on foliations, mapping classes groups, and differential geometry.
Graduate students and researchers interested in geometric topology, geometric group theory, low-dimensional topology, and dynamical systems of rational maps.
Collected Works, Volume: 27
2021; Hardcover
MSC: Primary 57; 53;
Print ISBN: 978-1-4704-6388-5
Product Code: CWORKS/27.1
CWORKS/27.2
CWORKS/27.3
CWORKS/27.4
Paperback ISBN: 9780128241608
Differential Equations with Mathematica, Fifth Edition uses the fundamental concepts of the popular platform to solve (analytically, numerically, and/or graphically) differential equations of interest to students, instructors, and scientists. Mathematica’s diversity makes it particularly well suited to performing calculations encountered when solving many ordinary and partial differential equations. In some cases, Mathematica’s built-in functions can immediately solve a differential equation by providing an explicit, implicit, or numerical solution. In other cases, Mathematica can be used to perform the calculations encountered when solving a differential equation. Because one goal of elementary differential equations courses is to introduce students to basic methods and algorithms so that they gain proficiency in them, nearly every topic covered this book introduces basic commands, also including typical examples of their application. A study of differential equations relies on concepts from calculus and linear algebra, so this text also includes discussions of relevant commands useful in those areas. In many cases, seeing a solution graphically is most meaningful, so the book relies heavily on Mathematica’s outstanding graphics capabilities.