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A Short Course in Differential Topology
This book offers a concise and modern introduction to differential topology, the study of smooth manifolds and their properties, at the advanced undergraduate/beginning graduate level. The treatment throughout is hands-on, including many concrete examples and exercises woven into the text with hints provided to guide the student.
Cyclic Cohomology at 40: Achievements and Future Prospects
This volume contains the proceedings of the virtual conference on Cyclic Cohomology at 40: Achievements and Future Prospects, held from September 27–October 1, 2021 and hosted by the Fields Institute for Research in Mathematical Sciences, Toronto, ON, Canada. Cyclic cohomology, since its discovery forty years ago in noncommutative differential geometry, has become a fundamental mathematical tool with applications in domains as diverse as analysis, algebraic K-theory, algebraic geometry, arithmetic geometry, solid state physics and quantum field theory. The reader will find survey articles providing a user-friendly introduction to applications of cyclic cohomology in such areas as higher ca...
The Local Structure of Algebraic K-Theory
Algebraic K-theory encodes important invariants for several mathematical disciplines, spanning from geometric topology and functional analysis to number theory and algebraic geometry. As is commonly encountered, this powerful mathematical object is very hard to calculate. Apart from Quillen's calculations of finite fields and Suslin's calculation of algebraically closed fields, few complete calculations were available before the discovery of homological invariants offered by motivic cohomology and topological cyclic homology. This book covers the connection between algebraic K-theory and Bökstedt, Hsiang and Madsen's topological cyclic homology and proves that the difference between the the...
Motivic Homotopy Theory
This book is based on lectures given at a summer school on motivic homotopy theory at the Sophus Lie Centre in Nordfjordeid, Norway, in August 2002. Aimed at graduate students in algebraic topology and algebraic geometry, it contains background material from both of these fields, as well as the foundations of motivic homotopy theory. It will serve as a good introduction as well as a convenient reference for a broad group of mathematicians to this important and fascinating new subject. Vladimir Voevodsky is one of the founders of the theory and received the Fields medal for his work, and the other authors have all done important work in the subject.
Spaces of PL Manifolds and Categories of Simple Maps
Since its introduction by Friedhelm Waldhausen in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing Waldhausen's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a "desingularization," improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.
Geometry and Topology: Aarhus
This volume includes both survey and research articles on major advances and future developments in geometry and topology. Papers include those presented as part of the 5th Aarhus Conference - a meeting of international participants held in connection with ICM Berlin in 1998 - and related papers on the subject. This collection of papers is aptly published in the Contemporary Mathematics series, as the works represent the state of research and address areas of future development in the area of manifold theory and geometry. The survey articles in particular would serve well as supplemental resources in related graduate courses.
Structure and Regularity of Group Actions on One-Manifolds
This book presents the theory of optimal and critical regularities of groups of diffeomorphisms, from the classical work of Denjoy and Herman, up through recent advances. Beginning with an investigation of regularity phenomena for single diffeomorphisms, the book goes on to describes a circle of ideas surrounding Filipkiewicz's Theorem, which recovers the smooth structure of a manifold from its full diffeomorphism group. Topics covered include the simplicity of homeomorphism groups, differentiability of continuous Lie group actions, smooth conjugation of diffeomorphism groups, and the reconstruction of spaces from group actions. Various classical and modern tools are developed for controlling the dynamics of general finitely generated group actions on one-dimensional manifolds, subject to regularity bounds, including material on Thompson's group F, nilpotent groups, right-angled Artin groups, chain groups, finitely generated groups with prescribed critical regularities, and applications to foliation theory and the study of mapping class groups. The book will be of interest to researchers in geometric group theory.
Noncommutative Geometry and Physics 3
Noncommutative differential geometry has many actual and potential applications to several domains in physics ranging from solid state to quantization of gravity. The strategy is to formulate usual differential geometry in a somewhat unusual manner, using in particular operator algebras and related concepts, so as to be able to plug in noncommutativity in a natural way. Algebraic tools such as K-theory and cyclic cohomology and homology play an important role in this field.
Advances in Noncommutative Geometry
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.
From Categories to Homotopy Theory
Bridge the gap between category theory and its applications in homotopy theory with this guide for graduate students and researchers.
