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| Author | : Bekka M Bachir La Harpe Pierre de Valette Alain |
| Publisher | : |
| Total Pages | : 488 |
| Release | : 2014-05-14 |
| Genre | : Mathematics |
| ISBN | : 9780511395116 |
A comprehensive introduction to the role of Property (T), with applications to an amazing number of fields within mathematics.
| Author | : Alan L. T. Paterson |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 474 |
| Release | : 1988 |
| Genre | : Mathematics |
| ISBN | : 0821809857 |
The subject of amenability has its roots in the work of Lebesgue at the turn of the century. In the 1940s, the subject began to shift from finitely additive measures to means. This shift is of fundamental importance, for it makes the substantial resources of functional analysis and abstract harmonic analysis available to the study of amenability. The ubiquity of amenability ideas and the depth of the mathematics involved points to the fundamental importance of the subject. This book presents a comprehensive and coherent account of amenability as it has been developed in the large and varied literature during this century. The book has a broad appeal, for it presents an account of the subject based on harmonic and functional analysis. In addition, the analytic techniques should be of considerable interest to analysts in all areas. In addition, the book contains applications of amenability to a number of areas: combinatorial group theory, semigroup theory, statistics, differential geometry, Lie groups, ergodic theory, cohomology, and operator algebras. The main objectives of the book are to provide an introduction to the subject as a whole and to go into many of its topics in some depth. The book begins with an informal, nontechnical account of amenability from its origins in the work of Lebesgue. The initial chapters establish the basic theory of amenability and provide a detailed treatment of invariant, finitely additive measures (i.e., invariant means) on locally compact groups. The author then discusses amenability for Lie groups, "almost invariant" properties of certain subsets of an amenable group, amenability and ergodic theorems, polynomial growth, and invariant mean cardinalities. Also included are detailed discussions of the two most important achievements in amenability in the 1980s: the solutions to von Neumann's conjecture and the Banach-Ruziewicz Problem. The main prerequisites for this book are a sound understanding of undergraduate-level mathematics and a knowledge of abstract harmonic analysis and functional analysis. The book is suitable for use in graduate courses, and the lists of problems in each chapter may be useful as student exercises.
| Author | : Pierre-Emmanuel Caprace |
| Publisher | : Cambridge University Press |
| Total Pages | : 367 |
| Release | : 2018-02-08 |
| Genre | : Mathematics |
| ISBN | : 1108349544 |
This collection of expository articles by a range of established experts and newer researchers provides an overview of the recent developments in the theory of locally compact groups. It includes introductory articles on totally disconnected locally compact groups, profinite groups, p-adic Lie groups and the metric geometry of locally compact groups. Concrete examples, including groups acting on trees and Neretin groups, are discussed in detail. An outline of the emerging structure theory of locally compact groups beyond the connected case is presented through three complementary approaches: Willis' theory of the scale function, global decompositions by means of subnormal series, and the local approach relying on the structure lattice. An introduction to lattices, invariant random subgroups and L2-invariants, and a brief account of the Burger–Mozes construction of simple lattices are also included. A final chapter collects various problems suggesting future research directions.
| Author | : Pieter Moree |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 360 |
| Release | : 2020-02-12 |
| Genre | : Education |
| ISBN | : 147045100X |
This volume contains the proceedings of the conference Dynamics: Topology and Numbers, held from July 2–6, 2018, at the Max Planck Institute for Mathematics, Bonn, Germany. The papers cover diverse fields of mathematics with a unifying theme of relation to dynamical systems. These include arithmetic geometry, flat geometry, complex dynamics, graph theory, relations to number theory, and topological dynamics. The volume is dedicated to the memory of Sergiy Kolyada and also contains some personal accounts of his life and mathematics.
| Author | : R. Diestel |
| Publisher | : Elsevier |
| Total Pages | : 392 |
| Release | : 2016-06-06 |
| Genre | : Mathematics |
| ISBN | : 148329479X |
This book has arisen from a colloquium held at St. John's College, Cambridge, in July 1989, which brought together most of today's leading experts in the field of infinite graph theory and combinatorics. This was the first such meeting ever held, and its aim was to assess the state of the art in the discipline, to consider its links with other parts of mathematics, and to discuss possible directions for future development. This volume reflects the Cambridge meeting in both level and scope. It contains research papers as well as expository surveys of particular areas. Together they offer a comprehensive portrait of infinite graph theory and combinatorics, which should be particularly attractive to anyone new to the discipline.
| Author | : Terence Tao |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 319 |
| Release | : 2015-04-16 |
| Genre | : Mathematics |
| ISBN | : 1470421968 |
Expander graphs are an important tool in theoretical computer science, geometric group theory, probability, and number theory. Furthermore, the techniques used to rigorously establish the expansion property of a graph draw from such diverse areas of mathematics as representation theory, algebraic geometry, and arithmetic combinatorics. This text focuses on the latter topic in the important case of Cayley graphs on finite groups of Lie type, developing tools such as Kazhdan's property (T), quasirandomness, product estimates, escape from subvarieties, and the Balog-Szemerédi-Gowers lemma. Applications to the affine sieve of Bourgain, Gamburd, and Sarnak are also given. The material is largely self-contained, with additional sections on the general theory of expanders, spectral theory, Lie theory, and the Lang-Weil bound, as well as numerous exercises and other optional material.
| Author | : Eli Glasner |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 402 |
| Release | : 2015-01-09 |
| Genre | : Mathematics |
| ISBN | : 1470419513 |
This book introduces modern ergodic theory. It emphasizes a new approach that relies on the technique of joining two (or more) dynamical systems. This approach has proved to be fruitful in many recent works, and this is the first time that the entire theory is presented from a joining perspective. Another new feature of the book is the presentation of basic definitions of ergodic theory in terms of the Koopman unitary representation associated with a dynamical system and the invariant mean on matrix coefficients, which exists for any acting groups, amenable or not. Accordingly, the first part of the book treats the ergodic theory for an action of an arbitrary countable group. The second part, which deals with entropy theory, is confined (for the sake of simplicity) to the classical case of a single measure-preserving transformation on a Lebesgue probability space.
| Author | : Karl Endel Petersen |
| Publisher | : Cambridge University Press |
| Total Pages | : 452 |
| Release | : 1995 |
| Genre | : Ergodic theory |
| ISBN | : 0521459990 |
Tutorial survey papers on important areas of ergodic theory, with related research papers.
| Author | : B. Hasselblatt |
| Publisher | : Elsevier |
| Total Pages | : 1231 |
| Release | : 2002-08-20 |
| Genre | : Mathematics |
| ISBN | : 0080533442 |
Volumes 1A and 1B.These volumes give a comprehensive survey of dynamics written by specialists in the various subfields of dynamical systems. The presentation attains coherence through a major introductory survey by the editors that organizes the entire subject, and by ample cross-references between individual surveys.The volumes are a valuable resource for dynamicists seeking to acquaint themselves with other specialties in the field, and to mathematicians active in other branches of mathematics who wish to learn about contemporary ideas and results dynamics. Assuming only general mathematical knowledge the surveys lead the reader towards the current state of research in dynamics.Volume 1B will appear 2005.
| Author | : Yves Cornulier |
| Publisher | : European Mathematical Society |
| Total Pages | : 248 |
| Release | : 2016 |
| Genre | : Mathematics |
| ISBN | : 9783037191668 |
The main aim of this book is the study of locally compact groups from a geometric perspective, with an emphasis on appropriate metrics that can be defined on them. The approach has been successful for finitely generated groups and can be favorably extended to locally compact groups. Parts of the book address the coarse geometry of metric spaces, where ``coarse'' refers to that part of geometry concerning properties that can be formulated in terms of large distances only. This point of view is instrumental in studying locally compact groups. Basic results in the subject are exposed with complete proofs; others are stated with appropriate references. Most importantly, the development of the theory is illustrated by numerous examples, including matrix groups with entries in the the field of real or complex numbers, or other locally compact fields such as $p$-adic fields, isometry groups of various metric spaces, and last but not least, discrete groups themselves. The book is aimed at graduate students, advanced undergraduate students, and mathematicians seeking some introduction to coarse geometry and locally compact groups.
