Ergodic Theory

Author: David Kerr
Publisher: Springer
ISBN: 9783319498478
Release Date: 2017-02-09
Genre: Mathematics

This book provides an introduction to the ergodic theory and topological dynamics of actions of countable groups. It is organized around the theme of probabilistic and combinatorial independence, and highlights the complementary roles of the asymptotic and the perturbative in its comprehensive treatment of the core concepts of weak mixing, compactness, entropy, and amenability. The more advanced material includes Popa's cocycle superrigidity, the Furstenberg-Zimmer structure theorem, and sofic entropy. The structure of the book is designed to be flexible enough to serve a variety of readers. The discussion of dynamics is developed from scratch assuming some rudimentary functional analysis, measure theory, and topology, and parts of the text can be used as an introductory course. Researchers in ergodic theory and related areas will also find the book valuable as a reference.

Fractal Geometry and Stochastics IV

Author: Christoph Bandt
Publisher: Springer Science & Business Media
ISBN: 3034600305
Release Date: 2010-01-08
Genre: Mathematics

Over the last fifteen years fractal geometry has established itself as a substantial mathematical theory in its own right. The interplay between fractal geometry, analysis and stochastics has highly influenced recent developments in mathematical modeling of complicated structures. This process has been forced by problems in these areas related to applications in statistical physics, biomathematics and finance. This book is a collection of survey articles covering many of the most recent developments, like Schramm-Loewner evolution, fractal scaling limits, exceptional sets for percolation, and heat kernels on fractals. The authors were the keynote speakers at the conference "Fractal Geometry and Stochastics IV" at Greifswald in September 2008.

Ramsey Theory for Product Spaces

Author: Pandelis Dodos
Publisher: American Mathematical Soc.
ISBN: 9781470428082
Release Date: 2016-05-16
Genre: Combinatorial analysis

Ramsey theory is a dynamic area of combinatorics that has various applications in analysis, ergodic theory, logic, number theory, probability theory, theoretical computer science, and topological dynamics. This book is devoted to one of the most important areas of Ramsey theory—the Ramsey theory of product spaces. It is a culmination of a series of recent breakthroughs by the two authors and their students who were able to lift this theory to the infinite-dimensional case. The book presents many major results and methods in the area, such as Szemerédi's regularity method, the hypergraph removal lemma, and the density Hales–Jewett theorem. This book addresses researchers in combinatorics but also working mathematicians and advanced graduate students who are interested in Ramsey theory. The prerequisites for reading this book are rather minimal: it only requires familiarity, at the graduate level, with probability theory and real analysis. Some familiarity with the basics of Ramsey theory would be beneficial, though not necessary.

First European Congress of Mathematics Paris July 6 10 1992

Author: Anthony Joseph
Publisher: Nelson Thornes
ISBN: 3764327995
Release Date: 1994-07
Genre: Mathematics

Table of Contents: D. Duffie: Martingales, Arbitrage, and Portfolio Choice • J. Fröhlich: Mathematical Aspects of the Quantum Hall Effect • M. Giaquinta: Analytic and Geometric Aspects of Variational Problems for Vector Valued Mappings • U. Hamenstädt: Harmonic Measures for Leafwise Elliptic Operators Along Foliations • M. Kontsevich: Feynman Diagrams and Low-Dimensional Topology • S.B. Kuksin: KAM-Theory for Partial Differential Equations • M. Laczkovich: Paradoxical Decompositions: A Survey of Recent Results • J.-F. Le Gall: A Path-Valued Markov Process and its Connections with Partial Differential Equations • I. Madsen: The Cyclotomic Trace in Algebraic K-Theory • A.S. Merkurjev: Algebraic K-Theory and Galois Cohomology • J. Nekovár: Values of L-Functions and p-Adic Cohomology • Y.A. Neretin: Mantles, Trains and Representations of Infinite Dimensional Groups • M.A. Nowak: The Evolutionary Dynamics of HIV Infections • R. Piene: On the Enumeration of Algebraic Curves - from Circles to Instantons • A. Quarteroni: Mathematical Aspects of Domain Decomposition Methods • A. Schrijver: Paths in Graphs and Curves on Surfaces • B. Silverman: Function Estimation and Functional Data Analysis • V. Strassen: Algebra and Complexity • P. Tukia: Generalizations of Fuchsian and Kleinian Groups • C. Viterbo: Properties of Embedded Lagrange Manifolds • D. Voiculescu: Alternative Entropies in Operator Algebras • M. Wodzicki : Algebraic K-Theory and Functional Analysis • D. Zagier: Values of Zeta Functions and Their Applications

C algebras and Finite dimensional Approximations

Author: Nathanial Patrick Brown
Publisher: American Mathematical Soc.
ISBN: 9780821843819
Release Date: 2008
Genre: Mathematics

$\mathrm{C}^*$-approximation theory has provided the foundation for many of the most important conceptual breakthroughs and applications of operator algebras. This book systematically studies (most of) the numerous types of approximation properties that have been important in recent years: nuclearity, exactness, quasidiagonality, local reflexivity, and others. Moreover, it contains user-friendly proofs, insofar as that is possible, of many fundamental results that were previously quite hard to extract from the literature. Indeed, perhaps the most important novelty of the first ten chapters is an earnest attempt to explain some fundamental, but difficult and technical, results as painlessly as possible. The latter half of the book presents related topics and applications--written with researchers and advanced, well-trained students in mind. The authors have tried to meet the needs both of students wishing to learn the basics of an important area of research as well as researchers who desire a fairly comprehensive reference for the theory and applications of $\mathrm{C}^*$-approximation theory.

Groups St Andrews 2005

Author: C. M. Campbell
Publisher: Cambridge University Press
ISBN: 9780521694698
Release Date: 2007-01-04
Genre: Mathematics

Selected papers from 'Groups St Andrews 2005' cover a wide spectrum of modern group theory.

Ergodic Theory via Joinings

Author: Eli Glasner
Publisher: American Mathematical Soc.
ISBN: 9781470419516
Release Date: 2015-01-09

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.

Global Aspects of Ergodic Group Actions

Author: A. S. Kechris
Publisher: American Mathematical Soc.
ISBN: 9780821848944
Release Date: 2010
Genre: Mathematics

The subject of this book is the study of ergodic, measure preserving actions of countable discrete groups on standard probability spaces. It explores a direction that emphasizes a global point of view, concentrating on the structure of the space of measure preserving actions of a given group and its associated cocycle spaces. These are equipped with canonical topological actions that give rise to the usual concepts of conjugacy of actions and cohomology of cocycles. Structural properties of discrete groups such as amenability, Kazhdan's property (T) and the Haagerup Approximation Property play a significant role in this theory as they have important connections to the global structure of these spaces. One of the main topics discussed in this book is the analysis of the complexity of the classification problems of conjugacy and orbit equivalence of actions, as well as of cohomology of cocycles. This involves ideas from topological dynamics, descriptive set theory, harmonic analysis, and the theory of unitary group representations. Also included in this title is a study of properties of the automorphism group of a standard probability space and some of its important subgroups, such as the full and automorphism groups of measure preserving equivalence relations and connections with the theory of costs. The book contains nine appendices that present necessary background material in functional analysis, measure theory, and group representations, thus making the book accessible to a wider audience.