Softcover ISBN:  9780821841372 
Product Code:  ULECT/40 
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eBook ISBN:  9781470421847 
Product Code:  ULECT/40.E 
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Softcover ISBN:  9780821841372 
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Product Code:  ULECT/40.B 
List Price:  $134.00 $101.50 
MAA Member Price:  $120.60 $91.35 
AMS Member Price:  $107.20 $81.20 
Softcover ISBN:  9780821841372 
Product Code:  ULECT/40 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $55.20 
eBook ISBN:  9781470421847 
Product Code:  ULECT/40.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 
Softcover ISBN:  9780821841372 
eBook ISBN:  9781470421847 
Product Code:  ULECT/40.B 
List Price:  $134.00 $101.50 
MAA Member Price:  $120.60 $91.35 
AMS Member Price:  $107.20 $81.20 

Book DetailsUniversity Lecture SeriesVolume: 40; 2006; 192 ppMSC: Primary 05; 22; 43; 46; Secondary 20; 28; 37
The “infinitedimensional groups” in the title refer to unitary groups of Hilbert spaces, the infinite symmetric group, groups of homeomorphisms of manifolds, groups of transformations of measure spaces, etc. The book presents an approach to the study of such groups based on ideas from geometric functional analysis and from exploring the interplay between dynamical properties of those groups, combinatorial Ramseytype theorems, and the phenomenon of concentration of measure.
The dynamics of infinitedimensional groups is very much unlike that of locally compact groups. For instance, every locally compact group acts freely on a suitable compact space (Veech). By contrast, a 1983 result by Gromov and Milman states that whenever the unitary group of a separable Hilbert space continuously acts on a compact space, it has a common fixed point.
In the book, this new fastgrowing theory is built strictly from wellunderstood examples up. The book has no close counterpart and is based on recent research articles. At the same time, it is organized so as to be reasonably selfcontained. The topic is essentially interdisciplinary and will be of interest to mathematicians working in geometric functional analysis, topological and ergodic dynamics, Ramsey theory, logic and descriptive set theory, representation theory, topological groups, and operator algebras.
ReadershipGraduate students and research mathematicians interested in representation theory, dynamical systems, geometric functional analysis, Ramsey theory, and descriptive set theory.

Table of Contents

Chapters

Introduction

Chapter 1. The Ramsey–Dvoretzky–Milman phenomenon

Chapter 2. The fixed point on compacta property

Chapter 3. The concentration property

Chapter 4. Lévy groups

Chapter 5. Urysohn metric space and its group of isometries

Chapter 6. Minimal flows

Chapter 7. Further aspects of concentration

Chapter 8. Oscillation stability and distortion


Additional Material

Reviews

This is a very wellwritten and lively exposition, with a number of basic examples worked out in detail. In comparison to the original version, while the set of topics treated is essentially the same, some chapters have been reorganized, updated and largely expanded.
Zentralblatt MATH


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The “infinitedimensional groups” in the title refer to unitary groups of Hilbert spaces, the infinite symmetric group, groups of homeomorphisms of manifolds, groups of transformations of measure spaces, etc. The book presents an approach to the study of such groups based on ideas from geometric functional analysis and from exploring the interplay between dynamical properties of those groups, combinatorial Ramseytype theorems, and the phenomenon of concentration of measure.
The dynamics of infinitedimensional groups is very much unlike that of locally compact groups. For instance, every locally compact group acts freely on a suitable compact space (Veech). By contrast, a 1983 result by Gromov and Milman states that whenever the unitary group of a separable Hilbert space continuously acts on a compact space, it has a common fixed point.
In the book, this new fastgrowing theory is built strictly from wellunderstood examples up. The book has no close counterpart and is based on recent research articles. At the same time, it is organized so as to be reasonably selfcontained. The topic is essentially interdisciplinary and will be of interest to mathematicians working in geometric functional analysis, topological and ergodic dynamics, Ramsey theory, logic and descriptive set theory, representation theory, topological groups, and operator algebras.
Graduate students and research mathematicians interested in representation theory, dynamical systems, geometric functional analysis, Ramsey theory, and descriptive set theory.

Chapters

Introduction

Chapter 1. The Ramsey–Dvoretzky–Milman phenomenon

Chapter 2. The fixed point on compacta property

Chapter 3. The concentration property

Chapter 4. Lévy groups

Chapter 5. Urysohn metric space and its group of isometries

Chapter 6. Minimal flows

Chapter 7. Further aspects of concentration

Chapter 8. Oscillation stability and distortion

This is a very wellwritten and lively exposition, with a number of basic examples worked out in detail. In comparison to the original version, while the set of topics treated is essentially the same, some chapters have been reorganized, updated and largely expanded.
Zentralblatt MATH