Analysis and Logic

This volume comprises articles from four outstanding researchers who work at the cusp of analysis and logic. The emphasis is on active research topics; many results are presented that have not been published before and open problems are formulated. Considerable effort has been made by the authors to integrate their articles and make them accessible to mathematicians new to the area.

Preface

ix

Introduction

xi

Ultraproducts in Analysis

1

(114)

C. Ward Henson

Jose Iovino

Introduction

3

(7)

Normed Space Structures

10

(3)

Signatures

13

(4)

Ultrapowers of Normed Space Structures

17

(4)

Positive Bounded Formulas

21

(10)

Basic Model Theory I

31

(3)

Quantifier-Free Formulas

34

(4)

Ultraproducts of Normed Space Structures

38

(5)

Basic Model Theory II

43

(12)

Isomorphic Ultrapowers

55

(17)

Alternative Formulations of the Theory

72

(3)

Homogeneous Structures

75

(4)

More Model Theory

79

(17)

Types

96

(19)

References

107

(4)

Index of Notation

111

(1)

Index

112

(3)

Actions of Polish Groups and Classification Problems

115

(74)

Alexander S. Kechris

Introduction

117

(3)

The General Glimm-Effros Dichotomy

120

(3)

Actions of Polish Groups

123

(3)

Actions of Countable Groups

126

(5)

Actions of Locally Compact Groups

131

(2)

Actions of the Infinite Symmetric Group

133

(4)

Turbulence I: Overview

137

(6)

Turbulence II: Basic Facts

143

(3)

Turbulence III: Induced Actions

146

(4)

Turbulence IV: Some Examples

150

(2)

Turbulence V: Calmness

152

(5)

Turbulence VI: The First Main Theorem

157

(6)

Turbulence VII: The Second Main Theorem

163

(26)

References

183

(3)

Index

186

(3)

On Subspaces, Asymptotic Structure, and Distortion of Banach Spaces; Connections with Logic

189

Edward Odell

Introduction

191

Background material: The 60's and 70's

194

The unconditional basic sequence problem and connections with distortion

208

Gowers' dichotomy: A block Ramsey Theorem

217

Distortion

223

Asymptotic structure

233

Ordinal Indices

244

The homogeneous Banach space problem

252

Concluding remarks

254

References

256

Index

265

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