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Publisher:

Chapman & Hall/CRC

Publication Date:

2009

Number of Pages:

383

Format:

Hardcover

Series:

Pure and Applied Mathematics 293

Price:

89.95

ISBN:

9781584887935

Category:

Monograph

[Reviewed by , on ]

Leon Harkleroad

03/16/2009

Blend together some real analysis and point-set topology. Add a handful of group theory. Mix well with model theory and computability theory from mathematical logic. Spice with a dash of graph theory and other flavors. The field known as invariant descriptive set theory contains many more ingredients — both in its tools and in its objects of study — than its name suggests, and Gao’s book provides a generous helping.

This area focuses on equivalence relations that arise in a variety of mathematical contexts. Typical examples include:

- for real numbers x ≈ y iff x – y is rational;
- for subsets of the free group on two generators, A≈ B iff for some g, A = gB;
- for countable locally finite trees, T
_{1}≈ T_{2}iff they are graph isomorphic.

Given two equivalence relations ≈ and ≈' that occur naturally in disparate mathematical contexts, often one can “nicely” (i.e., uniformly, via a Borel function) convert questions about ≈ into questions that ≈' can answer. Invariant descriptive set theory examines and compares equivalence relations from this viewpoint, thereby in Gao’s words, “making new connections between mathematical fields.”

This book, which grew out of lecture notes from a short course that the author gave as a visitor at the University of Notre Dame in 2005, was written to serve as a graduate text. Accordingly, Gao supplies necessary background material, detailed proofs, and plenty of exercises. In 2000, Greg Hjorth had also taught a short course in this area while a visitor at Notre Dame. Hjorth’s lecture notes evolved into a 43-page tutorial “Countable Models and the Theory of Borel Equivalence Relations,” one quarter of the compilation *The Notre Dame Lectures* [1].

Working on a larger canvas, Gao gets to develop the subject in more breadth and depth, as well as in a more leisurely fashion. On the other hand, Hjorth’s relative compactness sometimes helps make the forest more easily visible for the trees. Together, the two presentations make nice companion pieces, each in its own way providing a helpful and readable introduction to a young and developing area of research.

**References**

[1] *The Notre Dame Lectures*, ed. Peter Cholak. Lecture Notes in Logic 18, Association for Symbolic Logic/A K Peters, 2005.

Leon Harkleroad did his graduate work in computability theory at Notre Dame. He is glad that mathematical logic is flourishing there even more energetically than during his student days, which is more than can be said for football.

Polish spaces

The universal Urysohn space

Borel sets and Borel functions

Standard Borel spaces

The effective hierarchy

Analytic sets and Σ 1/1 sets

Coanalytic sets and Π 1/1 sets

The Gandy–Harrington topology

Metrics on topological groups

Polish groups

Continuity of homomorphisms

The permutation group

Universal Polish groups

The Graev metric groups

Polish

The Vaught transforms

Borel

Orbit equivalence relations

Extensions of Polish group actions

The logic actions

Strong Choquet spaces

Change of topology

Finer topologies on Polish

Topological realization of Borel

Borel reductions

Faithful Borel reductions

Perfect set theorems for equivalence relations

Smooth equivalence relations

The equivalence relation

Orbit equivalence relations embedding

The Harrington–Kechris–Louveau theorem

Consequences of the Glimm–Effros dichotomy

Actions of cli Polish groups

Generalities of countable Borel equivalence relations

Hyperfinite equivalence relations

Universal countable Borel equivalence relations

Amenable groups and amenable equivalence relations

Actions of locally compact Polish groups

Hypersmooth equivalence relations

Borel orbit equivalence relations

A jump operator for Borel equivalence relations

Examples of

Examples of Π

The Burgess trichotomy theorem

Definable reductions among analytic equivalence relations

Actions of standard Borel groups

Wild Polish groups

The topological Vaught conjecture

Homomorphisms and generic ergodicity

Local orbits of Polish group actions

Turbulent and generically turbulent actions

The Hjorth turbulence theorem

Examples of turbulence

Orbit equivalence relations and

A review of first-order logic

Model theory of infinitary logic

Invariant Borel classes of countable models

Polish topologies generated by countable fragments

Atomic models and

Elements of the Scott analysis

Borel approximations of isomorphism relations

The Scott rank and computable ordinals

A topological variation of the Scott analysis

Sharp analysis of

Countable graphs

Countable trees

Countable linear orderings

Countable groups

Standard Borel structures on hyperspaces

Classification versus nonclassification

Measurement of complexity

Classification notions

Classification problems up to essential countability

A roadmap of Borel equivalence relations

Orbit equivalence relations

General Σ 1/1 equivalence relations

Beyond analyticity

The Gandy basis theorem

The Gandy–Harrington topology on

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