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.