New Directions in Descriptive Set Theory

@article{Kechris1999NewDI,
  title={New Directions in Descriptive Set Theory},
  author={Alexander S. Kechris},
  journal={Bulletin of Symbolic Logic},
  year={1999},
  volume={5},
  pages={161 - 174}
}
  • A. Kechris
  • Published 1 June 1999
  • Mathematics
  • Bulletin of Symbolic Logic
§1. I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical examples are ℝ n , ℂ n , (separable) Hilbert space and more generally all separable Banach spaces, the Cantor space 2ℕ, the Baire space ℕℕ, the infinite symmetric group S∞, the unitary group (of the Hilbert space), the group of measure preserving transformations of the… 

Linear algebraic groups and countable Borel equivalence relations

This paper is a contribution to the study of Borel equivalence relations on standard Borel spaces (i.e., Polish spaces equipped with their Borel structure). In mathematics one often deals with

Borel and analytic sets in locales.

We systematically develop analogs of basic concepts from classical descriptive set theory in the context of pointless topology. Our starting point is to take the elements of the free complete Boolean

Polish metric spaces: their classification and isometry groups

In this communication we present some recent results on the classification of Polish metric spaces up to isometry and on the isometry groups of Polish metric spaces. A Polish metric space is a

2 8 Ja n 20 00 How Many Turing Degrees are There ?

A Borel equivalence relation on a Polish space is countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the

Rigidity Theorems For Actions Of Product Groups And Countable Borel Equivalence Relations

This Memoir is both a contribution to the theory of Borel equivalence relations, considered up to Borel reducibility, and measure preserving group actions considered up to orbit equivalence. Here E

On the classification of Polish metric spaces up to isometry

We study the classification problem of Polish metric spaces up to isometry and the isometry groups of Polish metric spaces. In the framework of the descriptive set theory of definable equivalence

Borel and Countably Determined Reducibility in Nonstandard Domain

Abstract. We consider, in a nonstandard domain, reducibility of equivalence relations in terms of the Borel reducibility ≤B and the countably determined (CD, for brevity) reducibility ≤CD. This

The effective theory of Borel equivalence relations

Uniformity, universality, and computability theory

A number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations are proved, including the existence of refinements of Martin's ultrafilter on Turing invariant Borel sets to the invariant borel sets of equivalence Relations that are much finer than Turing equivalence.
...

References

SHOWING 1-10 OF 37 REFERENCES

Descriptive Set Theory

Descriptive Set Theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary

Analytic Ideals

  • S. Solecki
  • Mathematics
    Bulletin of Symbolic Logic
  • 1996
§1. Introduction. Ideals and filters of subsets of natural numbers have been studied by set theorists and topologists for a long time. There is a vast literature concerning various kinds of

New Dichotomies for Borel Equivalence Relations

Two new dichotomy theorems for Borel equivalence relations are announced and presented in context by giving an overview of related recent developments.

The Descriptive Set Theory of Polish Group Actions: BETTER TOPOLOGIES

In this book the authors present their research into the foundations of the theory of Polish groups and the associated orbit equivalence relations. The particular case of locally compact groups has

The structure of hy-per nite Borel equivalence relations

We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not

The classification of hypersmooth Borel equivalence relations

This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which

Ergodic theory of amenable group actions. I: The Rohlin lemma

Classically, ergodic theory began with the study of flows or actions of R. Later, for technical reasons, much of the theory was first developed for actions of Z. More recently, there has been

Noncommutative Geometry

Noncommutative Spaces It was noticed a long time ago that various properties of sets of points can be restated in terms of properties of certain commutative rings of functions over those sets. In

Classical descriptive set theory

Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the