Borel Reducibility of Equivalence Relations

Spring Meeting 2017

About Borel Reducibility of Equivalence Relations

Classification has always been a central theme in mathematics. The study of Borel Reducibility of Equivalence Relations deals with the classification of points of standard Borel spaces up to equivalence relations by explicit, or Borel, mappings between such spaces. This idea gives rise to a notion of complexity of equivalence relations, and tools from Descriptive Set Theory are used to compare such relations and measure their complexities.

Abstracts and Slides

Julien Melleray

An Introduction to the Theory of Borel Complexity of Classification Problems

The aim of the talk is to give a gentle introduction to Borel reducibility theory, discuss a few examples and survey some results and open problems.

Andrew Brooke-Taylor

Borel Completeness of a Knot Invariant

The algebraic structures called "quandles" constitute a complete invariant for knots: there is a uniform way to assign a quandle to any tame knot, in such a way that different knots give rise to different quandles. However, knot theorists are generally unimpressed with quandles, complaining that they seem to be too hard to tell apart. I will talk about joint work with Sheila Miller using the concepts of Borel reducibility to show just how difficult this problem is. If time permits, I will also discuss ongoing work with Miller and Filippo Calderoni extending these results by considering the embeddability relation of quandles.

Luca Motto Ros

When Borel Reducibility is not Enough

In the last 30 years, Borel reducibility has proven to be an invaluable tool for tackling (and often solving, in a way or another) various classification problems arising in mathematics. However, there are situations in which such reducibility is not sufficiently strong to capture the essence of the problem or to give a satisfactory solution to it. In this talk we will discuss some strengthenings of the notion of Borel reducibility that have been proposed in the literature, compare them to one another, and mention some applications which motivated their introduction.

Raphaƫl Carroy

Topological Embeddability Between Functions

A function f embeds topologically in a function g if there are two topological embeddings b and c such that bf=gc. This notion was considered by Sławomir Solecki to find finite bases for non sigma-continuous functions between Polish spaces. Since then, finite bases have been found for several other classes of functions, including a finite basis for non Baire class one functions (joint work with Benjamin Miller). As a quasi-order, topological embeddability is however far from being always "simple". As a matter of fact, even on spaces of continuous functions it is very often analytic complete. I will pinpoint exactly when topological embeddability is an analytic complete quasi-order on continuous functions between Polish 0-dimensional spaces. This gives that on these spaces of functions, topological embeddability is either analytic complete or well-founded and without infinite antichains (joint work with Yann Pequignot and Zoltan Vidnyanszky).

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