Alain Louveau (CNRS and Université Paris 6, Paris, France)
In the seventies, a general program was proposed in Descriptive Set Theory (in its most explicit form by H. Friedman), to the effect of understanding simply definable (in particular Borel) mathematical structures. The motivation, similar to the one at the origin of the Descriptive Theory of simply definable sets, was to try to draw a dividing line between well-behaved and pathological structures, the existence of those being hopefully due only to the too wide generality of the notion of arbitray set or structure.
Since then, this program has been developed in various directions. In this talk, I will report on some of the progress made on understanding a specific type of mathematical structures, equivalence relations. I will mainly consider Borel equivalence relations, i.e. equivalence relations on a Polish space $X$ which are Borel as subsets of $X^2$, which are best understood. But I will also indicate some (usually weaker) extensions of the results to the more general class of analytic (continuous images of Borel) equivalence relations. This extended class contains in particular all orbit equivalence relations induced by Borel actions of Polish groups on Polish spaces. Many "symmetry" groups are naturally Polish groups, like the unitary group on the separable Hilbert space, the group of isometries of a complete separable metric space, the group of homeomorphisms of a compact metrizable space, the group $S_\infty({\bf N})$ of permutations of ${\bf N}$, etc. Hence many interesting isomorphism relations are analytic (but not necessarily Borel), such as the isomorphism relation between countable models of a given first-order theory in Model Theory.
The main question I will discuss will be that of the possible "size" of the quotient space $X/E$, for such an equivalence relation $E$ on a Polish space $X$. In the first part, I will discuss the cardinality of $X/E$, and present the dichotomy results of Silver (which implies that for Borel $E$, $X/E$ is either countable, or has the cardinality of the continuum), and of Burgess (that for $E$ analytic, $\omega_1$ is the only other possibility). I will also relate these results to the still open conjecture of Baught in Model Theory.
In the second part, I will introduce and study an alternative notion of size: for equivalence relations $E$ and $F$ on Polish spaces $X$ and $Y$ respectively, say that $E$ is Borel reducible to $F$ if there exists a Borel map $f\colon X\to Y$ such that for all $x$ and $y$ in $X$, $xEy\Rightleftarrow f(x)Ff(y)$ (the induced map$: $X/E\to Y/F$ is then one-to-one). The associated equivalence relation of Borel bi-reducibility can then heuristically be viewed as a "Borel" version of cardinality for the quotient spaces.
In contrast with the above results about cardinality of quotients, there are many possible "Borel sizes" of quotients, even in the case of Borel equivalence relations. However, there are still dichotomy results for this new notion of size, and I will discuss those due to Harrington, Kechris and myself (extending earlier work of Glimm and Effros) in the Borel case, and to Hjorth and Kechir in the analytic case.