Essential countability of treeable equivalence relations

We establish a dichotomy theorem characterizing the circumstances under which a treeable Borel equivalence relation E is essentially countable. Under additional topological assumptions on the treeing, we in fact show that E is essentially countable if and only if there is no continuous embedding of E1 into E. Our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations, and allow us to show that up to continuous Kakutani embeddability, there is a minimum Borel function which is not essentially countable-to-one

Granite : a planetary response to liquid water

Inaugural lecture delivered at Stellenbosch University on 7 October 2008.Granites are coarse-grained igneous rocks, rich in quartz and feldspars and containing one or more hydrous minerals, such as micas and amphiboles. They have crystallised from silica-rich magmas that contained significant amounts of dissolved H2O. Most such magmas are created when the pressures and temperatures, in hydrated rocks deep in the planet’s crust, exceed those of the solidus, producing melt and crystalline residue. During this process H2O need not be present in a free fluid, but the planet’s near-surface environments do need to have abundant liquid water to produce weathered and hydrated rocks that ultimately melt to make the magmas. Liquid water in sufficient amounts (oceans) to trigger the chain of processes that leads to the formation of granites occurs on only one terrestrial planet, namely Earth. This explains why only Earth of all the planets in the solar system has plate tectonics, granites, continents and terrestrial life

Relative Primeness and Borel Partition Properties for Equivalence Relations

We introduce a notion of relative primeness for equivalence relations, strengthening the notion of non-reducibility, and show for many standard benchmark equivalence relations that non-reducibility may be strengthened to relative primeness. We introduce several analogues of cardinal properties for Borel equivalence relations, including the notion of a prime equivalence relation and Borel partition properties on quotient spaces. In particular, we introduce a notion of Borel weak compactness, and characterize partition properties for the equivalence relations 2 and 1. We also discuss dichotomies related to primeness, and see that many natural questions related to Borel reducibility of equivalence relations may be viewed in the framework of relative primeness and Borel partition properties

Complemented sets, difference sets, and weakly wandering sequences

We consider the descriptive complexity of several sets of sequences of natural numbers, and show that the following are all complete analytic sets: the set of complemented sequences, the set of sequences containing an infinite difference set, the set of sequences which are weakly wandering sequences for some transformation, and several variants of these. We then use the same techniques to produce weakly wandering sequences with special properties, such as a sequence which is exhaustive weakly wandering for some transformation but which is not weakly wandering for any ergodic transformation. In this paper we consider descriptive aspects of weakly wandering sequences. These sequences are isomorphism invariants for measure-preserving transformations or Borel automorphisms introduced by Hajian and Kakutani in [6]. We first consider how difficult it is to determine whether some sequence can be a weakly wandering sequence for some transformation, a

Dichotomy Theorems for Families of Non-Cofinal Essential Complexity

We prove that for every Borel equivalence relation $E$, either $E$ is Borel reducible to $\mathbb{E}\_0$, or the family of Borel equivalence relations incompatible with $E$ has cofinal essential complexity. It follows that if $F$ is a Borel equivalence relation and $\cal F$ is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence relation $E$, either $E\in {\cal F}$ or $F$ is Borel reducible to $E$, then $\cal F$ consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a known theorem

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 complete separable metric space (X,d). Our first goal is to determine the exact complexity of the classification problem of general Polish metric spaces up to isometry. This work was motivated by a paper of Vershik [1998], where he remarks (in the beginning of Section 2): "The classification of Polish spaces up to isometry is an enormous task. More precisely, this classification is not 'smooth' in the modern terminology." Our Theorem 2.1 below quantifies precisely the enormity of this task. After doing this, we turn to special classes of Polish metric spaces and investigate the classification problems associated with them. Note that these classification problems are in principle no more complicated than the general one above. However, the determination of their exact complexity is not necessarily easier. The investigation of the classification problems naturally leads to some interesting results on the groups of isometries of Polish metric spaces. We shall also present these results below