In the probability measure preserving (p.m.p.) setup, a well-known consequence of Rokhlin's lemma is that any two aperiodic p.m.p. bijections are approximately conjugate, meaning that up to conjugacy, they only differ on sets of arbitrarily small measure. In the infinite measure-preserving (i.m.p.) setup, we will see why this is false and characterize approximate conjugacy for aperiodic i.m.p. bijections. This is joint work with Fabien Hoareau.
A characterisation of isometry topological groups of Polish ultrametric spaces is provided. This answers in the context of Polish spaces a problem of Krasner's. A finer analysis allows also to characterise isometry topological groups of subclasses of Polish ultrametric spaces, providing a solution to questions raised by Gao and Kechris. The groups obtained involve various kinds of generalised wreath products proposed in the literature by Hall, Holland, and Malicki. This is a joint work with A. Marcone and L. Motto Ros.
Given a class of groups, say the class of all countably infinite groups, it is natural to ask: what group properties are generic in the sense of Baire category in this class? For this question to make sense, we need a topology. In the case of countably infinite groups, a natural choice is the following: fix \(\mathbb{N}\) as a common universe and identify every group on \(\mathbb{N}\) with the group operation that defines it. These group operations form a Polish subspace in \(\mathbb{N}^{\mathbb{N}\times\mathbb{N}}\). I will talk about generic group properties in this setting, and, if time permits, I will also say a few words about other settings and generic properties in various classes of topological groups.
We show that there exists an equidecomposition between a closed disk and a closed square of the same area in \(\mathbb{R}^2\) by translations with algebraic irrational coordinates. Our proof uses a new method for bounding the discrepancy of product sets in the \(k\)-torus using only the Erdős–Turán inequality. This resolves a question of Laczkovich from 1990. We also obtain an improved upper bound on the number of pieces required to square the circle, by proving effective bounds on such discrepancy estimates for translations by certain algebraic irrational numbers. This builds on an idea of Frank Calegari for bounding certain sums of products of fractional parts of algebraic numbers, and some computer assistance. This is joint work with Spencer Unger.