The periodic tiling conjecture (PTC) in \(\mathbb{Z}^d\), that was recently disproved by Greenfeld and Tao, asserts that if a finite set \(F\subseteq \mathbb{Z}^d\) tiles \(\mathbb{Z}^d\) by translations, then it also tiles by (possibly different) translations that are periodic. On the other hand, the case \(d=2\) has an affirmative answer which was proved earlier by Bhattacharya. The analogous question in \(\mathbb{R}^2\) is open even for polygonal sets.Here, the most general result is by Kenyon, who proved that PTC holds for topological disks. In an ongoing work with de Dios Pont, Greenfeld and Madrid, we obtain a structure result about translational tilings by polygonal sets with rational slopes by connecting the ideas from the discrete and continuous case. Our main result states that if such a tiling is topologically minimal, then it is weakly periodic and satisfies a weak version of PTC. In the talk I will discuss the main differences between tilings in \(\mathbb{Z}^2\) and \(\mathbb{R}^2\) as well as the main ideas how to apply the structure theory of Greenfeld and Tao in the continuous setting.
There are two natural arithmetic operations on the class of linear orders, the sum \(+\) and lexicographic product \(\times\). These operations generalize the sum and product of ordinals.The arithmetic laws obeyed by the sum were uncovered in the pre-forcing days of set theory and are surprisingly nice. For example, while the right cancellation law \(A + X \cong B + X \Rightarrow A \cong B\) is not true for linear orders in general, its failure can be completely characterized: an order \(X\) fails to cancel in some such isomorphism if and only if there is a non-empty order \(R\) such that \(R + X \cong X\). Left cancellation is symmetrically characterized.Tarski and Aronszajn characterized the commuting pairs of linear orders, i.e. the pairs \(X\) and \(Y\) such that \(X + Y \cong Y + X\). Lindenbaum showed that \(X + X \cong Y + Y\) implies \(X \cong Y\) for linear orders \(X\) and \(Y\). More generally, we have the finite cancellation law \(nX \cong nY \Rightarrow X \cong Y\). There is even a sense in which the Euclidean algorithm holds for sums of linear orders.On the other hand, the arithmetic of the lexicographic product is much less well understood. The lone totally general classical result is due to Morel, who characterized the orders \(X\) for which the right cancellation law \(A \times X \cong B \times X \Rightarrow A \cong B\) holds. Morel showed that an order \(X\) fails to cancel in some such isomorphism if and only if there is a non-singleton order \(R\) such that \(R \times X \cong X\), in analogy with the additive case.In this talk we consider the question of whether Morel's cancellation theorem is true on the left. We'll show that, while the literal left-sided version of Morel’s theorem is false, an appropriately reformulated version is true. Our results suggest that a complete characterization of left cancellation in lexicographic products is possible. We’ll also discuss how our work might help in proving multiplicative versions of Tarski and Aronszajn's and Lindenbaum's additive laws.This is joint work with Eric Paul.
In his thesis, Arant introduced and studied the notion of Borel graphability: an analytic equivalence relation \(E\) on \(X\) is Borel graphable if there is a Borel graph on \(X\) whose connectivity relation is equal to \(E\). We will discuss two instances of Borel graphability with surprising properties. First, we will consider an equivalence relation arising from computability theory which is Borel graphable if and only if there is a non-constructible real. Second, we will explore the question of which Polish group actions have Borel graphable orbit equivalence relations and we will show that this is the case whenever the group is connected. This is joint work with Tyler Arant and Alekos Kechris.
Given sets \(X, Y\) and \(P \subseteq X \times Y\) with \({\rm proj}_X (P) = X\), a uniformization of \(P\) is a function \(f: X \to Y\) such that \((x,f(x)) \in P\) for all \(x \in X\). If now \(E\) is an equivalence relation on \(X\), we say that \(P\) is \(E\)-invariant if \(x_1 E x_2 \implies P_{x_1} = P_{x_2}\), where \(P_x = \{y : (x,y) \in P\}\) is the \(x\)-section of \(P\). In this case, an \(E\)-invariant uniformization is a uniformization \(f\) such that \(x_1 E x_2 \implies f(x_1) = f(x_2)\).Consider now the situation where \(X, Y\) are Polish spaces and \(P\) is a Borel subset of \(X \times Y\). In this case, standard results in descriptive set theory provide conditions which imply the existence of Borel uniformizations. These fall mainly into two categories: "small section" and "large section" uniformization results.Suppose now that \(E\) is a Borel equivalence relation on \(X\), \(P\) is \(E\)-invariant, and \(P\) has "small" or "large" sections. In this talk, we address the following question: When does there exist a Borel \(E\)-invariant uniformization of \(P\)?We show that for a fixed \(E\), every such \(P\) admits a Borel \(E\)-invariant uniformization iff \(E\) is smooth. Moreover, we compute the minimal definable complexity of counterexamples when \(E\) is not smooth. Our counterexamples use category, measure, and Ramsey-theoretic methods.We also consider "local" dichotomies for such pairs \((E, P)\). We give two new proofs of a dichotomy of Miller in the case where \(P\) has countable sections, the first using Miller's \((G_0, H_0)\) dichotomy and Lecomte's \(\aleph_0\)-dimensional \(G_0\) dichotomy, and the second using a new \(\aleph_0\)-dimensional analogue of the \((G_0, H_0)\) dichotomy. We also prove anti-dichotomy results for the "large section" case. We discuss the "\(K_\sigma\) section" case, which is still open.This is joint work with Alexander Kechris.
Fractional graph theory studies modifications of classical graph-theoretic problems (colorings, matchings, etc.) in which the answer is no longer required to be an integer. The study of fractional graph theory in the Borel setting was initiated by Meehan in 2018. In this talk, we will discuss "geometric" assumptions under which Borel and ordinary fractional combinatorics coincide. This is joint work with Felix Weilacher.
A colouring rule is a way to colour the points \(x\) of a probability space according to the colours of finitely many measures preserving tranformations of \(x\). The rule is paradoxical if the rule can be satisfied a.e. by some colourings, but by none whose inverse images are measurable with respect to any finitely additive extension for which the transformations remain measure preserving. We demonstrate paradoxical colouring rules defined via u.s.c. convex valued correspondences (if the colours \(b\) and \(c\) are acceptable by the rule than so are all convex combinations of \(b\) and \(c\)). This connects measure theoretic paradoxes to problems of optimization and shows that there is a continuous mapping from bounded group-invariant measurable functions to itself that doesn't have a fixed point (but does has a fixed point in non-measurable functions).
A conjecture of Hjorth states that the only countable Borel equivalence relations reducible to orbit equivalence relations of Polish abelian group actions are the hyperfinite ones. This conjecture was recently refuted by Allison, where he showed that every treeable countable Borel equivalence relation reduces to such an orbit equivalence relation. We show that this holds for more general equivalence relations, including all countable Borel equivalence relations. This is joint with Josh Frisch.
We give new examples of topological groups that do not have non-trivial continuous unitary representations, the so-called exotic groups. We prove that all groups of the form \(L^0(\phi, G)\), where \(\phi\) is a pathological submeasure and \(G\) is a topological group, are exotic. This result extends, with a different proof, a theorem of Herer and Christensen on exoticness of \(L^0(\phi,{\mathbb R})\) for \(\phi\) pathological.In our arguments, we introduce the escape property, a geometric condition on a topological group, inspired by the solution to Hilbert's fifth problem and satisfied by all locally compact groups, all non-archimedean groups, and all Banach--Lie groups. Our key result involving the escape property asserts triviality of all continuous homomorphisms from \(L^0(\phi, G)\) to \(L^0(\mu, H)\), where \(\phi\) is pathological, \(\mu\) is a measure, \(G\) is a topological group, and \(H\) is a topological group with the escape property.This is joint work with F. Martin Schneider.