Generalized descriptive set theory at uncountable cardinals & actions of monoids in combinatorics

Descriptive set theory studies those subsets of topological spaces (called pointsets) which can be defined, by means of a list of specified operations including, e.g., complement, countable union and intersection, projection, beginning with open sets of the space. Classical descriptive set theory (DST) considers mainly sets in Polish (that is, separable metric) spaces, this is why we shall identify it here as Polish descriptive set theory.

A space of lexicographic preferences

This thesis is divided into three parts, the first and second ones focused on combinatorics and classification problems on discrete and geometrical objects in the context of descriptive set theory, and the third one on generalized descriptive set theory at singular cardinals of countable cofinality.Descriptive Set Theory (briefly: DST) is the study of definable subsets of Polish spaces, i.e., separable completely metrizable spaces. One of the major branches of DST is Borel reducibility, successfully used in the last 30 years to solve and compare many classification problems. One of our goals is the classification of knots, very familiar and tangible objects in everyday life, which also play an important role in modern mathematics. The study of knots and their properties is known as knot theory. Our plan is to gain insight into knots using discrete objects, such as linear and circular orders. This approach was already exploited in [6]. The first part of this work is therefore devoted to countable linear orders and the study of the quasi-order of convex embeddability and its induced equivalence relation. We obtain both combinatorial and descriptive set-theoretic results. We also expand our research to the case of circular orders.Another objective of this first part is to extend the notion of convex embeddability on countable linear orders. We provide a family of quasi-orders of which embeddability is a particular case as well. We study these quasi-orders from a combinatorial point of view and analyse their complexity with respect to Borel reducibility. Furthermore, we extend the analysis of these quasi-orders to the set of uncountable linear orders.The second part of the project deals with classification problems on knots and $3$ -manifolds. The goal here is to apply the results obtained in the first part to the study of proper arcs and knots, establishing lower bounds for the complexity of some natural relations between these geometrical objects. We also obtain some combinatorial results which are particularly interesting when we restrict to the set of wild proper arcs and wild knots, classes which haven’t received much attention so far. These parts are included in the two preprints [4, 5] in collaboration with my supervisor Alberto Marcone, Luca Motto Ros (University of Torino), and Vadim Weinstein (University of Oulu).The second part of this work also includes the classification of non-compact $3$ -manifolds up to homeomorphism (the case of compact $3$ -manifolds has already been solved: indeed, there are only countably many $3$ -manifolds up to homeomorphism; see [7]), and that of Cantor sets of $\mathbb {R}^3$ up to conjugation (answering to Question 5.5 of [3]). Here we resort to algebraic tools. Stone duality gives a neat way to go back-and-forth between totally disconnected Polish spaces and countable Boolean algebras (see [1]). The main ingredient is the Stone space of all ultrafilters on a Boolean algebra. In this work we introduce a weaker concept which we call “blurry filter”. Using blurry filters instead of ultrafilters enables one to extend the class of spaces under consideration beyond totally disconnected. As an application of this method, we show that both homeomorphism on non-compact $3$ -manifolds and conjugation of Cantor sets in $\mathbb {R}^3$ are completely classifiable by countable structures. These results are part of an upcoming paper in collaboration with Vadim Weinstein.The last part of this thesis concerns the natural generalization of descriptive set theory that occurs when countable is replaced by uncountable, called Generalized Descriptive Set Theory (briefly: GDST). In particular, we focus on the case of GDST for a singular cardinal $\kappa $ of countable cofinality. The goal here is to study the generalizations of the Perfect Set Property and the Baire Property to subsets of spaces of higher cardinalities, like the power set $\mathcal {P}(\kappa )$ . We consider the question under which large cardinal hypotheses classes of definable subsets of these spaces possess such regularity properties, focusing on rank-into-rank axioms, like I2, and classes of sets definable by $\Sigma _1$ -formulas with parameters from various collections of sets. The obtained results are included in [2], a joint work with my co-supervisor Vincenzo Dimonte and Philipp Lücke (University of Hamburg).Abstract prepared by Martina IannellaE-mail: martina.iannella@tuwien.ac.at.URL: https://air.uniud.it/handle/11390/1262824.

Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the different traditions of the subject. It includes a wide variety of examples, exercises (over 400), and applications, in order to illustrate the general concepts and results of the theory. This text provides a first basic course in classical descriptive set theory and covers material with which mathematicians interested in the subject for its own sake or those that wish to use it in their field should be familiar. Over the years, researchers in diverse areas of mathematics, such as logic and set theory, analysis, topology, probability theory, etc., have brought to the subject of descriptive set theory their own intuitions, concepts, terminology and notation.

§1. Introduction. This paper considers several complexity questions regarding Borel automorphisms of a Polish space. Recall that a Borel automorphism is a bijection of the space with itself whose graph is a Borel set (equivalently, the inverse image of any Borel set is Borel). Since the inverse of a Borel automorphism is another Borel automorphism, as is the composition of two Borel automorphisms, the set of Borel automorphisms of a given Polish space forms a group under the operation of composition. We can also consider the class of automorphisms of all Polish spaces. We will be primarily concerned here with the following notion of equivalence:Definition 1.1. Two Borel automorphisms f and g of the Polish spaces X and Y are said to be Borel isomorphic, f ≅ g, if they are conjugate, i.e. there is a Borel bijection φ: X → Y such that φ ∘ f = g ∘ φ.We restrict ourselves to automorphisms of uncountable Polish spaces, as the Borel automorphisms of a countable space are simply the permutations of the space. Since any two uncountable Polish spaces are Borel isomorphic, any Borel automorphism is Borel isomorphic to some automorphism of a fixed space. Hence, up to Borel isomorphism we can fix a Polish space and represent any Borel automorphism as an automorphism of this space. We will use the Cantor space 2ω (with the product topology) as our representative space.

Admissible Sets and Infinitary Logic

A remarkable result of Henson and Ross [HR] states that if a function whose graph is Souslin in the product of two hyperfinite sets in an ℵ1 saturated nonstandard universe possesses a certain nice property (capacity) then there exists an internal subfunction of the given one possessing the same property. In particular, they prove that every 1-1 Souslin function preserves any internal counting measure, and show that every two internal sets A and B with ∣A∣/∣B∣ ≈ 1 are Borel bijective. As a supplement to the last-mentioned result of Henson and Ross, Keisler, Kunen, Miller and Leth showed [KKML] that two internal sets A and B are bijective by a countably determined bijection if and only if ∣A∣/∣B∣ is finite and not infinitesimal.In this paper we first show that injective Borel functions map Borel sets into Borel sets, a fact well known in classical descriptive set theory. Then, we extend the result of Henson and Ross concerning the Borel bijectivity of internal sets whose quotient of cardinalities is infinitely closed to 1. We prove that two Borel sets, to which we may assign a counting measure not equal to 0 or ∞, are Borel bijective if and only if they have the same counting measure ≠0, ∞. This, together with the similar characterization for Souslin and measurable countably determined sets, extends the above-mentioned results from [HR] and [KKML].

The separation, uniformization, and other properties of the Borel and projective hierarchies over hyperfinite sets are investigated and compared to the corresponding properties in classical descriptive set theory. The techniques used in this investigation also provide some results about countably determined sets and functions, as well as an improvement of an earlier theorem of Kunen and Miller.

The separation, uniformization, and other properties of the Borel and projective hierarchies over hyperfinite sets are investigated and compared to the corresponding properties in classical descriptive set theory. The techniques used in this investigation also provide some results about countably determined sets and functions, as well as an improvement of an earlier theorem of Kunen and Miller.

Classical descriptive set theory as a refinement of effective descriptive set theory

This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the question turns out to be closely related to porous sets in Banach spaces, it provides a bridge between descriptive set theory and the classical topic of existence of derivatives of vector-valued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces. The book opens several new research directions in this area of geometric nonlinear functional analysis. The new methods developed here include a game approach to perturbational variational principles that is of independent interest. Detailed explanation of the underlying ideas and motivation behind the proofs of the new results on Fréchet differentiability of vector-valued functions should make these arguments accessible to a wider audience. The most important special case of the differentiability results, that Lipschitz mappings from a Hilbert space into the plane have points of Fréchet differentiability, is given its own chapter with a proof that is independent of much of the work done to prove more general results. The book raises several open questions concerning its two main topics.

Descriptive Set Theory: Projective Sets

A Borel isomorphism that, together with its inverse, maps ℱσ-sets to ℱσ-sets will be said to be a Borel isomorphism at the first level. Such a Borel isomorphism will be called a first level isomorphism, for short. We study such first level isomorphisms between Polish spaces and between their Borel and analytic subsets.

A major recent theme in Descriptive Set Theory has been the study of countable Borel equivalence relations on standard Borel spaces, including their structure under the partial ordering of Borel reducibility. We shall contribute to this study by proving Borel incomparability results for the orbit equivalence relations arising from Bernoulli, profinite, and linear actions of certain subgroups of $PSL_2(mathbb R)$. We employ the techniques and general strategy pioneered by Adams and Kechris in cite{AK}, and develop purely Borel versions of cocycle superrigidity results arising in the dynamical theory of semisimple groups. Specifically, using Zimmer's cocycle superrigidity theorems cite{zim}, we will prove Borel superrigidity results for suitably chosen actions of groups of the form $PSL_2(mathcal{O})$, where $mathcal{O}$ is the ring of integers inside a multi-quadratic number field. In particular, for suitable primes $pne q$, we prove that the orbit equivalence relations arising from the natural actions of $PSL_2(mathbb Z[sqrt{q}])$ on the $p$-adic projective lines are incomparable with respect to Borel reducibility as $p$, $q$ vary. Furthermore, we also obtain Borel non-reducibility results for orbit equivalence relations arising from Bernoulli actions of the groups $PSL_2(mathcal{O})$. In particular, we show that if $E_p$ denotes the orbit equivalence relation arising from a nontrivial Bernoulli action of $PSL_2(mathbb Z[sqrt{p},])$, then $E_p$ and $E_q$ are incomparable with respect to Borel reducibility whenever $pne q$.%%%%

Isomorphisms for Convergence Structures: II. Borel Spaces and Infinite Measures