FROM REAL-LIFE TO VERY STRONG AXIOMS Classification problems in Descriptive Set Theory & regularity properties in Generalized Descriptive Set Theory

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 branch of DST is Borel reducibility, introduced in a seemingly independent way in [FS89] and [HKL90] and successfully used in the last thirty years to solve and compare many classification problems. One of our goal is the classification of knots. Knots are objects very familiar and tangible in everyday life, and they also play an important role in modern mathematics. The study of knots and their properties is known as knot theory (see e.g. [BZ03]). Our plan is to gain insight into knots using discrete objects, such as linear and circular orders. This approach was already exploited in [Kul17]. The first part of this work is hence 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 further 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-order 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 will be included in two forthcoming papers 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 study of the homeomorphism between 3-manifolds and the conjugation of Cantor spaces of R3. 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 [CG01]). 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 the “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 3-manifolds and conjugation of Cantor sets in R3 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 (GDST). In particular, we focus on the case of GDST for a singular cardinal κ of countable cofinality. The goal here is to study when some regularity properties, as the κ+-perfect set property and the κ+-Baire property, hold for non-κ+-analytic subsets of spaces defined in this context. The results obtained are included in a forthcoming paper in collaboration with my co-supervisor Vincenzo Dimonte and Philipp Lücke (University of Barcelona).