Two persistent directions in the study of the properties of the, so-called, combinatorial or extremal sets of reals, sets like maximal eventually different families of functions, maximal cofinitary groups or maximal independent families, are the study of their spectra and their projective complexity. In this talk, we will discuss some recent progress in the area, and point out towards interesting remaining open problems.

For an ideal $\mathcal{I}$ on $\omega$, let $m_{\mathcal{I}^+}$ be the $\sigma$-ideal on $\omega^\omega$ generated by sets of the form \[ m_\phi := \{x\in\omega^\omega \mid \forall n<\omega\, (x(n) \in \phi(x \upharpoonright n))\}, \] where $\phi\colon\omega^{<\omega}\to\mathcal{I}$. Sabok and Zapletal showed that the poset of Borel $m_{\mathcal{I}^+}$-positive sets is forcing equivalent to the poset of $\mathcal{I}^+$-branching Miller trees. We investigate cardinal invariants of $m_{\mathcal{I}^+}$ for various ideals $\mathcal{I}$, obtaining both ZFC theorems and consistency results. This is joint work with Aleksander Cieślak, Arturo Martínez-Celis, and Takashi Yamazoe.

We investigate the descriptive complexity of analytic quasi-orders and their associated equivalence relations arising in countable model theory. A foundational result by Louveau and Rosendal (early 2000s) established that the embeddability relation on countable combinatorial trees is analytic complete—i.e., every analytic quasi-order Borel reduces to it. As a consequence, the associated bi-embeddability equivalence relation is also analytic complete. This talk will survey recent advances by Paolini and Shelah on these relations in the class of torsion-free abelian groups, as well as the speaker’s work in the class of graphs, and present new findings concerning the class of linear orderings. We show that the embeddability relation on linear orderings Borel reduces to the elementary embeddability relation, but not vice versa. Moreover, neither of these quasi-orders is analytic complete. Structural properties of the elementary embeddability relation on linear orderings suggest that the associated equivalence relation of elementary bi-embeddability may be a candidate for analytic completeness, though establishing such a result would require fundamentally new techniques.

Proof mining is a research program that aims to utilize tools and insights from mathematical logic to extract computational information from proofs in mainstream mathematics. The fundamental logical 'substrates' of proof mining are the so-called general logical metatheorems on bound extractions. These metatheorems employ well-known proof interpretations, such as Gödel's functional interpretation, to yield general results that quantify and facilitate the extraction of computational content from broad classes of theorems and proofs relevant to their intended areas of application. This talk will focus on aspects of newly developed formal systems and metatheorems for extracting computational content from proofs in probability theory. While these systems involve many subtle details and features, we will concentrate on the key achievements they offer. This includes a novel extension of Bezem’s majorizability, which explains the uniformities in the extracted bounds of previous case studies and the formalisation of a strategy that transfers quantitative deterministic results to their corresponding probabilistic analogues. This is joint work with Nicholas Pischke.

Imaginary elements represent equivalence classes of definable equivalence relations. The use of imaginaries yields the existence of minimal (i.e. canonical) parameters for definable sets. In many classical theories, equivalence classes are already coded in the models of the theory and therefore imaginaries can be eliminated. In this talk, we will present a general criterion that yields the failure of elimination of imaginaries in various classical theories. The failure is due to equivalence classes arising as cosets of subgroups. We will illustrate the main ideas by considering several examples.

Given a topological space $X$ and a cardinal $\kappa$ recall that a set $A \subseteq X$ is called $\kappa$-dense if it has intersection of size $\kappa$ with every non-empty open subset of $X$. The Baumgartner axiom (for $X$ and $\kappa$), denoted $\mathsf{BA}_\kappa(X)$, is the statement that for every pair $A, B \subseteq X$ which are $\kappa$-dense there is a homeomorphism $h:X \to X$ which maps $A$ to $B$. It's easily seen that $\mathsf{BA}_{\aleph_1}(\mathbb R)$ is equivalent to Baumgartner's celebrated axiom about $\aleph_1$-dense linear orders. An interesting open conjecture of Steprans and Watson states that $\mathsf{BA}_{\aleph_1}(\mathbb R)$ implies $\mathsf{BA}_{\aleph_1}(\mathbb R^n)$ for $n > 1$. The converse is false by theorems of Abraham-Shelah and Steprans-Watson. A related interesting question is whether $\mathsf{BA}_{\aleph_1}(\mathbb R)$ implies $\mathfrak{p} > \aleph_1$. More generally, there are many seemingly difficult, interesting open problems regarding the relation between $\mathsf{BA}_\kappa(X)$ and $\mathsf{BA}_\kappa(Y)$ for different Polish spaces $X$ and $Y$ as well as the effect these axioms have on cardinal invariants. In this talk we will discuss some recent attempts to unravel these problems. Specifically we will look at some weakenings of Baumgartner's Axiom where these issues are clearer as well as some strengthenings involving Lipschitz functions in which more implications and applications become more readily provable.