Abstracts Arctic 7 David Aspero: (Short) memory iterations and side conditions Abstract: I presented a technique for building forcing iterations incorporating systems of models as side conditions and exhibiting a certain non-linear behaviour. Using this type of iterations we obtain models of forcing-axiom like statements together with large continuum. This is joint work with Golshani. --- Dia Basak: The Shape of Reason: The Role of Topology in Large Cardinals Abstract: We will investigate the effect of adding topological assumptions to collections of embeddings between models of set theory. --- Tom Benhamou: Generating Ultrafilters Abstract: This talk is motivated by several questions revolving special sets which generates \kappa-complete ultrafilters on measurable cardinals. After presenting these question, we will develop some theory to measure how complex are the bases of a given ultrafilter. In particular we will present the Tukey and Depth spectrum, and prove several general results regarding them. In the second part of the talk, we will give specific examples of ultrafilters and see how these examples are connected to the motivating questions. --- Natasha Dobrinen: On parametrized Ramsey theorems. Abstract: This is joint work with Jing Zhang. The first parametrization of an infinite-dimensional Ramsey theorem was proved independently by A. Miller and by S. Todorcevic. They proved a Ramsey theorem for the product of perfect subsets of the real numbers and the Ellentuck space. Since then, this line of work has been extended in various directions by J. Mijares, S. Todorcevic, T. Trujillo, and Y.Y. Zheng. In Zheng's thesis, she proved a general parametrized Ramsey theorem for infinite products of Sacks forcing and Ramsey spaces satisfying a condition she calls (L4). She proved that (L4) is a necessary property of a Ramsey space to have such a parametrized Ramsey theorem. However, checking (L4) is a non-trivial task, and the question remains whether every topological Ramsey space satisfies (L4). We define a transparent condition (L4-) which is satisfied by all of the standard Ramsey spaces in the literature. This along with a strengthened form of the finite A.4 (Pigeonhole Principle) for Ramsey spaces imply (L4). This yields new examples of topological Ramsey spaces with parametrized Ramsey theorem, yielding new ultrafilters that are preserved by product Sacks forcing. On the other hand, some Ramsey spaces do not satisfy our strengthened A.4, so it remains open whether every topological Ramsey space has a parametrized Ramsey theorem. ---- Vera Fischer: Good witnesses Abstract: Two persistent directions in the study of the properties of the, so called, combinatorial or extremal sets of reals, sets like maximal cofinitary groups or maximal ideal independent families, are the study of their spectra and their projective complexity. In this talk, we will discuss some recent progress and point out towards interesting remaining open problems. ---- Takehiko Gappo: Exact large cardinal strength of determinacy of games of fixed countable length Abstract: We discuss the large cardinal strength of determinacy of games on natural numbers of length \omega \cdot \alpha with payoff in <\omega^2 - \Pi^1_1 for arbitrary countable ordinal \alpha. This is joint work with Juan P. Aguilera. --- Derek Levinson: Derived Models and PFA Abstract: Wilson conjectured that PFA + ``kappa is a limit of Woodin cardinals'' implies the derived model at kappa satisfies Theta_0 < kappa^+. We prove this under the additional assumption that the derived model satisfies mouse capturing and discuss some related results. This is joint work with Nam Trang. --- Andreas Lietz: Separating Maximality Principles Abstract: (Part of this is joint work with Takehiko Gappo) We separate various versions of the Maximality Principle restricted to $\Sigma_n$ or $\Pi_n$-formulas for the usual suspects of forcing classes and natural allowed sets of parameters. We also consider the unusual suspect of the class of $\omega_1$-preserving forcings. Finally, we present an equiconsistency involving the $\Gamma$-Correct Proper Forcing Axiom, defined by Goodman, and strong forms of supercompactness. ---- Miguel Angel Mota: On the consistency strength of $MM(\omega_1)$ Abstract: We prove that the consistency of Martin's Maximum restricted to partial orders of cardinality $\omega_1$ follows from the consistency of ZFC. This is joint work with Dobrinen, Krueger, Marun and Zapletal. --- Ralf Schindler: The number of Woodin cardinals in core models. Abstract: By a core model we mean a fully iterable pure extender model which satisfies a form of weak covering and which embeds into any other such model. There is no core model with 155 Woodin cardinals. In fact by a joint result with G. Sargsyan from 2017 the number of Woodin cardinals in a core model K is either 0, or 1, or at least a strong cardinal in K. We discuss extensions of this result. --- Ted Slaman: Set Theoretic Aspects of Hausdorff Dimension Abstract: We will discuss some set theoretic aspects of Hausdorff dimension: (1) a consistency result generalizing the Borel Conjecture, (2) a solution to a question of C. A. Rogers on the gauge measures of closed sets and (3) a family of questions on the possible gauge dimensions of closed, Borel and unrestricted sets of real numbers --- Philip Welch: Nestings and Burrows We try to formulate tighter definitions that are sufficient to prove the existence of models of $\Pi^1_n$-Monotone Induction, and for strategies in Boolean combinations of $\Pi^0_3$-games following on work of Montalban-Shore. This is joint work with Juan Aguilera, and can be viewed as low-level constructibility theory together some admissibility theory. --- Ur Ya'ar: Inner models from extended logics and the Delta-operation We explore the relationship between two operations on an abstract logic L - the constructibility operation which produces the model of L-constructible sets, C(L); and the Delta-operation producing a logic Delta(L) extending L, which satisfies the Delta-interpolation property. We will show that the former is, in a sense, independent of the latter. ---- Taichi Yasuda: The cofinality of universally Baire sets problem Abstract: Aspéro and Schindler have solved the MM^{++} versus the Axiom (＊) problem completely. The result of Aspéro and Schindler leads us to the MM^{++} versus the Axiom (＊)^{+} problem. And to solve the problem, it is necessary to understand the cofinality of universally Baire sets. In this talk, first we introduce the cofinality of universally Baire sets problem and present known results. Second, we give new partial results.