Abstract. Suppose that we are given a finite set of tiles, where each tile is a $d$-dimensional rectangle with integer side lengths. The tiling problem posed by Gao, Jackson, Krohne, and Seward asks when there exists a continuous equivariant map from the free part of full shift (i.e. $\{0,1\}^{\mathbb{Z}^d}$) into the space of tilings by such a tileset. A more general question in continuous combinatorics, known as the subshift problem, asks the same question with tiling spaces replaced by subshifts of finite type.

Finitely dependent processes generalize independent processes by requiring that restrictions of the process to sufficiently separated subsets of $\mathbb{Z}^d$ are independent. The subshift problem in continuous combinatorics is closely related to the problem of constructing finitely dependent processes satisfying prescribed local constraints. We will see that cocycles or height functions on subshifts provide useful tools for answering this question and, in particular, help us to give a clean answer to the tiling problem in dimension two.

This is joint work with Nishant Chandgotia, Jan Grebík, and Filip Kucerák.

Abstract. The central theme of this talk will be applications of ideals on omega to the study of separable Banach spaces. Ideal Schauder bases replace ordinary convergence of partial sums with convergence governed by an ideal, creating a natural connection between Banach space theory and the combinatorial and descriptive properties of ideals.

I will discuss how the choice of an ideal affects the resulting basis theory. One of the main topics will be the continuity of coordinate functionals, including a positive result for analytic ideals that answers a question of Vladimir Kadets. I will also present examples and critical ideals that illustrate the variety of phenomena arising in the field. The results are joint work with Tomasz Kania, Noé de Rancourt, Adam Kwela and Piotr Koszmider.

Abstract. The classical countable random graph is obtained by independently tossing the same possibly biased coin for each pair of vertices and joining the pair whenever the outcome is heads. With probability one, this procedure produces the Rado graph. This talk asks how the picture changes when, instead of using the same coin for every potential edge, we use a family of coins with different biases. The main theme is the extent to which a prescribed countable graph can arise almost surely from such a non-uniform random construction. I will discuss conditions ensuring that the Rado graph still appears, as well as examples showing that other structures may also occur. These results reveal that non-uniform probabilities lead to a considerably richer family of almost-sure countable graphs, while preserving a close connection between probabilistic arguments and structural properties of graphs. The results are joint work with Leonardo Coregliano, Ziemowit Kostana and Jaroslaw Swaczyna.

Abstract. The constraint satisfaction problem CSP(D) associated with a finite relational structure D is the algorithmic problem of deciding whether a given input structure X admits a homomorphism to D. Deciding whether a (finite) graph has a proper vertex coloring is an example. The celebrated CSP Dichotomy Theorem (proved in 2017) says that every CSP is either in P, or it is NP-complete, splitting CSP's into easy and hard problems (modulo P≠NP). In set theory, one can naturally associate a compactness principle K(D) to CSP(D), which can be viewed as a weak form of the Axiom of Choice. It turns out that the strength of K(D) (over ZF) reflects the above-mentioned split of CSP's exactly. Viewed as an infinite counterpart of the CSP Dichotomy Theorem, this theorem reveals an interesting connection between CS and (choiceless) set theory.

This talk is based on joint work with László Tóth and Zoltán Vidnyánszky.

Abstract. We discuss a characterization of the Borel and analytic subsets of the Cantor space via infinite circuits, which can be used to reframe and reprove many classical results in descriptive set theory using purely combinatorial arguments. We focus on a 1983 proof by Michael Sipser that the Borel hierarchy is strictly increasing at all finite levels, extending his argument to the full transfinite hierarchy. We obtain new proofs of some Ramsey-like results about Borel sets along the way, and we discuss connections to circuit depth and the classes P, NP and coNP in computational complexity theory.

Abstract. We will survey some of the concepts and structures that relate methods from inner model theory at the level of Woodin cardinals to classical descriptive set theoretic questions about $L({\mathbb R})$ under determinacy.

Abstract. Suppose that a countable group $\Gamma$ is acting continuously on a compact Hausdorff space $X$. A point $x \in X$ is called aperiodic if its stabilizer under this action is trivial, and hyperaperiodic if every point in the closure of the orbit of $x$ is aperiodic. In this talk we will address the following question of Gao, Jackson, and Seward: If $X$ has "many" aperiodic points, must it also have a hyperaperiodic one? Along the way, we'll discover an intriguing difference between measure and Baire category.

Abstract. Suppose we are interested in understanding the behavior of sets of reals in some collection Gamma. This is really not a question about the whole V: all the information we need for this study should be contained in L(R, Gamma). Of course, it is not reasonable to expect to have a coherent descriptive set theory of an arbitrary Gamma, so some restrictions are in order. It turns out that one very natural restriction is to require that L(R, Gamma) satisfies the Axiom of Determinacy; indeed, in this case, one can say a lot about the sets of reals in Gamma: for example, they are all Lebesgue measurable, have the Baire property, have the perfect set property etc. This motivates the study of the models of the form L(R, Gamma). In my talk, I will be interested in the collection of all such models, ordered by the inclusion, rather than any model in particular. Some of the questions I will address are: What is the structure of this order? How to pass "algorithmically" from smaller to bigger models?

Abstract. A graph property is elusive (or evasive) if any algorithm testing it by asking questions of the form "Is there an edge between vertices x and y?" must, in the worst case, examine all pairs of vertices. Elusive properties of finite graphs have been extensively studied since the 70s. For infinite graphs, they were first studied by Csernák and Soukup in 2021. I will give a brief introduction to elusive properties via games, and then I will talk about some of our new results in the countably infinite case.

Joint work with Márton Elekes and Anett Kocsis. 80% of the talk requires only very elementary knowledge in graph theory.

Abstract. We characterize Martin-Löf randomness and Schnorr randomness in terms of the merging of opinions, along the lines of the Blackwell-Dubins Theorem. After setting up a general framework for defining notions of merging randomness, we focus on finite horizon events, that is, on weak merging in the sense of Kalai-Lehrer. In contrast to Blackwell-Dubins and Kalai-Lehrer, we consider not only the total variational distance but also the Hellinger distance and the Kullback-Leibler divergence. Our main result is a characterization of Martin-Löf randomness and Schnorr randomness in terms of weak merging and the summable Kullback-Leibler divergence. The main proof idea is that the Kullback-Leibler divergence between μ and ν, at a given stage of the learning process, is exactly the incremental growth, at that stage, of the predictable process of the Doob decomposition of the ν-submartingale L(σ)=−lnμ(σ)ν(σ). These characterizations of algorithmic randomness notions in terms of the Kullback-Leibler divergence can be viewed as global analogues of Vovk's theorem on what transpires locally with individual Martin-Löf μ- and ν-random points and the Hellinger distance between μ,ν. Preprint at: https://arxiv.org/abs/2504.00440

Abstract. ε-Talks are short informal talks on matters related to mathematical logic.

Ely Jrade: Hypergraph generalizations of the G_0 dichotomy

Miller's classical proof of the KST G_0 dichotomy was instrumental in facilitating other graph-dichotomy results. In particular, it streamlined the ability to generalize the G_0 dichotomy to hypergraphs. We present a timeline beginning at the successful generalization to uniform finite-dimensional hypergraphs, through to the obstacle to a natural extension to omega-dimensional hypergraphs, and concluding at the modification that admitted a KST-style dichotomy for this infinitary case.

Xiang Li: Infinite vector space without infinite proper subspace

The method of permutation models gives counterexamples to certain consequences of the Axiom of Choice. I will introduce this method and give a proof of Läuchli's result that, in the absence of the Axiom of Choice, there may exist an infinite vector space with no infinite proper subspace.

Calvin Osborne: Another prisoner problem

In this ε-talk, I will discuss one of the more niche "prisoner problems" that appear across logic (not the 3 prisoners problem, nor the 100 prisoners problem, nor the prisoners hat problem!). In addition to discussing the problem itself, a prime example of an unintuitive consequence of the Axiom of Choice, I will solve a few of the problem's generalizations.

Abstract. Motivated by Magidor's problem of obtaining the tree property at all regular cardinals above $\aleph_1$, we examine the extent to which the tree property can hold at successors of singulars of many different cofinalities simultaneously. This is trivial to obtain for limits of large cardinals, but presents much more of a challenge at small cardinals. Specifically we build a model in which the tree property holds at $\aleph_{\omega+\omega+1}$ and at $\aleph_{\omega_n+1}$ for all $n<\omega$ simultaneously. In fact, we can obtain the same result for the strong tree property. A similar result can also be obtained for arbitrarily many target cofinalities at once.