Abstract. TBA

Abstract. TBA

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.

Abstract. I will discuss a logic-based theory of explainable AI that has emerged over the last few years, which is geared towards explaining the decisions of classifiers like ones based on Bayesian Networks, random forests and some types of neural networks. The theory is based on compiling the input-output behavior of such classifiers into symbolic form and then using sophisticated machinery from symbolic logic to explain decisions. The theory employs newly-introduced logical operators for abstracting instances (i.e., classifier input) into necessary and sufficient conditions for triggering the decision on that instance. These conditions, called the "reasons behinds decisions," are then used to answer questions such as: What minimal aspects of an instance guarantee the decision, and what minimal aspects must be changed (and in what way) to change the decision. I will further illustrate the theory using concrete examples and case studies.

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

Ely Jrade: Applying the G_0 dichotomy to prove a result in Descriptive Set Theory

In 1999, Kechris, Solecki, and Todorcevic proved an integral graph-theoretic result in descriptive set theory known as the G_0 Dichotomy. It reveals that finding whether an analytic graph G fails to have a countable Borel coloring is equivalent to finding whether the canonical G_0 continuously homomorphs into G. Remarkably, despite there being non-graph theoretic approaches to proving, for instance, the Luzin-Novikov Theorem, we provide a result due to Miller which produces an argument utilizing the G_0 dichotomy.

Xiang Li: Keisler measures and stable regularity lemma

A stable graph is a finite graph omitting large half-graphs. The regularity lemma for stable graphs (Malliris and Shelah, 2011) strengthens Szemerédi's regularity lemma by eliminating the need for "irregular pairs" in the partition of such graphs. We will prove the stable regularity lemma using Keisler measures and nonstandard models of set theory.

Abstract. This is a Caltech Logic Seminar talk, happening live at UCLA.

A graphing of an equivalence relation $E$ is a graph $G$ whose connectivity equivalence relation is equal to $E$. In previous joint work with Alekos Kechris and Patrick Lutz, we studied analytic equivalence relations which have Borel graphings. In this talk, we will discuss new results about when arithmetical equivalence relations have definable graphings which are lower down in the arithmetical hierarchy. In particular, we will see that for any computable relational language $L$, computable isomorphism of $L$-structures presented on the natural numbers (a $\Sigma^0_3$ equivalence relation) has a $\Pi^0_2$ graphing. We will also prove a result on how to arithmetically construct a graphing of the Friedman-Stanley jump of E from a graphing of E.

Abstract. In 1925, Tarski asked whether a disk in the plane can be partitioned into finitely many pieces that can then be rearranged to form a square. This became known as Tarski's circle squaring problem. It was solved positively by Laczkovich in 1990, and in 2017 Marks and Unger gave a constructive (i.e., Borel) solution. A key role in the Marks--Unger approach is played by bounded Borel flows in graphs associated to finitely generated subgroups of the torus. We undertake a systematic analysis of such flows under certain natural geometric regularity assumptions. In particular, we prove a Hodge decomposition theorem for them and extend the scope of the theory from random subgroups to ones generated by sufficiently independent algebraic elements. This is joint work with Anush Tserunyan and Spencer Unger.

Abstract. We show how to construct an $\aleph_1$-Suslin tree which is indestructible under forcing with a given c.c.c. poset of size $\aleph_1$, in $L(x)$ for any real $x$. This answers a question of Woodin from his 2023 VIG talk, and relates to his work on generic MA models.

Abstract. The connectivity relation of a Borel graph is always analytic, but it turns out that not every analytic equivalence relation can be realized as a connectivity relation of a Borel graph. We call an analytic equivalence relation Borel graphable if it is the connectedness relation of a Borel graph. We will examine two important types of equivalence relations for the theory of Borel graphability: (1) equivalence relations coming from recursion theory (Church-Kleene ordinals, in particular) that provide interesting examples and counter-examples; and (2) orbit equivalence relations from Polish group actions. In particular, we will see that all connected Polish groups have Borel graphable orbit equivalence relations. This is joint work with Alekos Kechris and Patrick Lutz.

Abstract. A collection K of certain structures (e.g., graphs, groups, topological spaces) has a universal object M in K if every member of K embeds into M. By a fundamental result in model theory, if K is the collection of all models of a certain complete first order theory T and of some regular size kappa, then the continuum hypothesis implies the existence of a universal object for K. I will describe several results concerning universality problems for collections of second order structures involving Aronszajn trees. This is joint work with Siiri Kivimäki, Menachem Magidor and Jouko Vananen.

Abstract. Stationary sets are a fundamental concept in set theory, but their definition only makes sense at regular cardinals. We will describe mutual stationarity, a property that can be viewed as an analog of stationarity for singular cardinals, and discuss how it interacts with other combinatorial properties at or near $\aleph_\omega$. In particular, we will discuss the tree property and the failure of the Singular Cardinal Hypothesis.