Several recent methods for proving pointwise ergodic theorems for pmp actions of free groups critically use weak mixing properties of Markov measures on the boundary of a free group of finite rank. However, it was not known exactly which Markov measures are weak mixing. In joint work with Jenna Zomback, we give a complete characterization of such measures. It turns out that, under mild non-degeneracy assumptions, they are exactly the Markov measures arising from strictly irreducible transition matrices — a condition introduced by Bufetov in 2000 for a different purpose. The proof of this characterization goes through proving equivalences with a new dynamical condition on the action that we call chaining, which is interesting in its own right.

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For any fixed n and \(e > 0\), given a sufficiently long sequence of events in a probability space all of measure at least \(e\), some \(n\) of them will have a common intersection. This follows from the inclusion-exclusion principle. A more subtle pattern: for any \(0 < p < q < 1\), we can't find events \(A_i\) and \(B_i\) so that the measure of \(A_i\ \cap B_j\) is less that \(p\) and of \(A_j \cap B_i\) is greater than \(q\) for all \(1 < i < j < n\), assuming \(n\) is sufficiently large. This is closely connected to a fundamental model-theoretic property of probability algebras called stability. We will discuss these and more complicated patterns that arise when our events are indexed by multiple indices. In particular, how such results are connected to higher arity generalizations of de Finetti's theorem in probability, structural Ramsey theory, hypergraph regularity in combinatorics, and model theory.

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The Becker-Kechris topological realization theorem shows that given a Borel action of a Polish group, a set is open in some compatible Polish realization of the action iff it is Borel and open in the quotient topology on each orbit. We will present a result showing that conversely, every standard Borel space equipped with an abstract such "Borel family of orbitwise Polish topologies" (on a Borel equivalence relation, or more generally a Borel groupoid) may be represented in terms of a Polish group action. Along the way, we will introduce and survey some of the machinery that is useful in proving the Becker-Kechris theorem and other "Borel piecewise Polish" contexts, such as the Kunugi-Novikov uniformization theorem and Baire category quantifiers.

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We discuss some new combinatorial characterizations of when a class of finite graphs is monadically stable. The ultimate goal of this area is to prove a dichotomy theorem about first-order model-checking. That is, for which graph classes is there an efficient algorithm to determine, given an \(n\)-vertex graph \(G\) in the class and a first-order sentence \(\phi\), whether or not \(G\) models \(\phi\)? We discuss several recent results in the area which, combined, imply that such algorithms exist for monadically stable classes. It is conjectured that the right boundary is monadic NIP.

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The ordinal-valued Scott rank of a countable structure gives a robust measure of the complexity of that structure. Given a sentence T of infinitary logic, we can think of it as a theory defining a class of models. The Scott spectrum of T is the set of Scott ranks of countable models of T. We are particularly interested in gaps in Scott spectra where either T has no models of small Scott rank, or where T has models of various different ranks but has a large gap of missing ranks in between. I will talk both about older work on the general case as well as more recent work with Gonzalez on theories of linear orders.

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In this talk we will discuss an old question of Kechris from the 60s. The question asks whether there can be long sequences of distinct \({\boldsymbol \Sigma}^1_{2n+2}\) sets. Building on a work of Hjorth, who showed that there cannot be long sequences of distinct \({\boldsymbol \Sigma}^1_2\) sets, we will establish that there cannot be such long sequences of \({\boldsymbol \Sigma}^1_{2n+2}\) sets. Then we will discuss generalizations of this theorem to larger pointclasses (due to Neeman, Levinson and the speaker).

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We consider the problem of finding a 2-regular subgraph of a given regular graph with no odd cycles. We show that this is always possible in the BP context. As a consequence, odd-regular bipartite Borel graphs on Polish spaces admit perfect matchings with the property of Baire, in contrast with recent examples of Kun in the measure-theoretic setting. Analogous results in the measure-theoretic context hold for hyperfinite graphs. This is joint work with Matt Bowen and Felix Weilacher, building upon prior joint work with Kechris and with Miller.

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Number words are used in two distinct ways in natural languages like English. Sometimes they are used as adjectives, as in "there are three stooges". And sometimes they are used as nouns, as in "the number of stooges is three". Adjective uses can be analyzed using numerical quantifiers, but noun uses require the ontology of arithmetic. Despite this, it is natural to think that there is a sense in which noun uses of number words can be implicitly defined using adjective uses of number words — that is, using numerical quantifiers. Something like this strategy was even pursued by Russell and Whitehead in Principia Mathematica. Unfortunately, in the absence of a special axiom of infinity, the strategy only generates arithmetic if it is applied in a controversial, impredicative fashion. Ramsey argued that such a procedure is justified if and only if a metaphysics of platonism is assumed. Against this now standard position, I argue that lessons from contemporary metaontology allow for the justification of impredicative definitions, and thus arithmetic, without any objectionable metaphysical assumptions. I end by arguing that this move helps us to answer broad philosophical questions about the very nature of numbers and other mathematical objects.

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Algorithmic fractal dimensions, developed at the beginning of this century, use the theory of computing to assign dimensions to individual points in Euclidean spaces and other metric spaces. Subsequent research produced several indications of the geometric significance of these dimensions of points, culminating in the 2017 proof of the Point-to-Set Principle, which completely characterizes the Hausdorff and packing dimensions of arbitrary subsets of Euclidean spaces in terms of the relativized algorithmic dimensions of their constituent points. Since that time, mathematicians and theoretical computer scientists have used the Point-to-Set Principle to prove various new theorems in and around geometric measure theory. These theorems are classical (i.e., their statements do not involve computability or related aspects of logic), and some have solved well-known open problems. This talk will survey these developments.

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We introduce a new method called Poset Shuffling, which, given a poset \({\mathbb P}\), constructs a related poset that preserves essential generic sets added by \({\mathbb P}\) while avoiding certain undesirable ones. This technique is applied in the context of square principles, addressing questions posed by Jensen, and by Cummings and Friedman, regarding the failure of the global square principle and the specific points at which square principles fail. This work is a joint project with Daniel Iosub.

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A Borel equivalence relation \(E\) on a standard Borel space is hyperfinite if it is the increasing union of Borel equivalence relations with finite classes. The hyperfinite Borel equivalence relations are the simplest nontrivial class of Borel equivalence relations, by the Glimm-Effros dichotomy of Harrington-Kechris-Louveau. Yet many questions about hyperfiniteness remain open. For example, it is an open problem of Weiss (1984) whether the orbit equivalence relation of a Borel action of a countable amenable group is hyperfinite.

Some researchers have hoped we can use soft tools from Borel graph combinatorics and metric geometry to attack this problem, rather than relying on a sophisticated understanding of the structure of Folner sets and their tilings which have been key to much partial progress on Weiss's question. Recently, Bernshteyn and Yu made a significant advance in this direction by showing that every graph of polynomial growth is hyperfinite. Their result parallels the 2002 theorem of Jackson-Kechris-Louveau that Borel actions of polynomial growth groups are hyperfinite. We extend Bernshteyn and Yu's result to show there is a constant \(0 < c < 1\) such that every graph of growth less than \(\exp(n^c)\) is hyperfinite. This is joint work with Jan Grebik, Vaclav Rozhon, and Forte Shinko.

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