We investigate maximal almost disjoint families for \(\kappa>\omega\). Neeman and Norwood showed under AD^+ that there are no infinite maximal almost disjoint families for \(\kappa=\omega\), and Schrittesser and Tornquist showed this from all sets being Ramsey plus Ramsey uniformization. We show under AD^+ that there are no mad families on any \(\kappa <\Theta\) for the ideal \(B(\kappa)\) of bounded subsets of \(\kappa\). For the ideal \(P_\kappa(\kappa\)) of sets of size less than \(\kappa\), we prove the corresponding result except in the case \(\mathrm{cof}(\kappa)=\omega\), which is left open. In particular, this answers a question of S. Muller. This is joint work with William Chan and Nam Trang.

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The Galvin-Prikry theorem states that Borel subsets of the Baire space are Ramsey. Silver extended this to analytic sets, and Ellentuck gave a topological characterization of Ramsey sets in terms of the property of Baire in the Vietoris topology.

We present work extending these theorems to several classes of countable homogeneous structures. An obstruction to exact analogues of Galvin-Prikry or Ellentuck is the presence of big Ramsey degrees. We will discuss how different properties of the structures affect which analogues have been proved. Presented is work of the speaker for \(\mathbb{Q}\)-like structures, and joint work with Zucker for binary finitely constrained FAP classes. A feature of the work with Zucker is showing that we can weaken one of Todorcevic's four axioms guaranteeing a Ramsey space, and still achieve the same conclusion.

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Expanding ideas of Holy and Schlicht, Welch proposed a two person game of ordinal length. If \(\kappa\) is weakly compact, then Player II has a winning strategy in the game of length \(\omega\). If \(\kappa\) is measurable, then Player II wins the game of length \(2^\kappa\). The results of the talk show that if there is no \(\kappa^+\)-saturated ideal on \(\kappa\) and for a regular \(\gamma>\omega\), Player II winning the game of length \(\gamma\) implies that there is a precipitous ideal on \(\kappa\) whose quotient algebra has a dense \(\gamma\)-closed set. The converse is immediate. Moreover by forcing over a fine structure model one can find models where II wins the game of a given length \(\gamma\) but no regular \(\gamma'>\gamma\).

However the following question remains completely open: Is the Welch game determined?

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The Banach-Tarski paradox says that you can take a ball in 3 dimensional space, partition it into finitely many pieces, and rearrange those pieces in such a way that you obtain two balls of equal volume as the first. This leads naturally to the question of how complicated these pieces need to be. A version of this question arose recently in model theory, where it was asked if all groups definable in combinatorially simple mathematical structures must necessarily be definably amenable or, in other words, must carry an invariant probability measure on their definable subsets. In recent work, joint with several authors, we answered this in the negative, producing simple structures which carry a paradoxical decomposition. We will describe this work and, along the way, we will explain the fundamentals of the model theory of measures.

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Assume AD. A least branch hod pair is a pair \((P,\Sigma)\) such that \(P\) is a premouse constructed from a coherent sequence of extenders together with a predicate for \(\Sigma\), and \(\Sigma\) is an iteration strategy for \(P\) having certain regularity properties. Such pairs can be used to analyze HOD fine structurally, provided that there are enough of them. More precisely, one needs that every Suslin, co-Suslin set of reals is Wadge reducible to the codeset of some lbr hod pair. HOD Pair Capturing (HPC) asserts that this is the case.

HPC is the fundamental open question in the theory of HOD in models of AD which do not have iteration strategies for mice with long extenders. We shall discuss what is known about it.

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It turns out that model theoretic theorems about infinite objects can have interesting consequences for finite objects—not only because of compactness or pseudofiniteness, but because of certain resonances between the way structure changes across integers and across infinite cardinals. The talk will discuss various examples from recent work and the core work in model theory supporting them.

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We'll show that assuming AD+, if \(\kappa < \Theta\) is a regular cardinal and \((\kappa^+)^{\mathrm{HOD}}\) has uncountable cofinality, then HOD satisfies "\(\kappa\) is \(\kappa^+\)-supercompact." We'll then discuss some variants of this theorem and their bearing on the inner model problem for supercompact cardinals.

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A new argument is offered which proceeds through epistemic possibility (for all \(S\) knows, \(p\)), cutting a trail from modality to John Stuart Mill's account of proper names as lacking in connotation. New definitions are provided for various epistemic modal notions. An unexpected theorem about epistemic modality is proved: \(p\) is epistemically necessary for \(S\) iff either \(S\) knows \(p\) or \(S\) knows \(\lnot\lnot p\), or both. The identity relation is well-behaved in metaphysically possible worlds but goes rogue in epistemically possible worlds. Whereas it can be epistemically possible for an agent \(S\) that Lewis Carroll is not Charles Dodgson, this is not epistemically possible in the manner that anti-Millianism needs.

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The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation \(T\) has emerged. This picture included substantial evidence that pointed to these groups (for a generic \(T\)) being all topologically isomorphic to a single group, namely, \(L^0\)—the topological group of all Lebesgue measurable functions from \([0,1]\) to the circle. In fact, Glasner and Weiss asked if this is the case.

We will describe the background touched on above. We will indicate a proof of the following theorem that answers the Glasner–Weiss question in the negative: for a generic measure preserving transformation \(T\), the closed group generated by \(T\) is not topologically isomorphic to \(L^0\). The proof rests on an analysis of unitary representations of the non-locally compact group \(L^0\).

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Is it always possible to increase the Hausdorff dimension of a set by taking its image under a continuous function? More concretely, suppose \(A\) is a subset of \(\mathbb{R}\) of Hausdorff dimension \(1/2\). Is it possible to find a continuous function \(f \colon \mathbb{R} \to \mathbb{R}\) such that the Hausdorff dimension of \(f(A)\) is greater than \(1/2\)? It is not hard to show that this is possible whenever \(A\) is analytic. However, Joe Miller and I have shown that if we assume CH then there is a set for which this is not possible. Moreover, there is a subset of \(\mathbb{R}^2\) of Hausdorff dimension \(1\) whose image under any continuous function \(f\colon \mathbb{R}^2 \to \mathbb{R}\) has Hausdorff dimension \(0\). The proofs rely on ideas from the study of effective Hausdorff dimension and the point-to-set principle of Jack and Neil Lutz.

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We define a "lightface" version of the Pmax model. The resulting model of MA satisfies that \(\omega_1\) is not inaccessible to reals but nevertheless has a "canonical" theory. One application is that even assuming MA, V = HOD is not equivalent to the assertion that every finite OD set contains only OD members, answering a question of Enayat and Kanovei. We also (but now using very strong large cardinal hypotheses) produce an exotic model of MM++ which exhibits features not possible in all the previously known models of MM, or even of PFA. This answers questions of Foreman and of Viale.

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