Abstract. Assume the axiom of determinacy. Martin's partition relations on \(\omega_1\) imply that the \(\epsilon\)-partition measure on \([\omega_1]^{\epsilon}\) are countably complete ultrafilter for each \(\epsilon \leq \omega_1\).

An almost everywhere (according to the partition measures) selection principle for club subsets of \(\omega_1\) will be discussed.

From this uniformization, one can show that every function \(\Phi : [\omega_1]^{\omega_1} \rightarrow \omega_1\) is continuous almost everywhere with respect to the strong partition measures.

Since the \(\epsilon\)-partition measure is an ultrafilter, for any sentence of set theory, for almost all increasing \(\epsilon\)-sequence of countable ordinals, \(L[f]\) satisfies this sentence or its negation. The \(\epsilon\)-stable theory is the collection of statements that hold for almost all \(\epsilon\)-sequences. We will discuss some natural statements which belong to the various stable theory. For example, for all \(\epsilon \leq \omega_1\), there is a club \(C\) subset of \(\omega_1\) so that for all \(\epsilon\)-length increasing sequences \(f\) through \(C\), \(L[f]\) satisfies the generalize continuum hypothesis. The club selection principle is important for studying the \(\omega_1\)-stable theory.

This is joint work with Jackson and Trang.

Abstract. In this talk we will consider \(*\)-reductions between orbit equivalence relations. These are Baire measurable reductions which preserve generic notions, i.e., preimages of comeager sets are comeager. In short, \(*\)-reductions are weaker than Borel reductions in the sense of definability, but much more sensitive to the dynamics of the orbit equivalence relations in question.

Based on a past joint work with M. Lupini we will introduce a notion of dimension for Polish \(G\)-spaces. This dimension is always \(0\) whenever the group \(G\) admits a complete and left invariant metric, but in principle, it can take any value $n$ within \(\{0,1,\ldots\}\cup\{\infty\}\). For each such \(n\) we will produce a free action of \(S_{\infty}\) which is generically \(n\)-dimensional and we will deduce that the associated orbit equivalence relations are pairwise incomparable with respect to \(*\)-reductions.

This is a joint work with A. Kruckman.

Abstract. Sealing is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing. generic-LSA is the statement that the Largest Suslin Axiom (LSA) holds in all generic extensions. Under a mild large cardinal hypothesis, we show that Sealing is equiconsistent with generic-LSA. As a consequence, Sealing is weaker than the theory "ZFC+there is a Woodin cardinal which is a limit of Woodin cardinals". Sealing's consistency being weak represents an obstruction to the current program of descriptive inner model theory. Going beyond this bound in core model induction applications seems challenging and requires us to construct third order objects (subsets of the universally Baire sets). We will state the precise theorems and explain their impact on current developments of inner model theory. Time allowed, we'll say a bit about how to overcome the obstructions imposed by Sealing. This is joint work with G. Sargsyan.

Abstract. A computable quotient presentation of a mathematical structure \(\mathcal A\) consists of a computable structure on the natural numbers \(\langle \mathbb{N},\star,\ast,\dots \rangle\), meaning that the operations and relations of the structure are computable, and an equivalence relation \(E\) on \(\mathbb{N}\), not necessarily computable but which is a congruence with respect to this structure, such that the quotient \(\langle \mathbb{N},\star,\ast,\dots \rangle\) is isomorphic to the given structure \(\mathcal{A}\). Thus, one may consider computable quotient presentations of graphs, groups, orders, rings and so on. A natural question asked by B. Khoussainov in 2016, is if the Tennenbaum Thoerem extends to the context of computable presentations of nonstandard models of arithmetic. In a joint work with J.D. Hamkins we have proved that no nonstandard model of arithmetic admits a computable quotient presentation by a computably enumerable equivalence relation on the natural numbers. However, as it happens, there exists a nonstandard model of arithmetic admitting a computable quotient presentation by a co-c.e. equivalence relation. Actually, there are infinitely many of those. The idea of the proof consists in simulating the Henkin construction via finite injury priority argument. What is quite surprising, the construction works (i.e. injury lemma holds) by Hilbert's Basis Theorem. During the talk I'll present ideas of the proof of the latter result, which is joint work with T. Slaman and L. Harington.

Abstract. Weakly compact cardinals are equivalent to the statement that every \(\kappa\)-complete filter on a Boolean algebra \({\mathcal B}\) of size \(\kappa\) can be extended to a \(\kappa\)-complete ultrafilter on \({\mathcal B}\). One can continue this finitely many times. Can it be continued transfinitely?

Fix a cardinal \(\kappa\) and consider the following game \(\mathcal G_\gamma\) of ordinal length \(\gamma\): Player I plays a a sequence of collections \(\mathcal S_\alpha\subseteq P(\kappa)\) of size \(\kappa\) and player II plays an increasing sequence of \(\kappa\)-complete ultrafilters \(U_\alpha\) on \(\bigcup_{\beta\le \alpha}\mathcal S_\beta\). Player II wins if she can continue playing until stage \(\gamma\).

Clearly if \(\kappa\) is measurable then II wins the game of any length. Welch asked whether the property that &quot II has a winning strategy in \(\mathcal G_\gamma\)&quot can hold at a non-measurable cardinal.

The main result in this talk is that if II wins \(\mathcal G_{\omega_1}\) then there is a precipitous ideal on \(\kappa\) whose quotient has a countably closed dense subset. Hence the answer to Welch's question, at least for \(\gamma\ge \omega_1\), is no.

In joint work with Magidor, we prove that it is consistent at a non-measurable cardinal for II to have a winning strategy in \(\mathcal G_{\omega_1}\), hence the theorem is not vacuous.

Abstract. In their 1985 paper, Abraham, Rubin, and Shelah studied a number of combinatorial principles about \(\aleph_1\)-sized objects. One such axiom is the so-called "ARS Open Coloring Axiom" (hereafter \(\mathsf{OCA}_{ARS}\)), which concerns decompositions of \(\aleph_1\)-sized sets of reals into homogeneous sets for clopen colorings. One of the main open questions from their paper is whether or not \(\mathsf{OCA}_{ARS}\) is consistent with a value of the continuum greater than \(\aleph_2\) (it implies that the continuum is at least \(\aleph_2\)).

There are two additional theorems which complicate the situation. First, Moore has shown that Todorcevic's Open Coloring Axiom (hereafter \(\mathsf{OCA}_T\)) together with \(\mathsf{OCA}_{ARS}\) decides the value of the continuum to be exactly \(\aleph_2\), but second, Farah has shown that (a restricted version of) \(\mathsf{OCA}_T\) is consistent with an arbitrarily large value of the continuum. It is therefore of interest whether or not \(\mathsf{OCA}_{ARS}\) on its own decides the value of \(2^{\aleph_0}\).

Recently Gilton and Itay Neeman have answered this question, showing that \(\mathsf{OCA}_{ARS}\) is in fact consistent with \(2^{\aleph_0}=\aleph_3\). As in the original ARS paper, we need to build so-called preassignments of colors in order to add the requisite homogeneous sets. However, these can only be built over models satisfying the \(\mathsf{CH}\). To get around this difficulty, we build preassignments with very strong symmetry conditions, which allow us to combine them in many different ways, using a new type of poset called a partition product. In this talk, we will motivate and define partition products, sketch the construction of these preassignments, and show how, as a result, we can obtain a model of \(\mathsf{OCA}_{ARS}\) with a large value of the continuum.

Abstract. A measure-theoretic sheaf is constructed over an arbitrary sigma-finite measure space. The collection of such sheaves forms a topos, and is an instance of a Boolean-valued model of ZFC. In this talk, adopting a variant of the sheaf-theoretic approach, we present semantics of some fundamental structures in this model such as vector spaces, topological spaces, and measure spaces. Interestingly, it turns out that these Boolean-valued structures allow for a meaningful interpretation in a standard model as well which often can be characterized by kernel type objects. This opens a way for applications of Boolean-valued techniques to analysis and its applications. We present details on two of these applications in probability and ergodic theory.

Abstract. The connection between finitely additive probability measures and NIP theories was first noticed by Keisler. Around 20 years later, the work of Hrushovski, Peterzil, Pillay, and Simon greatly expanded this connection. Out of this research came the concept of generically stable measures. In the context of NIP theories, these particular measures exhibit stable behavior. In particular, Hrushovski, Pillay, and Simon demonstrated that generically stable measures admit a natural finite approximation. In this talk, we will discuss generically stable measures in the local setting. We will describe connections between these measures and concepts in functional analysis as well as show that this interpretation allows us to derive an approximation theorem.

Abstract. An active line of research in modern combinatorics is extending classical results from the dense setting (e.g., Szemeredi's theorem) to the sparse random setting. These results state that a property of a given "dense" structure is inherited by a randomly chosen "sparse" substructure. A recent breakthrough tool for proving such statements is the Balogh-Morris-Samotij and Saxton-Thomason hypergraph containers method, which bounds the number of independent sets in finite hypergraphs. In a joint work with A. Bernshteyn, M. Delcourt, and H. Towsner, we give a new — elementary and nonalgorithmic — proof of the containers theorem for finite hypergraphs. Our proof is inspired by considering hyperfinite hypergraphs in the setting of nonstandard analysis, where there is a notion of dimension capturing the logarithmic rate of growth of finite sets. Applying this intuition in another setting with a notion of dimension, namely, algebraically closed fields, A. Bernshteyn, M. Delcourt, and I prove an analogous theorem for "dense" algebraically definable hypergraphs: any Zariski-generic low-dimensional subset of such hypergraphs is itself "dense" (in particular, not independent).

Abstract. We will discuss a variant of the uniform Martin's conjecture for the many-one degrees.

Abstract. We show the consistency of $\square_{\kappa, 2}$ plus $SATP(\kappa^+)$ for regular $\kappa$ assuming a weakly compact. Using methods of Golshani-Hayut we also establish a global consistency result for successors of regulars from class many supercompacts.