This talk will be concerned with compactness phenomena in set theory. Compactness is the phenomenon by which the local properties of a mathematical structure determine its global behaviour. This phenomenon is intrinsic to the architecture of the mathematical universe and manifests in various forms. Over the past fifty years, the study of compactness phenomena has been one of the flagships of research in set theory. This talk will present recent discoveries spanning classical themes like the tree property and stationary reflection while also forging new connections with other topics, such as Woodin's HOD Conjecture.
We prove from ZF + AD + DC that there is no sequence of distinct $\Gamma_{1,m}$ sets of length $\aleph_{m+2}$. This is the optimal result for the pointclass $\Gamma_{1,m}$ by earlier work of Hjorth. We also get a bound on the length of sequences of $\Gamma_{2n+1,m}$ sets using the same techniques.
We examine distal theories and structures in the context of continuous logic, providing several equivalent definitions.
By studying the combinatorics of fuzzy VC-classes, we find continuous versions of (strong) honest definitions and distal cell decompositions.
By studying generically stable Keisler measures in continuous logic, we apply the theory of continuous distality to analytic versions of graph regularity.
We will also present some examples of distal metric structures, including dual linear continua and a continuous version of o-minimality.
This is the first of a series of talks that start by introducing weakly compact cardinals, and goes to "super ineffable" cardinals. It focusses on ineffability properties and the differences between "super ineffable" and "completely ineffable" cardinals.
We show that (a) PFA is consistent with having that NS_{\omega_1} is \Pi_1 definable and that (b) MM proves that NS_{\omega_1} is not \Pi_1 definable. Yet another time this shows that MM is the right generalization of MA. This is joint work with D. Asperó, S. Hoffelner, P. Larson, X. Sun, L. Wu.
We will show there is no sequence of distinct Sigma^2_1 sets of length (delta^2_1)^+ in L(R). We also discuss how to prove an analogous result for any inductive-like pointclass in L(R). This is joint work with Itay Neeman and Grigor Sargsyan.
