A major project in set theory aims to explore the connection between large cardinals and so-called generic absoluteness principles, which assert that forcing notions from a certain class cannot change the truth value of (projective, for instance) statements about the real numbers. For example, in the 80s Kunen showed that absoluteness to ccc forcing extensions is equiconsistent with a weakly compact cardinal. More recently, Schindler showed that absoluteness to proper forcing extensions is equiconsistent with a remarkable cardinal. (Remarkable cardinals will be defined in the talk.) Schindler's proof does not resemble Kunen's, however, using almost-disjoint coding instead of Kunen's innovative method of coding along branchless trees. We show how to reconcile these two proofs, giving a new proof of Schindler's theorem that generalizes Kunen's methods and suggests further investigation of non-thin trees.

Forcing and elementary embeddings are central topics in set theory. Most of what set theorists have focused on are the study of forcing and elementary embeddings over models of ZFC. In this talk, we focus on forcing and elementary embeddings over models of the Axiom of Determinacy (AD). In particular, we focus on answering the following questions: work in V which models AD. Let P be a forcing poset and g ⊆ P be V -generic.

1) Does V [g] model AD?

2) Is there an elementary embedding from V to V [g]?

Regarding question 1, we want to classify what forcings preserve AD. We show that forcings that add Cohen reals, random reals, and many other well-known forcings do not preserve AD. Regarding question 2, an analogous statement to the famous Kunen’s theorem for models of ZFC, can be shown: suppose V = L(X) for some set X and V models AD, then there is no elementary embedding from V to itself. We conjecture that there are no elementary embeddings from V to itself. We present some of the results discussed above. There is still much work to do to completely answer questions 1 and 2. This is an ongoing joint work with D. Ikegami.

In this talk, I’ll sketch a way of unifying a wide variety of set theoretic approaches for generating new models from old models. The underlying methodology will draw from techniques in Sheaf Theory and the theory of Boolean Ultrapowers.

We describe a number of related questions at the interface of set theory and homology theory, centering on (1) the additivity of strong homology, and (2) the cohomology of the ordinals. In the first, the question is, at heart: To how general a category of topological spaces may classical homology theory be continuously extended? And in the tension between various potential senses of continuity lie a number of delicate set-theoretic questions. These questions led to the consideration of the Cech cohomology of the ordinals; the surprise was that this is a meaningful thing to consider at all. It very much is, describing or suggesting at once (i) distinctive combinatorial principles associated to the nth infinite cardinal, for each n, holding in ZFC, (ii) rich connections between cofinality and dimension, and (iii) higher-dimensional extensions of the method of minimal walks.

One of the primary theoretical tools in machine learning is Vapnik-Chervonenkis dimension (VC dimension), which measures the maximum number of distinct data points a hypothesis set can distinguish. This concept is primarily used in assessing the effectiveness of training classification algorithms from data, and it is established that having finite VC dimension guarantees uniform versions of the various laws of large numbers, in the sense of e.g. Dudley (2014). An important result of Laskowski (1992) showed that finite VC dimension corresponds to a logical notion independently developed by Shelah, known as the non-independence property, and in subsequent decades much work has been done on finite VC dimension within model theory under the aegis of so-called NIP theories (cf. Simon, 2015).

However, despite this deep connection to logic, there has been little done on the computable model theory of VC dimension (one recent exception being Andrews and Guingona, 2016). The basic questions here are the following: (1) how computationally difficult is it to detect that one is in a setting with finite VC dimension?, and (2) if one is in this situation, how hard is it to compute what the precise VC dimension is? The current paper aims to answer these questions and discuss their implications.

Reference

Andrews, U. and Guingona, V. (2016). A local characterization of VC-minimality. Proceedings of the American Mathematical Society, 144(5):2241–2256.

Dudley, R. M. (2014). Uniform central limit theorems, volume 142 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, New York, second edition.

Laskowski, M. C. (1992). Vapnik-Chervonenkis classes of definable sets. Journal of the London Mathematical Society, 45(2):377–384.

Simon, P. (2015). A Guide to NIP Theories. Lecture Notes in Logic. Cambridge University Press, Cambridge.