The majority of the talk won't assume much background beyond basic logic.

In this talk, after making sense of the first paragraph of this abstract, we will show that the interplay between hypergraphs and flat pregeometries runs deeper, and make some reckless conjectures.

An ongoing project of mine is to understand the general behavior of Medvedev reducibility on the ordinals. In this talk, I'll present one particular aspect of the problem: Given an ordinal α, how does the supremum of the ordinals Medvedev reducible to α (= Med(α)) compare with other notions of "least non-α-computable ordinal?" Specifically, I'll look at two such points emerging naturally from generalized computability theory, and show that in general Med(α) lies strictly between these two points. One of these inequalities is straightforward computability theory; the other requires a more set-theoretic argument, using forcing.

If time remains, I'll also outline a result on the degree structure of Medvedev reducibility on the ordinals; I'll show that it has "large antichains," the key difficulty being proving this result inside ZFC.

We show that the dimension of the partial order of all finite subsets of 2^ℵ₀ under set inclusion is ℵ₀. More generally, if κ ≤ 2^λ for infinite cardinals κ and λ such that λ is least such, then [κ]^<λ as a partial order under set inclusion has dimension λ.

We also show that the dimension of any locally countable partial ordering P of size κ⁺ for regular uncountable κ is at most κ. In particular, this implies that it is consistent with ZFC that the dimension of the Turing degrees under partial ordering can be strictly less than the continuum.

This is joint work with Higuchi (Nagoya) as well as Raghavan and Stephan (Singapore).

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.

No prior knowledge beyond basic model theory will be assumed. We will explain all of our definitions in the talk.

Using this result and a modified construction of an entangled set (due to Avraham) we will show that, under set theoretic assumption, some extra restriction can be given to these automorphisms.

We consider a generalization of the TP to a certain global property, the "super tree property" ITP. The ITP characterizes supercompactness, but can consistently hold for small cardinals. These are closely related to a family of guessing principles which have found a number of applications in inner model theory and infinitary combinatorics.

I will introduce these principles with as few prerequisites as possible, and discuss results on the extent to which the super tree property can hold near a singular. This is joint work with Dima Sinapova.

I will discuss some results and open questions on computable numberings and Rogers semilattices.