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 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.

There are many duality theorems in both finite and infinite graph theory that relate the maximum number of disjoint paths through a given graph G to the minimum size of a cut-set disconnecting G. The quintessential example is Menger's theorem, which says that the maximum number of vertex disjoint paths between two vertices x and y in a finite graph G is equal to the minimum size of a vertex cut disconnecting x and y.

We present a general approach to proving path/cut-set duality theorems for locally finite graphs. As one consequence of this approach, we give novel proofs of Menger's theorem and another classical duality result due to Halin. As another, we prove the following general existence theorem, which can be viewed as a maximal extension of König's lemma: given an infinite, locally finite connected graph G and a vertex x in G, there is a pruned tree T contained in G and rooted at x that, in a precise sense, splits as early and as often as possible.