Abstract. The connectivity relation of a Borel graph is always analytic, but it turns out that not every analytic equivalence relation can be realized as a connectivity relation of a Borel graph. We call an analytic equivalence relation Borel graphable if it is the connectedness relation of a Borel graph. We will examine two important types of equivalence relations for the theory of Borel graphability: (1) equivalence relations coming from recursion theory (Church-Kleene ordinals, in particular) that provide interesting examples and counter-examples; and (2) orbit equivalence relations from Polish group actions. In particular, we will see that all connected Polish groups have Borel graphable orbit equivalence relations. This is joint work with Alekos Kechris and Patrick Lutz.

Abstract. A collection K of certain structures (e.g., graphs, groups, topological spaces) has a universal object M in K if every member of K embeds into M. By a fundamental result in model theory, if K is the collection of all models of a certain complete first order theory T and of some regular size kappa, then the continuum hypothesis implies the existence of a universal object for K. I will describe several results concerning universality problems for collections of second order structures involving Aronszajn trees. This is joint work with Siiri Kivimäki, Menachem Magidor and Jouko Vananen.

Abstract. Stationary sets are a fundamental concept in set theory, but their definition only makes sense at regular cardinals. We will describe mutual stationarity, a property that can be viewed as an analog of stationarity for singular cardinals, and discuss how it interacts with other combinatorial properties at or near $\aleph_\omega$. In particular, we will discuss the tree property and the failure of the Singular Cardinal Hypothesis.