This is the second in a series of lectures on naive descriptive set theory based on an expository paper by Matt Foreman. The topics discussed will include tree representations, universality properties of Polish spaces, and subspaces of Polish spaces.

Tree properties are a family of combinatorial principles that characterize large cardinal properties for inaccessibles, but can consistently hold for "small" (successor) cardinals such as $\aleph_2$.  It is a classic theorem of Magidor and Shelah that if $\kappa$ is the singular limit of supercompact cardinals, then $\kappa^+$ has the tree property.  Neeman showed how to force $\kappa^+$ to become $\aleph_{\omega+1}$ while maintaining the tree property.  Fontanella generalized these results to the strong tree property.

We show (in ZFC) that if $\kappa$ is a singular limit of supercompact cardinals, then $\kappa^+$ has the super tree property (this jump from "strong" to "super" is analogous to the jump in strength from strongly to supercompact cardinals).  We remark on how to get the super tree property at $\aleph_{\omega+1}$, and on some interesting consequences for the existence of guessing models at successors of singulars.  This is joint work with Dima Sinapova.

In this first of two talks, I will explain the notion of definability in continuous logic and connect it with the notion of spectral gap in the theory of unitary representations and in ergodic theory.

In  Ramsey  Theory, ultrafilters often play an instrumental role.
By using nonstandard models of the integers, one can replace those
third-order objects (ultrafilters are families of subsets) by simple
points.

In this talk we present a nonstandard technique that is grounded
on the above observation, and show its use in proving some new results
in Ramsey Theory of Diophantine equations.
 

In this talk, we will continue with basics of measurable cardinals and their relationship to non-trivial elementary embeddings.  We proceed with basic facts about the constructible universe, L.  After laying this groundwork, we show L cannot have a measurable cardinal.  Time permitting, we will discuss the dichotomy introduced by Jensen's covering lemma: either L is a good approximation to V, or there is a non-trivial elementary embedding from L to L.