This is the fifth in a series of lectures on naive descriptive set theory based on an expository paper by Matt Foreman. We continue the discussion of universality properties of Polish spaces and subspaces of Polish spaces.

This is the third in a series of lectures on naive descriptive set theory based on an expository paper by Matt Foreman. We continue the discussion of universality properties of Polish spaces and subspaces of Polish spaces.

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.