Laver functions for supercompact cardinals appear in many forcing constructions, including all known constructions of models of strong forcing axioms. Viale proved that the Proper Forcing Axiom implies the existence of a "generic" Laver function from $\omega_2 \to H_{\omega_2}$. I will discuss his result and some recent work of mine on generic Laver functions.

We finish the presentation of Laver preparation from the last week.

The classical Theorem of Laver on making a supercompact cardinal indestructible will be presented.

We will present the combinatorial principles ITP and ISP, discuss how they strengthen the
tree property, and talk about their relative consistency

For each \alpha < \omega_1, let

X_\alpha = \{f : \omega^\alpha \rightarro\powerset_{\omega_1}(\mathbb{R})| f is increasing and continuous}

and \mu_\alpha be a normal fine measure on X_\alpha. We identify X_0 with \powerset_{\omega_1}(R). Martin and Woodin independently showed that these measures exist assuming (ZF + DC_\mathbb{R}) + AD + Every set is Suslin (\mu_0's existence was originally shown by Solovay from AD_\mathbb{R}). We sketch the proof of the derived model construction giving the existence of these measures (+ AD^+) from large cardinals and the Prikry forcing construction which gives back the exact large cardinal strength from AD^+ and the measure. If time allows, we will survey some theorems on the structure theory of the model L(\mathbb{R},\mu_\alpha) assuming the model satisfies \Theta > \omega_2 and \mu_\alpha is a normal fine measure on X_\alpha. Here the main theorem is that our assumption implies L(\mathbb{R},\mu_\alpha) satisfies AD^+