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^+

It is hard to find analogues of MA in which aleph_1 is replaced by the successor of a singular cardinal because
a) The consequences of MA-like axioms have large consistency strength
b) There is no satisfactory analogue of finite support ccc iteration

Dzamonja and Shelah found an ingenious approach to proving results of this general kind. I will outline their work and then describe some recent joint work with Dzamonja and Morgan, aimed at bringing results of this kind down to aleph_{omega+1}

Starting from a supercompact, we construct a model in which SCH fails at $\aleph_\omega$ and there is a bad scale at $\aleph_\omega$. The existence of a bad scale implies the failure of weak square. The construction uses two Prikry type forcings defined in different ground models and a suitably defined projection between them. This is joint work with Spencer Unger.