Starting with a cardinal which is both subcompact and measurable, we produce a model in which \square_{\kappa,2} holds but \square_\kappa fails at a singular cardinal \kappa.  We will discuss several of the essential tools used, and also several ways in which this result may be extended.
 

We present a theorem of Foreman that allows an exact characterization of what happens to the structure of precipitous ideals after suitable forcing. This theorem unifies several well-known results, giving as them quick corollaries. We will use it to show: forcing precipitous ideals from large cardinals, preservation theorems of Kakuda and Baumgartner-Taylor, and Solovay's consistency result on real-valued measurable cardinals.  We will also show some new applications due to the speaker.
 

Under the Axiom of Determinacy, the least uncountable cardinal omega_1 behaves like a large cardinal. We will present a theorem due to Woodin saying that from a strong form of determinacy, namely AD_R + "Theta is regular," one can force the Axiom of Choice together with the statement "there is an omega_1-dense ideal on omega_1." In essence, omega_1 retains a trace of its "large cardinal" nature that is consistent with AC.
 

We continue with an introduction to the principle ISP and its relatives as well as their connections to supercompact cardinals and the proper forcing axiom. In particular, we prove that PFA implies ISP.

We present some basic consequences of Martin's Maximum.