We present early applications of Martin's Maximum due to Foreman, Magidor and Shelah.

This is a continuation of the previous two talks, where we used large cardinals to get a normal, lambda-dense ideal on [lambda]^<kappa, where kappa is the successor of a regular cardinal, and GCH holds near kappa.  In this talk we show that the analogous statement for a successor of a singular cardinal is inconsistent.  If time permits, we will begin discussion of consistently separating certain properties at kappa>omega_1 that coincide at omega_1.

The topic of this talk is inspired by measure-theoretic questions raised by Ulam: What is the smallest number of countably additive, two valued measures on R such that every subset is measurable in one of them?  Under CH, the minimal answer to this question has several equivalent formulations, one of which is the maximal saturation property for ideals on aleph_1, aleph_1-density.  Our goal is to show that these equivalences are special to aleph_1.  In the first talk, we will show how to get normal ideals of minimal possible density on a variety of spaces from almost-huge cardinals.  This generalizes a result of Woodin.
 

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.