Basic theory of modified Prikry forcing will be presented. In particular, the proof of Prikry lemma for modified Prikry forcing will be discussed.

Given a regular cardinal $\kappa$, an uncountably cofinal ordinal $\nu<\kappa$ is a reflection point of the stationry set $S\subseteq\kappa$ just in the case where $S\cap\alpha$ is stationary in $\alpha$. Starting from ininitely many supercompact cardinals, Magidor constructed a model of set theory where every stationary $S\subseteq\aleph_{\omega+1}$ has a reflection point. In this series of talks we present a construction of a model of set theory where we obtain a large amount of stationary reflection (although not full) using a significantly weaker large cardinal hypothesis. We start from a quasicompact (quasicompactness is a large cardinal hypothesis significantly weaker than any nontrivial variant of supercompactness) cardinal $\kappa$ and use modified Prikry forcing to turn $\kappa$ into $\aleph_{\omega+1}$. We then show that in the resulting model every stationray $S\subeteq\aleph_{\omega+1}$ not concentrating on ordinals of ground model cofinality $\kappa$ has a reflection point.

Given a measurable cardinal \kappa, any \Sigma^1_3 statement is absolute for generic extensions via forcings of size <\kappa. A proof of this classical theorem will be presented.

We present a proof of a theorem of Gitik and Shelah that places limits on the structure of quotient algebras by sigma-additive ideals. We will start by showing connections between Cohen forcing and Baire category on the reals. Then by using generic ultrapowers, we will prove that no sigma-additive ideal yields an atomless algebra with a countable dense subset. We will discuss connections with Ulam's measure problem: How many measures does it take to measure all sets of reals?

In this series of two talks I will give an introduction to some of my recent research on the ineffable tree property. The ineffable tree property is a two cardinal combinatorial principle which can consistently hold at small cardinals. My recent work has been on generalizing results about the classical tree property to the setting of the ineffable tree property. The main theorem that I will work towards in these talks generalizes a theorem of Cummings and Foreman. From omega supercompact cardinals, Cummings and Foreman constructed a model where the tree property holds at all of the $\aleph_n$ with $1 < n < \omega$. I recently proved that in their model the $(\aleph_n,\lambda)$ ineffable tree property holds for all $n$ with $1 < n < \omega$ and $\lambda \geq \aleph_n$.