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 $\lambda \geq \aleph_n$.

Continuous logic is a relatively new logic better equipped for studying the model theory of structures based on complete metric spaces. There are continuous analogs of virtually every notion and theorem from classical model theory, often with equalities replaced by approximations. However, most of the work done in continuous logic has centered around sophisticated topics concerning stability and its generalizations. In this talk, I will discuss the more basic notion of definability in metric structures. More specifically, I will consider the question of which functions are definable in Urysohn's metric space. Urysohn's metric space is the unique (up to isometry) Polish space that is universal and ultrahomogeneous. In many ways, Urysohn's metric space is to continuous logic as the the infinite set is to classical logic. However, we will see that the task of understanding the definable functions in Urysohn's metric space involves some interesting topological considerations.

The construction of the Martin-Solovay tree will be presented

Basic theory of homogeneity systems and homogeneous trees will be presented.