# Tutorial Track

## M. Goldstern: Iterated forcing

I will present some old and well-known facts about iterated forcing, and also give a few more recent constructions. In the first hour I will define iterated forcing and present the proof of "preservation of properness for countable support iterations", and if time permits, also sketch a preservation theorem. In the second lecture I will talk about some techniques that are useful for dealing with products (continuous reading, bigness and halving of creatures), and in the third lecture I will talk about finite support iterations.

## T. Jech: Measure Algebras

In the 1930s, John von Neumann asked if measure algebras can be characterized algebraically, in particular, as Boolean $\sigma$-algebras that satisfy the countable chain condition and the weak distributivity law.

75 years – and extensive work by many mathematicians – later, we now know that the von Neumann conditions are not sufficient. However, in order to carry a sigma-additive measure it is sufficient (and necessary) that the algebra is weakly distributive and has the additional property that we call “uniformly concentrated”: there exists a choice function $F$ on finite antichains such that whenever $A_n$ is a sequence of antichains with $|A_n|\ge 2^n$ then the sequence ${F(A_n)}$ converges to $0$.

In the tutorial we give a complete self-contained proof of this result (for the lack of time we will use but not prove two classical mathematical results: the Hahn–Banach Theorem and Hall’s “Marriage” Theorem). No specialized knowledge is necessary; familiarity with convergence of sequences in a Boolean sigma-algebra would be helpful.

## Y. Moschovakis: Effective descriptive set theory, what it is about

The subject is a common generalization of classical Descriptive Set Theory and Recursion Theory and my aim in this tutorial is to give an elementary exposition of a few of its fundamental ideas, notions and techniques. I will assume only some basic facts about recursive functions on the natural numbers, set theory and (metric) topology, what would normally be covered in a semester of an introductory class in each of these topics.

## L. Zdomskyy: Menger spaces everywhere

The first lecture will be devoted to general properties of Menger and Hurewicz spaces. During the second one we'll discuss filters with these properties and the Mathias forcing associated to them. This lecture will be based on the article arxiv.org/pdf/1401.2283.pdf">Mathias Forcing and Combinatorial Covering Properties of Filters. The third lecture will be about topological Ramsey theory and be based on arxiv.org/pdf/1407.7437.pdf">Algebra, Selections and Additive Ramsey Theory.