Dr. Jinrich Zapletal from the University of Florida (USA) will give a short course on forcing. We will have four sessions:
Abstract: Given a sigma ideal I on a Polish space X, we will study the forcing P_I of Borel I-positive sets ordered by inclusion. In this lecture, I will provide the basic observations regarding the forcing properties of P_I in general and their translation into the language of Borel sets and functions.
Abstract: I will give a long list of sigma ideals I for which the factor forcing P_I is proper, and I will study their finer forcing properties. The investigation will take us to such fields as measure theory, dynamical systems, Fourier analysis, Borel equivalence relations and others. This will provide a counterpoint to Shelah's combinatorial approach to definable proper forcing.
Abstract: I will study the operations on forcings countable support iteration, side by side product, illfounded iteration and others and I will find the corresponding operations on ideals. This provides an analytic description of these operations and suggests many new possibilities.
Abstract: I will use the work from the previous lectures to prove various absoluteness theorems for forcing extensions (stating that certain well known forcing extensions are optimal for certain types of independence results) and preservation theorems for the countable support iteration quite different from Shelah's.