Mini-course (30.11.2023-25.01.2024, 6 lectures)- 1st lecture:
Cichoń's diagram describes the connections between combinatorial notions related to measure, category, and compactness of sets of irrational numbers. In the second part of the 2010's decade, Goldstern, Kellner and Shelah constructed a forcing model of Cichoń's Maximum (meaning that all non-dependent cardinal characteristics are pairwise different) by using large cardinals. Some years later, we eliminated this large cardinal assumption. In this mini-course, we explore the forcing techniques to construct the Cichoń's Maximum model and much more. Concretely, we discuss the following components: 1. Tukey connections and cardinal characteristics of the continuum 2. Review of FS (finite support) iterations and basic methods to modify cardinal characteristics. 3. Preservation theory for cardinal characteristics. 4. FS iterations with measures and ultrafilters on the natural numbers. 5. Boolean Ultrapowers. 6. Forcing Intersected with submodels.