Abstract: The set theoretic generalizations of algebras have been introduced in the 1960s to give a set theoretic interpretation of usual algebraic structures. The shift in perspective from algebra to set theory is that in set theory the focus is on the collection of possible algebras and sub-algebras on specific cardinals rather than on particular algebraic structures. The study of collections of algebras and sub-algebras has generated many well-known problems in combinatorial set theory (e.g., Chang’s conjecture and the existence of small singular Jonsson cardinals). In the 1990s Foreman and Magidor used algebras to initiate a study of Singular Stationarity, i.e., a study of alternative notions of stationarity for subsets of singular cardinals. They introduced and developed two notions of singular stationarity called Mutual Stationarity and Tight Stationarity, and used their findings to prove a fundamental result concerning generalized nonstationary ideals. The two notions of singular stationarity have been studied in the last two decades, and the main purpose of the talk to describe the related known and recent results. I will start by giving some background material on algebras and stationary sets, and describe the history of a well-known problem by Jonsson. We will then proceed to describe the work done on the notion of Mutually Stationary Sequences and sketch a recent proof which is based on the existence of special types of strong filters. In the second part of the talk, we will connect the notions of Singular Stationarity to results from PCF theory and Extender-based forcing methods.
Wed, 27/12/2017 - 11:00 to 13:00