Singular cardinals and cardinal characteristics
This project deals with the study of cardinal characteristics of the generalized Baire spaces κ^κ, in the specific case where κ is an uncountable singular cardinal. It is inspired by the work of Shimon Garti and Saharon Shelah on the study of cardinal characteristics of generalized Baire spaces, for an uncountable cardinal. Among others, we will study cardinals in the generalized Cichon's diagram and the consequences the singularity assumption will have on it, by using the already existing tools in the area of singular cardinals and our experience in the study of such generalizations.
Cardinal characteristics of the classical Baire space ω^ω are cardinals describing mostly the combinatorial or topological structure of the real line. They are usually defined in terms of ideals on the reals, or some very closely related structure such as P(ω)/fin and typically they assume values between ℵ_1, the first uncountable cardinal and c. Hence, they are uninteresting in models where the continuum hypothesis (2_0^ℵ=ℵ_1) holds. However, in models of set theory where CH fails they may assume different values and interact with each other in several ways. In the last years, special interest has been given to the study of these characteristics on the generalized Baire spaces κ^κ (the space of functions from κ to κ), when κ is an uncountable cardinal. By the time, the case where κ is additionally regular (or even a large cardinal) has been explored by many authors (included me) and nowadays it is possible to find many interesting ZFC and consistency results involving them.
On the other hand, singular cardinals represent another important area of study within set theory. They arose from the crucial concept of cofinality, which appeared after Julius König. These cardinals turn to be the source of many interesting problems and were the starting point of the outstanding PCF theory (PCF stands for possible cofinalities.) Particular interest has been given, for instance to the value of the continuum function for singular cardinals which have risen to the well-know singular cardinal hypothesis (SCH).
