Cardinal Invariants and Sets of Reals

These research projects concern the set theory of the real line, a part of descriptive set theory. The principal investigator studies various ideals of sets of real numbers. Cardinal characteristics of the continuum and the associated families of small sets are central to this research. Specifically, he focuses on problems concerning the consistency of various generalizations of the Borel Conjecture, a statement asserting that the families of sets in question consist entirely of countable sets. He also studies the extent of the duality between the two classical notions of smallness: measure zero and the first category. This problem concerns finding parallels between the measure concepts such as strong measure zero and universal measure zero, and their first category analogs. The study of the structure of the real numbers marks the origins of set theory and has been the object of systematic research since the beginning of the last century. Concepts of measure and category have been studied rigorously for about one hundred years, and have been successfully used in many areas of modern mathematics. Bartoszynski pursues several problems in this area. These problems may yield positive answers that are theorems of mathematics or they may turn out to be independent from the standard axioms of set theory. The positive answers involve new results in both finite and infinite combinatorics and have applications reaching beyond traditional set theory to real analysis, measure theory, and topology. On the other hand, the independence results, particularly ones using forcing, have applications within set theory itself.