RUI: Filtrations of Boolean algebras and related structures

Boise State University, Boise ID

Investigators

Abstract

Geschke plans to study applications of filtrations of Boolean algebras, Banach spaces and C-star-algebras to problems in measure theory, functional analysis, set-theoretic topology and topological dynamics. Filtrations can often be used to construct automorphisms of or homomorphisms from the structures under consideration. In other cases, interesting properties of a structure can be characterized in terms of the existence of a filtration of a specific type. The following topics will be considered: Cofinalities of Boolean algebras, C-star-algebras and Banach spaces, automorphisms of the Calkin algebra and other C-star- and Boolean algebras, Borel liftings for measure and category with a large size of the continuum, the structure of phase spaces of minimal dynamical systems. Many natural questions about properties of certain infinite structures cannot be decided using the usual axioms of mathematics. This is where logic and set theory have to be used to analyze the situation. Boolean algebras are sufficiently nice, so that set-theoretic methods can be applied very directly in order to answer questions about them. Understanding this interaction between Boolean algebras and set theory helps in getting the right intuition about applications of set theory to questions about more complicated structures. Large structures can often be analyzed by breaking them into smaller pieces that fit together nicely. This approach has been particularly successful with Boolean algebras. Therefore it should be applied systematically to study other infinite structures as well.