RUI: Filtrations of Boolean algebras and related structures

Geschke计划研究布尔代数、Banach空间和C-星代数的滤子在测度论、泛函分析、集合论拓扑和拓扑动力学中的应用。滤子通常可以用来构造所考虑的结构的自同构或同态。在其他情况下，结构的有趣性质可以通过特定类型的滤子的存在来刻画。将考虑以下主题：布尔代数、C-星代数和Banach空间的余定性，Calkin代数和其他C-星代数和布尔代数的自同构，具有大尺寸连续统的测度和范畴的Borel提升，最小动力系统的相空间的结构。许多关于某些无限结构的性质的自然问题不能用通常的数学公理来决定。这就是逻辑和集合论必须用来分析情况的地方。布尔代数足够好了，了解布尔代数和集合论之间的这种相互作用，有助于对集合论在更复杂的结构问题上的应用有一个正确的直觉。大型结构通常可以通过将它们分解成更小的碎片来分析，这种方法在布尔代数中尤其成功，因此它也应该系统地应用于其他无限结构的研究。

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.