Grant: Discovery Grants Program - Individual

My research focuses on general topology and analysis, recently more specifically approximation theory. Some ofthe research has a set-theoretic flavor, as both of these areas occasionally require the use of deep set-theoretictechniques. The method of forcing especially, since 1963, when it was developed by Paul Cohen for proving thatthe Continuum Hypothesis is not decidable on the basis of the usual axioms for set theory, has led to importantadvances in these areas.A theorem of Hoischen provides approximation of smooth functions and their derivatives by entire functions, with interpolation on a closed discrete set. This theorem has been useful in helping to analyze theexistence of entire functions restricting to order-isomorphisms of subsets of the real line which are 'thick' in a sense,analogous to the Barth-Schneider theorem, which gives entire functions restricting to order-isomorphisms ofcountable dense sets. The insights gained from this work have also led to variations on the Hoischen theoremthat incorporate the ability to control the values of the derivatives on a countable set while choosing theapproximating function so that the graphs of its derivatives are generic in the sense that they cut a meagersection through a given meager set. If we want the graphs to instead cut a nullsection through a given null set, then there are weaker results in the same direction. We will seekto better understand this setting. In work in progress, we areexamining the comonotone approximation of piecewise monotone functions of one variable by entire functions.Another theme of my proposal is the interaction of the family of Baire sets with the measure in a topologicalmeasure space. I am studying liftings for the algebra of bounded measurable functions on a probability space.(A lifting is an algebra homomorphism of this algebra into itself which selects a representative from eachequivalence class for the relation that two functions are equivalent if they are equal almost everywhere.) Thesewere first shown to exist, for complete probability spaces, by von Neumann and Maharam. They are fundamentalto understanding the measure algebra of a probability space. When the measure is not complete however, theyare in general not known to exist, except under special settheoreticassumptions. I propose to continue previouswork investigating the existence of Baire liftings in various forcing extensions, and also closely related questionof how liftings for product spaces interact with liftings for the factors.