Applications of Set Theory to Analysis

California Institute Of Technology, Pasadena CA

Investigators

Abstract

The Fields Institute in Toronto, Ontario is running an intensive program in Applications of Set Theory to Analysis from September to December 2002. This is an award for the support of junior U.S. participants in that international program. Long-term activities in the semester include graduate courses and seminars, and shorter activities include a workshop on Descriptive Set Theory and a workshop on Banach Spaces, Algebras, and Subspaces of Baire Class 1 Functions. Even people whose only contact with higher mathematics comes from watching reruns of "Star Trek" or "The Simpsons" will have encountered the phenomenon that the decimal expansion of Pi requires infinitely many digits. The ratio of the circumference of a circle to its diameter, therefore, is a manifestation of the actual, as opposed to just the potential, infinite. While the existence of the actual infinite has had a profound influence on mathematics, its acceptance was not immediate and the history of the subject is fraught with controversy. Two major branches of mathematics which devote much of their resources to the study of the infinite are analysis and set theory. Much of analysis is concerned with understanding limiting behaviour in various contexts. A simple prototype of this is found in considering the number Pi as the limit of increasingly longer finite decimal approximations to Pi. Many of the spaces which analysts study consist of limits of simpler objects. Set theory, on the other hand, studies the infinite in a more abstract setting, concentrating on the combinatorial and structural implications of the infinite. While the connections between set theory and analysis have always been acknowledged, too much of the work in these two areas has gone on largely unheeded by researchers in the other area. The chief purpose of the current proposal is to create an environment where leading set theorists and analysts can collaborate on problems which straddle the boundaries of their subjects.