Applications of Set Theory to Analysis

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The Fields Institute in Toronto, Ontario is running an intensiveprogram in Applications of Set Theory to Analysis from Septemberto December 2002. This is an award for the support of juniorU.S. participants in that international program. Long-termactivities in the semester include graduate courses and seminars,and shorter activities include a workshop on Descriptive SetTheory and a workshop on Banach Spaces, Algebras, and Subspacesof Baire Class 1 Functions.Even people whose only contact with higher mathematics comes fromwatching reruns of "Star Trek" or "The Simpsons" will have encounteredthe phenomenon that the decimal expansion of Pi requires infinitelymany digits. The ratio of the circumference of a circle to itsdiameter, therefore, is a manifestation of the actual, as opposed tojust the potential, infinite. While the existence of the actualinfinite has had a profound influence on mathematics, its acceptancewas not immediate and the history of the subject is fraught withcontroversy. Two major branches of mathematics which devote much oftheir resources to the study of the infinite are analysis and settheory. Much of analysis is concerned with understanding limitingbehaviour in various contexts. A simple prototype of this is found inconsidering the number Pi as the limit of increasingly longer finitedecimal approximations to Pi. Many of the spaces which analysts studyconsist of limits of simpler objects. Set theory, on the other hand,studies the infinite in a more abstract setting, concentrating on thecombinatorial and structural implications of the infinite. While theconnections between set theory and analysis have always beenacknowledged, too much of the work in these two areas has gone onlargely unheeded by researchers in the other area. The chief purposeof the current proposal is to create an environment where leading settheorists and analysts can collaborate on problems which straddle theboundaries of their subjects.