Applications of Set Theory to Analysis

位于安大略多伦多的菲尔兹研究所从2002年9月到12月正在进行一个集理论在分析中的应用的强化项目。 这是一个奖的支持，美国青年。 参加这个国际项目。 本学期的长期活动包括研究生课程和研讨会，短期活动包括描述性集合论研讨会和Banach空间、代数和Baire第一类函数的子空间研讨会。即使是那些只通过观看《星星迷航》或《辛普森一家》重播而接触高等数学的人也会理解圆周率的小数展开需要无限多位数的现象。因此，圆的周长与直径之比是实际的表现，而不仅仅是潜在的表现。虽然实无穷的存在对数学产生了深远的影响，但它的接受并不是立即的，这个主题的历史充满了争议。 数学的两个主要分支把大量的资源投入到无穷大的研究中，这两个分支是分析和集合论。很多分析都是关于理解各种情况下的限制行为。一个简单的原型，这是发现在考虑数Pi的限制越来越长的finitedecimal近似Pi。分析家研究的许多空间都是由简单物体的极限组成的。集合论，另一方面，研究无限在一个更抽象的设置，集中在thecombinatorial和结构的影响无限。虽然集合论和分析之间的联系一直被人们所承认，但这两个领域的太多工作在很大程度上没有被另一个领域的研究人员所注意。当前提案的主要目的是创造一个环境，让领先的集合理论家和分析师可以在跨越学科边界的问题上进行合作。

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.