Applications of Set Theory to Analysis

安大略省多伦多市的菲尔兹研究所正在从 2002 年 9 月到 12 月举办一个集合论在分析中的应用强化项目。该奖项是为了表彰美国大三学生的支持。 该国际计划的参与者。 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巴纳赫空间、代数和贝尔一类函数子空间研讨会。即使是那些只通过观看《星际迷航》或《辛普森一家》重播才接触到高等数学的人，也会遇到过圆周率的十进制展开需要无穷多个数字的现象。因此，圆的周长与其直径的比值是实际无限的表现，而不只是潜在的无限。虽然实际无限的存在对数学产生了深远的影响，但它并没有立即被接受，而且该学科的历史充满了争议。 数学的两个主要分支是分析和集合论，它们投入大量资源来研究无限。许多分析都涉及理解各种情况下的限制行为。一个简单的原型是在将数字 Pi 视为 Pi 的越来越长的有限十进制近似值的极限时发现的。分析师研究的许多空间都由简单物体的限制组成。另一方面，集合论在更抽象的背景下研究无限，专注于无限的组合和结构含义。虽然集合论和分析之间的联系一直被承认，但这两个领域的太多工作基本上没有被其他领域的研究人员注意到。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就跨越其学科边界的问题进行合作。

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.