Applications of Set Theory to Analysis

2002年9月至12月，安大略省多伦多的菲尔兹研究所（Fields Institute）正在开展“集合论在分析中的应用”的强化项目。这是一个支持美国青少年的奖项。国际项目的参与者。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理论和Baire 1类函数的巴拿赫空间、代数和子空间研讨会。即使是那些只通过观看《星际迷航》（Star Trek）或《辛普森一家》（The Simpsons）的重播才接触到高等数学的人，也会遇到这样的现象：圆周率的十进制展开需要无限多的数字。因此，圆的周长与其直径之比是实际无限的表现，而不仅仅是潜在无限的表现。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，而且这一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，它们投入了大量的资源来研究无穷。许多分析都是关于理解在各种情况下的限制性行为。这个问题的一个简单的原型是考虑π作为π的越来越长的有限十进制近似值的极限。分析者研究的许多空间都是由简单物体的极限组成的。另一方面，集合论在一个更抽象的环境中研究无限，集中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在很大程度上被其他领域的研究人员所忽视。当前提案的主要目的是创造一个环境，让领先的集合论者和分析师能够在跨越各自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.