Applications of descriptive set theory to functional analysis and topological dynamics

描述集合论在泛函分析和拓扑动力学中的应用

• 批准号: 0901405
• 负责人: Christian Rosendal
• 金额: $ 21.29万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901405&HistoricalAwards=false
• 关键词: Applications; descriptive; set; theory; functional

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The PI intends to work on three different topics involving descriptive set theory and its applications in Banach space theory and topological dynamics. Firstly, he intends to further elaborate on the structure theory of large Polish group and, in particular, automorphism groups of countable first order structures. The PI's main interest in this domain is the existence of generic representations of countable discrete groups in such automorphism groups and its uses in investigating the dynamics and algebraic structure of the automorphism group. Secondly, the PI will be working on the classical problem of J.P.R. Christensen of whether any universally measurable homomorphism between Polish groups is continuous. Again, this problem is tightly connected with the geometry and dynamical properties of Polish groups and very likely to leads to new techniques and concepts of value outside of the study of automatic continuity. Finally, Rosendal will be continuing his collaboration with V. Ferenczi on the rough classification of separable Banach spaces, a programme initiated by W. T. Gowers in the course of the solution to the homogeneous space problem. This programme heavily relies on Ramsey theoretical and game theoretical methods from descriptive set theory and thus enriches both descriptive set theory by developing new techniques and functional analysis by contributing to one of the most central problems of Banach space theory. In recent years, there has been a growing collaboration and sharing of ideas between researchers working in dynamical systems and descriptive set theory. Initially, this was largely fostered by work on Borel equivalence relations but has lately also come to include topological dynamics. The principal objectives of descriptive set theory consists in studying the definable subsets of the real line, showing that these are much better behaved than arbitrary subsets. This has developed into a large structure theory of such sets which gives a solid framework within which most of mathematical analysis takes place, and as such, the use of descriptive set theoretical techniques in parts of functional analysis is continuously expanding.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。PI打算从事三个不同的主题，涉及描述集理论及其在Banach空间理论和拓扑动力学中的应用。首先，他打算进一步阐述大波兰群的结构理论，特别是可数一阶结构的自同构群。PI在这一领域的主要兴趣是可数离散群在这种自同构群中的一般表示的存在及其在研究自同构群的动力学和代数结构中的用途。第二，私家侦探要解决J.P.R.的经典问题.克里斯滕森的问题是否任何普遍可测同态之间的波兰集团是连续的。同样，这个问题与波兰群的几何和动力学性质密切相关，并且很可能导致自动连续性研究之外的新技术和价值概念。最后，罗森达尔将继续他的合作与V. Ferenczi的粗糙分类的可分Banach空间，一个计划发起的W。T. Gowers在解决齐次空间问题的过程中。该计划在很大程度上依赖于拉姆齐理论和博弈论方法从描述集理论，从而丰富了描述集理论通过开发新技术和功能分析作出贡献的最核心的问题之一的Banach空间理论。近年来，在动力系统和描述集合论领域的研究人员之间的合作和思想共享越来越多。最初，这在很大程度上是由博雷尔等价关系的工作，但最近也来包括拓扑动力学。描述集合论的主要目的在于研究真实的直线的可定义子集，表明这些子集比任意子集表现得更好。这已经发展成为一个大的结构理论，这样的集，它提供了一个坚实的框架内，大部分的数学分析发生，因此，使用描述集理论技术的部分功能分析正在不断扩大。