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年的《美国复苏与再投资法》（公法111-5）资助的。PI打算在涉及描述性集理论及其在Banach Space理论和拓扑动态中的描述性集理论及其应用的三个不同主题上工作。首先，他打算进一步详细介绍大波兰群体的结构理论，尤其是可计数一阶结构的自动形态群体。 PI对该领域的主要兴趣是在此类自动形态组中存在可数离散组的通用表示及其在研究自动形态组的动力学和代数结构中的用途。其次，PI将解决J.P.R.的经典问题。克里斯滕森（Christensen）关于波兰群体之间是否有任何普遍可衡量的同态性能是连续的。同样，这个问题与波兰群体的几何形状和动力学特性密切相关，并且很可能会导致自动连续性研究以外的新技术和价值概念。最后，罗森达尔（Rosendal）将继续与V. Ferenczi在可分开的Banach空间的粗略分类中进行合作，这是W. T. Gowers在解决同质空间问题的过程中发起的该计划。该程序在很大程度上依赖于描述性集理论的Ramsey理论和游戏理论方法，因此通过开发新技术和功能分析来丰富描述性集理论，通过为Banach Space理论的最中心问题之一做出贡献。近年来，在动态系统和描述性集合理论的研究人员之间的合作和共享越来越多。最初，这在很大程度上是由Borel等效关系的工作促进的，但最近也包括拓扑动态。描述集理论的主要目标是研究真实行的可定义子集，表明这些子集比任意子集更好。这已经发展成为了此类集合的大结构理论，该理论赋予了大多数数学分析的稳固框架，因此，在功能分析的部分地区使用描述性集的理论技术是不断扩展的。