Descriptive set theory and its relations with functional and harmonic analysis

描述集合论及其与泛函分析和调和分析的关系

• 批准号: 1201295
• 负责人: Christian Rosendal
• 金额: $ 31.76万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-05-15 至 2016-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201295&HistoricalAwards=false
• 关键词: Descriptive; set; theory; its; relations

The PI intends to conduct research on the structure of Polish groups and applications of Descriptive Set Theory and Ramsey theory in Functional and Harmonic Analysis. Recently, a number of techniques originating in abstract harmonic analysis have been adopted to the setting of non-locally compact Polish groups, thus providing a rich tool set for furthering the understanding of the topological groups arising internally in descriptive set theory and the model theory of countable first-order structures. The PI is engaged is studying various aspect of these tools, both applying the resulting structure theory of Polish groups to problems in functional analysis, as witnessed in his recent collaboration with V. Ferenczi on the existence of optimal norms on Banach spaces, and calibrating properties of countable first-order structures M with topological properties of their automorphism group Aut(M). In addition, the PI is studying the block Ramsey theory originating in the work of W.T. Gowers, which has led to significant results in the geometric theory of Banach spaces, in particular, a classification program for separable Banach spaces based on infinite-dimensional Ramsey theory. As has become clearer recently, the underlying Ramsey theoretical results are best understood in terms of infinite games of perfect information with very strong winning strategies and to which usual determinacy does not immediately apply. The research described in the proposal is largely inter-disciplinary, connecting a variety of fields from mathematical logic to functional and harmonic analysis. The PI aims to further the continuous integration of descriptive set theory with other branches of mathematics and thus provide new venues for a field traditionally having benefited enormously from deep work in purer areas of set theory. In part through the PI's work, eventually, these results will hopefully trickle down to applications to rather tame problems in analysis, i.e., not a priori involving set theory or logic.

PI打算对波兰群的结构以及描述集理论和Ramsey理论在泛函和调和分析中的应用进行研究。最近，一些技术起源于抽象调和分析已被采用到设置非局部紧波兰群，从而提供了丰富的工具集，进一步了解的拓扑群内部产生的描述集理论和模型理论的可数一阶结构。PI正在研究这些工具的各个方面，既将波兰群的结构理论应用于泛函分析中的问题，就像他最近与V. Ferenczi在Banach空间上存在最优范数的合作中所见证的那样，又用其自同构群Aut（M）的拓扑性质校准可数一阶结构M的性质。此外，PI正在研究源于W.T. Gowers，他在Banach空间几何理论方面取得了重要成果，特别是基于无限维Ramsey理论的可分Banach空间的分类程序。正如最近变得更加清楚的那样，拉姆齐的理论结果最好用具有非常强的获胜策略的完美信息的无限博弈来理解，通常的确定性并不立即适用。提案中描述的研究在很大程度上是跨学科的，连接了从数理逻辑到泛函和谐波分析的各种领域。PI旨在进一步将描述性集合论与其他数学分支进行持续整合，从而为传统上从集合论更纯粹领域的深入工作中受益匪浅的领域提供新的场所。通过PI的部分工作，最终，这些结果将有望应用于分析中的问题，即，不涉及集合论或逻辑的先验的。