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拟对波兰群的结构以及描述集合论和拉姆齐理论在泛函和调和分析中的应用进行研究。最近，许多源自抽象调和分析的技术已被应用于非局部紧波兰群的设置，从而为进一步理解描述性集合论和模型论中内部出现的拓扑群提供了丰富的工具集。可数的一阶结构。 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问题，正如他最近与 V. Ferenczi 就 Banach 空间上最优范数的存在性进行合作所证明的那样，并校准属性可数一阶结构 M 及其自同构群 Aut(M) 的拓扑性质。此外，PI正在研究源于W.T. Gowers工作的分块Ramsey理论，该理论在Banach空间的几何理论方面取得了重大成果，特别是基于无限维Ramsey理论的可分离Banach空间的分类程序。最近变得更加清楚的是，拉姆齐的基本理论结果可以通过完美信息的无限博弈和非常强大的获胜策略来最好地理解，而通常的确定性并不立即适用。提案中描述的研究很大程度上是跨学科的，连接了从数理逻辑到泛函和调和分析的各个领域。 PI 旨在进一步将描述性集合论与其他数学分支不断整合，从而为传统上从集合论纯领域的深入研究中受益匪浅的领域提供新的场所。最终，部分通过 PI 的工作，这些结果将有望渗透到应用程序中，从而解决分析中的问题，即不涉及集合论或逻辑的先验问题。