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打算在功能和谐波分析中对波兰群体的结构以及描述性集理论和拉姆齐理论的应用进行研究。最近，已经采用了许多源自抽象谐波分析的技术，用于非局部紧凑的波兰群体的设置，从而提供了一种丰富的工具集，用于进一步了解对描述性集合理论的内部拓扑组的理解和可数的一阶结构的模型理论。 PI参与其中正在研究这些工具的各个方面，既将波兰群体的结构理论应用于功能分析中的问题，正如他最近与Ferenczi在Banach空间上存在最佳规范的合作所见证的那样，以及校准Banach的特性，以及具有自动化型组自动拓扑特性的可计数一阶结构M（M）。此外，PI还研究了源自W.T. Gowers的工作的Ramsey理论，该理论在Banach空间的几何理论中取得了重大结果，特别是基于无限二维Ramsey理论的可分离Banach空间的分类程序。正如最近变得越来越清楚的那样，从无限的完美信息游戏中，最好理解潜在的Ramsey理论结果，并具有非常强大的获胜策略，而通常的确定性并不立即适用。该提案中描述的研究主要是跨学科的，它将从数学逻辑到功能和谐波分析的各个领域连接起来。 PI旨在将描述性理论与其他数学分支的连续整合到不断的整合，从而为传统上从纯粹的纯粹理论领域的深层工作中受益匪浅的领域提供了新的场所。最终，在某种程度上，通过PI的工作，这些结果希望降低到应用程序中的应用程序，以驯服分析中的问题，即不是涉及集合理论或逻辑的先验问题。