Descriptive set theory and its relations with functional and harmonic analysis

PI计划研究波兰群的结构以及描述集理论和拉姆齐理论在泛函和调和分析中的应用。近年来，一些源自抽象调和分析的技术被应用于非局部紧的波兰群的设置，从而为进一步理解描述集合论和可数一阶结构模型论中内部产生的拓扑群提供了丰富的工具集。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，如他最近与V. Ferenczi合作研究Banach空间上最优范数的存在性，也将可数一阶结构M的性质与其自同构群Aut(M)的拓扑性质进行校准。此外，PI正在研究起源于W.T. Gowers工作的块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空间分类程序。最近越来越清楚的是，Ramsey理论的基本结果最好理解为具有非常强的获胜策略的完全信息的无限博弈，通常的确定性并不立即适用于这种博弈。提案中描述的研究在很大程度上是跨学科的，连接了从数学逻辑到泛函和谐波分析的各个领域。PI旨在进一步将描述性集合论与其他数学分支的持续整合，从而为传统上从集合论的更纯粹领域的深入工作中受益匪浅的领域提供新的场所。在某种程度上，通过PI的工作，最终，这些结果将有望逐步应用于分析中相当温和的问题，也就是说，不涉及集合论或逻辑的先验问题。

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.