The set theory of Polish groups

波兰群的集合论

• 批准号: 0556368
• 负责人: Christian Rosendal
• 金额: $ 12.12万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-05-15 至 2009-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0556368&HistoricalAwards=false
• 关键词: set; theory; Polish; groups

Rosendal proposes to pursue a general study of Polish groups from the point of view of descriptive set theory, model theory and topological dynamics. In the project Rosendal will investigate both algebraic and topological properties of these groups, such as the small index property, the Bergman property, phenomena of automatic continuity of homomorphisms and extreme amenability. The main problems investigated are to a great extent motivated by the model theoretical problem of reconstructing a countable structure from its group of automorphisms. One of the principal tools used in this connection is the automatic continuity of isomorphisms of automorphism groups, which motivated the study of the broader phenomenon of automatic continuity of arbitrary homomorphisms between classes of Polish groups. Rosendal also plans to apply these ideas to study the topological dynamics of uncountable discrete groups most notably in connection with the fixed point on metric compacta property. The study of Polish groups using the very diverse methods of several fields seems likely to promote the further integration of separate knowledge and deeper understanding of the objects considered.In a separate project, Rosendal intends to continue his work with V. Ferenczi on a question of G. Godefroy concerning the number of non-isomorphic subspaces of a non-Hilbertian Banach space. The methods that have proven useful at this moment have been highly set theoretical in nature and have reduced the problem to the case of minimal spaces. Probably more analytical tools will be of greater utility for the next steps. In connection with this, Rosendal plans to revisit Gowers' determinacy theorem and by using set theoretical tools extend its applications beyond its nominal reach of analytic sets. This should provide tools in Banach space theory not obtainable by classical geometric considerations. Also Rosendal will pursue another ongoing project with B.D. Miller of classifying Borel transformations up to a descriptive notion of Kakutani equivalence. This is a completely parallel project to the theory of Kakutani equivalence in ergodic theory, but due to the nature of the objects, the methods used are completely descriptive set theoretical and go back to works of Glimm and Effros in operator algebra. By stressing the interconnections of mathematical logic with other domains of mathematics, Rosendal hopes to enrich both logic itself and provide new insight into objects outside of mathematical logic. The main applications of his research will be in functional analysis (Banach space theory), topological groups, and ergodic theory.

罗森达尔建议从描述集合论、模型论和拓扑动力学的角度对波兰群进行一般性研究。在该项目中，罗森达尔将研究这些群的代数和拓扑性质，例如小指数性质、伯格曼性质、同态自动连续性现象和极端顺应性。研究的主要问题在很大程度上是由从自同构群重建可数结构的模型理论问题引发的。在这方面使用的主要工具之一是自同构群的同构的自动连续性，它激发了对波兰群类之间任意同态自动连续性的更广泛现象的研究。 Rosendal 还计划应用这些思想来研究不可数离散群的拓扑动力学，尤其是与度量紧致性质上的不动点相关的拓扑动力学。使用多个领域非常多样化的方法对波兰群体进行研究似乎可能会促进单独知识的进一步整合和对所考虑对象的更深入理解。在一个单独的项目中，罗森达尔打算继续与 V. Ferenczi 就以下问题进行合作： G. Godefroy 关于非希尔伯特巴纳赫空间的非同构子空间的数量。目前已被证明有用的方法本质上是高度理论化的，并将问题简化为最小空间的情况。也许更多的分析工具对于接下来的步骤会有更大的用处。与此相关，罗森达尔计划重新审视高尔斯确定性定理，并通过使用集合理论工具将其应用扩展到解析集合的名义范围之外。这应该为巴纳赫空间理论提供经典几何考虑无法获得的工具。 Rosendal 还将与 B.D. 一起开展另一个正在进行的项目。 Miller 将 Borel 变换分类为 Kakutani 等价的描述性概念。这是与遍历理论中的角谷等价理论完全平行的项目，但由于对象的性质，所使用的方法是完全描述性的集合理论，并且可以追溯到算子代数中 Glimm 和 Effros 的著作。通过强调数理逻辑与其他数学领域的相互联系，罗森达尔希望丰富逻辑本身，并为数理逻辑之外的对象提供新的见解。他的研究的主要应用是泛函分析（Banach 空间理论）、拓扑群和遍历理论。