Applications of Descriptive Set Theory to Ideals of Closed Sets and Indecomposable Continua

描述集合论在闭集理想和不可分解连续体中的应用

• 批准号: 0342318
• 负责人: Slawomir Solecki
• 金额: $ 3.46万
• 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: 2003
• 资助国家: 美国
• 起止时间: 2003-05-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0342318&HistoricalAwards=false
• 关键词: Applications; Descriptive; Set; Theory; Ideals

Solecki studies possible applications of descriptive set theory to indecomposable continua and to ideals of closed sets. The first part of the project is concerned with studying the composant equivalence relation on indecomposable continua using techniques and notions developed in the study of Borel equivalence relations. Solecki builds on his prior work on indecomposable continua. He primarily investigates the question whether on a comeager subset of an indecomposable continuum the composant equivalence relation is Borel isomorphic to one of two special Borel equivalence relations via an isomorphism preserving meager sets. The affirmative answer to this question would solve an old problem of Kuratowski and even partial results for special indecomposable continua would sharpen several theorems from the literature. In the second part of the project, Solecki studies ideals of closed subsets of a Polish space. He investigates a certain very concrete representation of simply definable ideals of compact sets. This is connected with several open problems in this area of mathematics. Additionally, he continues his study of the ideal of Haar null subsets of a Polish group. Particular aims here are to develop the theory for all non-abelian Polish groups (the theory works fine for the class of Polish groups with invariant metrics) and to fully understand the connection between Haar null sets in infinite products of locally compact groups and amenability of the factor groups.One of the themes of Solecki's project is the investigation of indecomposable continua. These are fascinating geometrical objects whose intricate topological properties attracted interest of mathematicians since the beginning of the (last) century. However, only quite recently it was realized how ubiquitous such continua are and how important a role they play in various contexts in dynamical systems and topology. There is an old conjecture, due to Kuratowski, which is still unresolved and whose confirmation would completely reveal the finer structure of indecomposable continua. Solecki works on particularly important instances of this hypothesis and other problems related to it. Another theme of Solecki's project is the study of certain notions of smallness. These are important in various branches of mathematics to measure the size of sets under consideration. The starting point here is his observation that a vast class of such families of small sets admit surprising and very concrete type of representations. The possibility of representing a family of small sets in this fashion has deep implications for the structure of such families and, if realized, answers some old questions regarding this structure. Solecki studies the extent to which such representations can be established, interconnections between these type of representations and properties of notions of smallness, and other problems related to notions of smallness.

Solecki研究了描述性集理论在不可分解的连续图和封闭式理想中的可能应用。该项目的第一部分与研究在鲍尔等效关系研究中开发的技术和概念研究不可分解的连续性的组合剂对等关系有关。 Solecki建立在他先前在不可分解的Continua方面的工作。他主要调查了一个问题，是否在不可分解的连续体的一部分子集上，组合剂对等关系是通过保存微小的同构相对于两个特殊的鲍尔等效关系之一的borel同构的同构。这个问题的肯定答案将解决库拉托夫斯基的旧问题，甚至是特殊不可分解的连续图的部分结果都会使文献中的几个定理提高。在项目的第二部分中，Solecki研究了波兰空间的封闭子集的理想。 他研究了简单的紧凑型套装理想的一定非常具体的表示。这与该数学领域的几个开放问题有关。此外，他继续研究波兰群体的Haar Null子集的理想。这里的特定目的是为所有非亚伯兰波兰群体（该理论对具有不变度指标的波兰群体的阶级都效果很好）开发理论，并充分了解本地紧凑型组无限产品中的Haar Null集合和对局部紧凑型组的性能的联系，并且可以理解。因子组。索莱基项目的主题之一是对不可分解的连续图的调查。这些是引人入胜的几何物体，其复杂的拓扑特性自（上个世纪初）引起了数学家的兴趣。但是，直到最近才意识到这种连续性的无处不在，以及它们在动态系统和拓扑中的各种情况下的重要性。由于Kuratowski仍未解决，其确认会完全揭示出更精细的不可塑性连续图的结构，因此有一个旧的猜想。 Solecki致力于该假设及其其他问题的特别重要实例。 Solecki项目的另一个主题是研究某些小小的概念。这些在数学的各个分支中很重要，以衡量所考虑的集合的大小。这里的起点是他的观察结果是，大量这样的小家族承认了令人惊讶且非常具体的表现形式。以这种方式代表小型家族的可能性对此类家庭的结构产生了深远的影响，如果意识到，则回答了有关这种结构的一些旧问题。 SOLECKI研究了可以建立这种表示的程度，这些类型的表示形式和小概念的性质之间的互连以及与小规模概念有关的其他问题。