Borel Equivalence Relations with Applications to Indecomposable Continua and Polish Group Actions

Borel 等价关系及其在不可分解的 Continua 和 Polish 群作用中的应用

• 批准号: 9803676
• 负责人: Slawomir Solecki
• 金额: $ 6.55万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-15 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803676&HistoricalAwards=false
• 关键词: Borel; Equivalence; Relations; Applications; Indecomposable

Solecki proposes several projects on the borderline of descriptive set theory, which is a branch of mathematical logic, and topology. He proposes to continue his work on applications of definable equivalence relations to indecomposable continua and to continuous actions of Polish groups. Specifically, he will investigate the structure of the equivalence relation induced on an indecomposable continuum by its partition into composants. His earlier work leads him to believe that a complete classification up to Borel isomorphism of these equivalence relations is within reach. If it can be accomplished, it should shed light on an old problem of Kuratowski on determining the size of Borel sets that are unions of families of composants. The second area of research proposed in the project is the study of the relation between topologies on groups and complexity of equivalence relations induced by their actions. The proposer will attempt to establish a characterization of subgroups of Polish groups which themselves carry Polish group topologies stronger than the subgroup topology. This characterization should be in terms of the complexity of the equivalence relation induced by the left translation action. Also he proposes to find a characterization of local compactness of Polish groups in terms of the equivalence relations induced by their continuous actions. This project has two parts. First, ``indecomposable continua'' will be studied. Even though indecomposable continua were initially discovered as paradoxical, exceptional examples of curves, their importance for today's research comes from the fact that they occur naturally and commonly in certain mathematical models. For instance, when studying the evolution of a physical or a biological system, one is particularly interested in describing families of states of such a system which have some sort of stability and to which other states evolve. Surprisingly, even for simple, natural systems, such ``attractors'' can have a very intricate geometric structure, in particular, they can be indecomposable continua. In this work, a mathematical discipline called descriptive set theory, which has no obvious connections with indecomposable continua, is being used to uncover certain deeper aspects of their structure. These new methods have already helped to solve some old problems, and it is expected that they will yield still new and exciting results and applications. The motivation for the second part of the project comes from the following considerations. An important part of a mathematician's or a physicist's work is classifying objects he/she is interested in so that objects that differ in an inessential way are not distinguished by the classification. In most situations, two object differ ``in an inessential way'' if one can be transformed into the other by a transformation taken from a suitable family of transformations called a group acting on the family of objects. There is a well-developed theory of classifying objects up to actions of groups that behave as if they were locally finite, the so-called locally compact groups. In many important instances, though, this theory is insufficient. This work will contribute to a larger, rapidly developing field, which investigates actions of non-locally compact groups.

索莱茨基提出了几个项目的边界上的描述集理论，这是一个分支的数理逻辑，拓扑结构。他建议继续他的工作应用可定义的等价关系，以不可分解的连续和连续行动的波兰团体。具体来说，他将调查结构的等价关系诱导的不可分解的连续其分区成composants。他早期的工作使他相信，一个完整的分类波莱尔同构的这些等价关系是触手可及的。如果它可以完成，它应该揭示了一个老问题的库拉托夫斯基对确定的大小博雷尔集是工会家庭的composants。该项目提出的第二个研究领域是研究群体拓扑结构之间的关系以及由其行为引起的等价关系的复杂性。提议者将试图建立一个波兰群的子群的特征，这些子群本身携带比子群拓扑更强的波兰群拓扑。这一特征应根据左平移动作所导致的等价关系的复杂性来描述。此外，他建议找到一个表征局部紧性的波兰群体的等价关系引起的连续行动。这个项目有两个部分。首先，我们将研究“不可分解连续统”。 尽管不可分解连续统最初被发现是自相矛盾的，特殊的曲线例子，但它们对今天的研究的重要性来自于它们在某些数学模型中自然和普遍存在的事实。例如，当研究物理或生物系统的演化时，人们特别感兴趣的是描述这样一个系统的状态族，这些状态族具有某种稳定性，并且其他状态演化到这些状态族。令人惊讶的是，即使对于简单的自然系统，这样的“吸引子”也可以有非常复杂的几何结构，特别是，它们可以是不可分解的连续统。在这项工作中，一个数学学科称为描述集理论，这与不可分解的连续统没有明显的联系，被用来揭示其结构的某些更深层次的方面。这些新方法已经帮助解决了一些老问题，预计它们将产生新的和令人兴奋的结果和应用。该项目第二部分的动机来自以下考虑。数学家或物理学家的工作的一个重要部分是对他/她感兴趣的对象进行分类，以便不通过分类来区分以非本质方式不同的对象。在大多数情况下，如果一个对象可以通过从一个适当的变换族（称为作用于对象族的群）中选取的变换变换而变换为另一个对象，则两个对象“以非本质的方式”不同。有一个发展完善的理论，将对象分类为行为好像它们是局部有限的群的作用，即所谓的局部紧群。然而，在许多重要的情况下，这一理论是不够的。这项工作将有助于一个更大的，迅速发展的领域，调查行动的非本地紧凑的群体。