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.

Solecki提出了关于描述性集理论界限的几个项目，这是数学逻辑和拓扑的一个分支。他建议继续在与不可分解的连续图和波兰群体的持续行动的可确定等效关系的应用上进行工作。具体而言，他将研究通过将其分配到成文厂对不可分解的连续体引起的等效关系的结构。他的较早作品使他相信，这些对等关系的完整分类与Borel同构相关。如果可以完成，它应该阐明库拉托夫斯基的旧问题，以确定鲍尔尔（Borel）的大小，即鲍尔（Borel）的大小，这些鲍尔（Borel）集合是合成商系列的工会。该项目提出的研究领域的第二个领域是研究拓扑对群体的关系以及其行为引起的等价关系的复杂性。提议者将尝试建立波兰群体的亚组的特征，这些亚组本身具有比亚组拓扑更强的波兰群体拓扑。该表征应根据左翻译作用引起的等价关系的复杂性。他还建议根据持续行动引起的等价关系来找到波兰群体局部紧凑性的特征。这个项目有两个部分。首先，将研究``不可塑性的连续图''。 即使最初发现不可分解的连续图是曲线的矛盾，异常的例子，但它们对当今研究的重要性源于它们在某些数学模型中自然而通常发生的事实。例如，在研究物理或生物系统的演变时，人们特别有兴趣描述具有某种稳定性以及其他州进化的系统状态的家庭。令人惊讶的是，即使对于简单的天然系统，``吸引者''也可以具有非常复杂的几何结构，尤其是它们也可以是不可塑性的连续性。在这项工作中，一种名为“描述性集”理论的数学学科，该理论与不可塑性的连续图没有明显的联系，用于揭示其结构的某些更深层次的方面。这些新方法已经有助于解决一些旧问题，预计它们将产生仍然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重要组成部分是对他/她感兴趣的对象进行分类，以便以不必要的方式不同的对象不会通过分类来区分。在大多数情况下，如果一个对象可以通过从适当的转换家族进行转换，称为对象家族的群体，则有两个对象``以一种不必要的方式''不同。有一个完善的理论，将对象分类为行为的行为，这些行为表现得好像是本地有限的，即所谓的当地紧凑型组。但是，在许多重要情况下，该理论不足。这项工作将有助于更大，快速发展的领域，该领域调查了非局部紧凑型组的行动。