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

索莱茨基提出了几个关于描述集合论（数学逻辑的一个分支）和拓扑学边缘的项目。他打算继续研究可定义等价关系在不可分解连续体和波兰群的连续作用中的应用。具体地说，他将研究在不可分解连续体上通过分割成组成部分而产生的等价关系的结构。他早期的工作使他相信这些等价关系的完全分类达到Borel同构是可以实现的。如果它能被完成，它将会给Kuratowski关于确定Borel集合大小的老问题带来启示，Borel集合是组成族的并集。该项目提出的第二个研究领域是研究群上拓扑之间的关系以及它们的作用所引起的等价关系的复杂性。提议者将尝试建立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。这种表征应该从左翻译行为所引起的对等关系的复杂性出发。他还提出用波兰群的连续作用所引起的等价关系来描述波兰群的局部紧性。这个项目有两个部分。首先，“不可分解连续体”将被研究。尽管不可分解连续体最初是作为矛盾的、特殊的曲线例子被发现的，但它们对今天研究的重要性在于它们在某些数学模型中自然而普遍地出现。例如，当研究一个物理或生物系统的进化时，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，其他状态也会向它们进化。令人惊讶的是，即使对于简单的自然系统，这样的“吸引子”也可以具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体。在这项工作中，一门被称为描述性集合理论的数学学科，与不可分解连续体没有明显的联系，正在被用来揭示它们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、令人兴奋的结果和应用。项目第二部分的动机来自以下考虑。数学家或物理学家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那些在本质上不同的对象。在大多数情况下，如果一个对象可以通过从一个合适的转换族（称为作用于对象族的群）中选取的转换转化为另一个对象，则两个对象“以一种无关紧要的方式”不同。有一种很成熟的理论，将物体分类为表现为局部有限的群的行为，即所谓的局部紧群。然而，在许多重要的例子中，这个理论是不够的。这项工作将有助于一个更大的、迅速发展的领域，即调查非本地紧凑群体的行动。

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.