Descriptive Set Theory and Its Applications

该项目涉及波兰群的可定义作用理论的发展，它们的轨道空间的结构和分类，以及与可定义等价关系密切相关的研究。 它的动机是基本的基础问题，如理解数学对象的完整分类的性质，直到一些等价的概念，通过不变量，并创建一个数学框架来衡量这种分类问题的复杂性。另一方面，鉴于其广泛的范围，它与许多其他数学领域，如模型论，可计算性理论，拓扑群及其表示理论，拓扑动力学，遍历理论，算子代数和组合学有着自然的相互作用。在这个一般性的计划中，我们建议研究：（i）可数结构的自同构群的拓扑动力学与有限Ramsey理论之间的新发展的联系，以及这种联系与无限Ramsey理论的相关半群框架;（ii）波兰群中的genericity和enough genericity的概念，以及它们与群的其他结构性质如小指数性质的关系，不可数共尾性、Bergman有限生成性质、树上作用的不动点性质和自动连续性;（iii）遍历群作用的全局理论，包括遍历理论中分类问题的复杂性研究;（四）在许多数学领域中出现的一个基本问题是对给定的组合进行分类，研究对象。这相当于提供了这些物体的“目录”或“列表”，原则上与生物学中的物种或天文学中的恒星和星系的编目没有什么不同。如果这样的分类是可能的，那么人们就对所涉及的数学结构有了“完整”的理解。 否则，或多或少的“混乱”行为是预期的。因此，了解在什么情况下可以进行分类非常重要。这个困难的基本问题进一步复杂化的事实是，什么构成一个可接受的分类是非常依赖于特定的数学领域的研究，所以标准的“好”的分类在一个领域可能不适合在另一个。 在其基本层面上，该项目旨在发展一种通用的数量理论，在许多情况下可以精确地衡量分类问题的复杂性，从而提供客观的手段，使人们可以在任何特定领域确定是否有可能对所涉物体进行令人满意的分类。在这个计划中出现的发展往往导致各种数学结构的对称性及其动力学的研究，这是这个项目的另一个重要方面。

This project deals with the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related study of definable equivalence relations. It is motivated by basic foundational questions, like understanding the nature of complete classification of mathematical objects, up to some notion of equivalence, by invariants, and creating a mathematical framework for measuring the complexity of such classification problems This theory is developed within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of its broad scope, it has natural interactions with many other areas of mathematics, such as model theory, computability theory, the theory of topological groups and their representations, topological dynamics, ergodic theory, operator algebras, and combinatorics. Within this general program it is proposed to study: (i) newly developed connections between the topological dynamics of automorphism groups of countable structures and finite Ramsey theory as well as a related semigroup framework for such connections with infinite Ramsey theory; (ii) the concepts of genericity and ample genericity in Polish groups and their relation to other structural properties of groups such as the small index property, uncountable cofinality, the Bergman finite generation property, fixed point properties for actions on trees and automatic continuity; (iii) the global theory of ergodic group actions, including the study of complexity of classification problems arising in ergodic theory; (iv) aspects of measurable combinatorics.A fundamental question that arises in many fields of mathematics is that of classifying a given collection of objects under study. This amounts to providing a "catalog" or "listing" of these objects, in principle not unlike that of cataloging species in biology or stars and galaxies in astronomy. If such a classification is possible, one has a "complete" understanding of the mathematical structures involved. Otherwise a more or less "chaotic" behavior is expected. It is thus very important to understand under what circumstances a classification is possible. This difficult foundational question is further complicated by the fact that what constitutes an acceptable classification is very much dependent on the particular field of mathematics studied, so the criteria for a "good" classification in one area might not be appropriate in another. At its basic level, this project aims to develop a general quantitative theory, which in many situations can precisely measure the complexity of a classification problem and thus provide objective means by which one can decide, in any given field, whether a satisfactory classification of the objects in question is possible. Developments arising in this program often lead to the study of the symmetries of various mathematical structures and their dynamics and this is another important aspect of this project.