Descriptive Set Theory and Its Applications

这个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，以及与之密切相关的可定义等价关系的研究。它是由基本的基础问题驱动的，比如理解数学对象的完全分类的本质，直到一些等价的概念，通过不变量，并创建一个数学框架来测量这些分类问题的复杂性。这个理论是在描述性集合理论的背景下发展起来的，它提供了基本的潜在概念和方法。另一方面，鉴于其广泛的范围，它与许多其他数学领域有自然的相互作用，如模型论、可计算论、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。在这一总体规划中，提出研究：(1)可数结构自同构群的拓扑动力学与有限Ramsey理论之间的新联系，以及这种联系与无限Ramsey理论的相关半群框架；（ii）波兰群的泛型和充足泛型的概念及其与群的其他结构性质的关系，如小指数性质、不可数共性、Bergman有限生成性质、树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，包括遍历理论中产生的分类问题的复杂性的研究；（iv）可测组合学的各个方面。在许多数学领域中出现的一个基本问题是对所研究对象的给定集合进行分类。这相当于为这些天体提供一个“目录”或“清单”，原则上与生物学上的物种编目或天文学上的恒星和星系编目没有什么不同。如果这样的分类是可能的，那么一个人就对所涉及的数学结构有了“完整”的理解。否则，预计会出现或多或少的“混乱”行为。因此，了解在什么情况下可以进行分类是非常重要的。这个困难的基础问题因以下事实而变得更加复杂：什么是可接受的分类在很大程度上取决于所研究的特定数学领域，因此在一个领域“好”分类的标准可能不适用于另一个领域。在其基本层面上，该项目旨在发展一种一般的定量理论，在许多情况下可以精确地测量分类问题的复杂性，从而提供客观的手段，通过这种手段，人们可以决定，在任何给定的领域，是否有可能对所讨论的对象进行令人满意的分类。这个项目的发展经常导致对各种数学结构及其动力学的对称性的研究，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.