Descriptive Set Theory and Its Applications

这个项目涉及波兰群体的可定义行动理论的发展，其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。它的动机是基本的基本问题，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，以及创建一个数学框架来衡量这种分类问题的复杂性。这个理论是在描述集合论的背景下发展起来的，它提供了基本的基本概念和方法。另一方面，鉴于其广泛的范围，它与许多其他数学领域有着天然的相互作用，如模型理论、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。在这一一般程序中，建议研究：(I)新发展的可数结构的自同构群的拓扑动力学与有限Ramsey理论之间的联系以及与无限Ramsey理论之间的联系的相关半群框架；(Ii)Polish群中的广泛性和充分广泛性的概念及其与群的其他结构性质的关系，如小指数性质、不可数余终结性质、Bergman有限生成性质、树上作用的不动点性质和自连续性；(Iii)遍历群作用的整体理论，包括遍历理论中分类问题的复杂性的研究；(Iv)可测组合的方方面面：在许多数学领域中出现的一个基本问题是对研究对象的给定集合进行分类的问题。这相当于提供这些天体的“目录”或“清单”，原则上与生物学中的物种编目或天文学中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，或多或少会出现一种“混乱”的行为。因此，了解在什么情况下分类是可能的是非常重要的。由于什么构成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，这一困难的基本问题变得更加复杂，因此一个领域的“好”分类的标准可能不适用于另一个领域。在基本层面上，这个项目的目的是发展一种通用的量化理论，在许多情况下，它可以准确地衡量分类问题的复杂性，从而提供客观的手段，使人们能够在任何给定的领域中决定是否可能对所涉对象进行令人满意的分类。在这个项目中出现的发展往往导致对各种数学结构的对称性及其动力学的研究，这是该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.