Descriptive Set Theory

描述性集合论

• 批准号: 0455285
• 负责人: Alexander Kechris
• 金额: $ 33.65万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-05-01 至 2010-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0455285&HistoricalAwards=false
• 关键词: Descriptive; Set; Theory

The general aim of this project is 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. This work 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, recursion 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) problems arising in the theory of countable Borel equivalence relations, particularly concerning hyperfiniteness and treeability, including descriptive aspects of free actions of free groups; (ii) 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; (iii) 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; (iv) complexity of classification problems concerning the isometric or topological classification of various kinds of metric, topological or Banach spaces.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. This is achieved by associating with each collection of objects to be classified an appropriate concept of "magnitude" or "size", which in a precise sense measures the difficulty of its classification problem. This new theory of "magnitude" as well as problems in different directions that arise in the course of the development of this theory are investigated in this project.

该项目的总体目标是发展波兰群的可定义作用理论、其轨道空间的结构和分类，以及可定义等价关系的密切相关研究。 这项工作的动机是基本的基础问题，例如理解数学对象的完整分类的本质，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此类分类问题的复杂性。该理论是在描述性集合论的背景下发展起来的，它提供了基本的概念和方法。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，例如模型论、递归理论、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。在这个总体计划中，建议研究：（i）可数 Borel 等价关系理论中出现的问题，特别是关于超有限性和可树性，包括自由群的自由行动的描述方面； (ii) 可数结构自同构群的拓扑动力学与有限拉姆齐理论之间新发展的联系，以及与无限拉姆齐理论联系的相关半群框架； (iii) 波兰群中的泛型和充足泛型的概念及其与群的其他结构特性的关系，例如小指数特性、不可数共尾性、伯格曼有限生成特性、对树的作用的不动点特性和自动连续性； (iv) 涉及各种度量、拓扑或巴纳赫空间的等距或拓扑分类的分类问题的复杂性。在许多数学领域中出现的一个基本问题是对所研究的给定对象集合进行分类。这相当于提供这些物体的“目录”或“列表”，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，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非常重要。这个困难的基础问题变得更加复杂，因为可接受的分类的构成很大程度上取决于所研究的特定数学领域，因此一个领域的“良好”分类标准可能不适用于另一个领域。 在基本层面上，该项目旨在发展一种通用的定量理论，该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，使人们可以在任何给定领域中决定是否对分类问题进行令人满意的分类。有问题的对象是可能的。这是通过将每个要分类的对象集合与适当的“大小”或“大小”概念相关联来实现的，这在精确意义上衡量了其分类问题的难度。本项目对这一新的“量”理论以及该理论发展过程中出现的不同方向的问题进行了研究。