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.

该项目的一般目的是发展波兰群体的可定义行动理论，其轨道空间的结构和分类以及对可定性等效关系的紧密相关研究。 这项工作是由基本的基础问题激励的，例如了解数学对象的完整分类的性质，直至对等效性，不变性的某种概念，并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理论的背景下发展的，该理论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具有自然相互作用，例如模型理论，递归理论，拓扑群体及其表示理论，拓扑动力学，阵程动力学，经营者代数和组合术。在此一般计划中，建议研究：（i）在可数的鲍尔等效关系理论中引起的问题，尤其是关于超强性和差异性的问题，包括自由群体自由行动的描述性方面； （ii）自动形态群体的拓扑动态与有限的拉姆西理论的拓扑动态以及与无限的Ramsey理论联系的相关半群框架之间的联系； （iii）波兰人群体中通用和充分通用性的概念及其与小组的其他结构特性的关系，例如小索引特性，无数的辅助性，伯格曼有限的生成特性，固定点在树上的作用和自动连续性； （iv）有关各种度量，拓扑或Banach空间的等距或拓扑分类的分类问题的复杂性问题。在许多数学领域中引起的基本问题是对所研究的对象集合进行分类。这相当于提供这些物体的“目录”或“清单”，从原则上讲，与生物学或天文学中的恒星和星系中的分类物种不同。如果可以进行这样的分类，则对所涉及的数学结构有一个“完全”的理解。 否则，预计或多或少会产生“混乱”行为。因此，了解在什么情况下可能会出现分类非常重要。这个困难的基础问题进一步复杂化，即构成可接受的分类的事实很大程度上取决于所研究的特定数学领域，因此在一个领域中“良好”分类的标准可能在另一个领域不合适。 从基本层面上讲，该项目旨在开发一种一般的定量理论，在许多情况下，该理论可以精确地衡量分类问题的复杂性，从而提供客观的手段，通过这些方式可以在任何给定的领域中决定对对象的令人满意的分类。这是通过与每个对象集合相关联来实现的，该集合被分类为“大小”或“大小”的适当概念，从精确的意义上讲，这可以衡量其分类问题的难度。在该项目中研究了这种“大小”的新理论以及该理论发展过程中出现的不同方向的问题。