Descriptive Set Theory and Its Applications

描述集合论及其应用

• 批准号: 1464475
• 负责人: Alexander Kechris
• 金额: $ 50.01万
• 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: 2015
• 资助国家: 美国
• 起止时间: 2015-05-15 至 2021-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1464475&HistoricalAwards=false
• 关键词: Descriptive; Set; Theory; Its; Applications

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 as well as connections to areas of discrete mathematics or probability and this is another important aspect of this project. 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 and graphs. 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, group theory, topological dynamics, ergodic theory, probability theory and combinatorics. Within this general program Kechris proposes to study: (i) newly developed connections between the topological dynamics and ergodic theory of automorphism groups of countable structures and finite Ramsey theory; (ii) descriptive aspects of the global theory of ergodic group actions and equivalence relations, including the study of complexity of classification problems arising in ergodic theory; (iii) descriptive graph combinatorics, including its relations to ergodic theory, geometric group theory and probability theory; (iv) structurability for countable Borel equivalence relations.

在许多数学领域中出现的一个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。该计划中产生的发展通常会导致研究各种数学结构的对称性及其动力学以及与离散数学或概率领域的联系，这是该项目的另一个重要方面。该项目的总体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和图形的紧密相关研究。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，群体理论，拓扑动态，千古理论，概率理论和组合学。在这个一般计划中，Kechris提出了研究：（i）可计数结构和有限的Ramsey理论的拓扑动力学与千古理论之间新开发的联系； （ii）全球群体行动和等价关系的全球理论的描述性方面，包括研究厄基德理论中出现的分类问题的复杂性； （iii）描述性图组合学，包括其与厄戈德理论的关系，几何群体理论和概率理论； （iv）可计数鲍尔等效关系的结构性。