Descriptive Set Theory and Its Applications

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。在这个项目中产生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，以及与离散数学或概率领域的联系，这是这个项目的另一个重要方面。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，以及与之密切相关的可定义等价关系和图的研究。这项工作的动机是基本的基础问题，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。这个理论是在描述集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，如模型论、群论、拓扑动力学、遍历论、概率论和组合学有自然的相互作用。在这个总体规划中，Kechris建议研究：(i)可数结构自同构群的拓扑动力学和遍历理论与有限Ramsey理论之间的新联系；（ii）遍历群作用和等价关系的全局理论的描述方面，包括遍历理论中出现的分类问题的复杂性的研究；（iii）描述图组合学，包括其与遍历论、几何群论和概率论的关系；（iv）可数Borel等价关系的结构性。

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.