Descriptive Set Theory and Its Applications

在数学的许多领域中出现的一个基本问题是对给定的研究对象集合进行分类。这相当于提供了这些物体的“目录”或“列表”，原则上与生物学中的物种或天文学中的恒星和星系的编目没有什么不同。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，或多或少的“混乱”行为是预期的。因此，了解在什么情况下可以进行分类非常重要。这个困难的基本问题进一步复杂化的事实是，什么构成一个可接受的分类是非常依赖于特定的数学领域的研究，所以标准的“好”的分类在一个领域可能不适合在另一个。 在其基本层面上，该项目旨在发展一种通用的数量理论，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，使人们可以在任何特定领域确定是否有可能对有关物体进行令人满意的分类。在这个程序中出现的发展往往导致各种数学结构的对称性及其动力学的研究，以及离散数学或概率领域的连接，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，以及可定义等价关系和图的密切相关研究。这项工作的动机是基本的基础问题，如理解数学对象的完整分类的性质，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，描述集合论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，如模型论，群论，拓扑动力学，遍历理论，概率论和组合学有着自然的相互作用。在这个一般程序凯克里斯建议研究：（一）新开发的连接拓扑动力学和遍历理论的自同构群的可数结构和有限拉姆齐理论;（二）描述方面的全球理论遍历群行动和等价关系，包括研究的复杂性分类问题所产生的遍历理论;（iii）描述图组合学，包括它与遍历理论、几何群论和概率论的关系;（iv）可数Borel等价关系的可结构性。

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.