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研究对象的给定集合进行分类的问题。这相当于提供这些天体的“目录”或“清单”，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，或多或少会出现一种“混乱”的行为。因此，了解在什么情况下分类是可能的是非常重要的。由于什么构成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，这一困难的基本问题变得更加复杂，因此一个领域的“好”分类的标准可能不适用于另一个领域。在基本层面上，这个项目的目的是发展一种通用的量化理论，在许多情况下，它可以准确地衡量分类问题的复杂性，从而提供客观的手段，使人们能够在任何给定的领域中决定是否可能对所涉对象进行令人满意的分类。这一计划的发展经常导致对各种数学结构的对称性及其动力学的研究，以及与离散数学、动力系统或概率学领域的联系，这是该计划的另一个重要方面。此外，该项目还为研究生和本科生提供研究培训机会。这个项目的总体目标是发展波兰群的可定义行动理论，发展其轨道空间的结构和分类，以及密切相关的可定义等价关系和图的研究。这项工作是由基本的基本问题推动的，比如理解数学对象完全分类的本质，直到某种等价的概念，通过不变量，以及创建一个衡量此类分类问题的复杂性的数学框架。这一理论是在描述集合论的背景下发展起来的，它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，如模型论、群论、拓扑动力学、遍历论、概率论和组合学有着天然的相互作用。在这个一般计划中，Kechris建议研究：(I)可数Borel等价关系理论；(Ii)保持群行动和等价关系的全球测度论的描述性方面；(Iii)描述图组合学，包括它与遍历理论、几何群理论和概率论的关系；(Iv)波兰非阿基米德群的分类问题；以及(V)波兰群的动力学。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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, dynamical systems or probability and this is another important aspect of this project. In addition the project also provides research training opportunities for graduate and undergraduate students. 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) the theory of countable Borel equivalence relations; (ii) descriptive aspects of the global theory of measure preserving group actions and equivalence relations; (iii) descriptive graph combinatorics, including its relation to ergodic theory, geometric group theory and probability theory; (iv) classification problems for Polish non-archimedean groups; and (v) dynamics of Polish groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.