Descriptive Set Theory, Geometric Group Theory, and Combinatorial Model Theory

提出的研究涉及数学逻辑技术的发展和应用，以解决广泛的数学问题。首先，描述集合论的经典主题已经被发现，与遍历理论和超刚性理论的技术相结合，在代数和其他地方的分类问题的分析中有广泛的应用。特别是Borel等价关系的主题将被应用于几何群论中的问题，围绕着准等距的中心概念。现在经典的Borel约简理论将通过考虑连续约简而得到强化，连续约简在许多情况下对应于“自然”或“不变”结构。其次，模型论的技术将被应用于解决组合学中的某些问题，特别是存在一个禁止子图的全称图的问题，以及度量齐次图的分类，这一问题最初由Cameron在1998年提出，随后在Kechris、Pestov和Todorcevic的工作中被认为与拓扑动力学有联系。在这两种情况下，同质结构的新例子将被识别（在图的适当细化类别中），然后与拓扑动力学的关系将涉及在有限结构的相关类中进一步研究结构拉姆齐理论。该提案包括一个访客计划，该计划基于世界上最杰出的数理逻辑学家之一萨哈伦·希拉和他的许多合作者的长期访问，他们将在拟议研究的各个方面进行合作，包括描述性集合论和组合学的应用。这项研究的重点是跨学科的。分类的概念是许多数学研究分支的基础，数学逻辑的技术，结合来自其他数学分支的更专业的工具，一些最近的突破的结果，使分析这种方法的范围和局限性成为可能，揭示了丰富的结构。数学逻辑技术也为组合问题提供了一个非常广阔的视角，提出了可计算性问题，并为解决有关有限结构类的复杂性的具体问题提供了新的和系统的技术，以及这种具有可处理结构的类的单一无限极限的存在问题，这为区分“混沌”和“高度结构化”类提供了一种方法。并对新病例进行识别，进行结构分析。

The research proposed involves the development and application of techniques of mathematical logic to a broad range of mathematical problems. First, the classical subject of descriptive set theory has been found, in combination with techniques of ergodic theory and superrigidity theory, to have extensive applications to the analysis of problems of classification in algebra and elsewhere. In particular the subject of Borel equivalence relations will be applied to problems in geometric group theory, revolving around the central notion of quasi-isometry. The now classical theory of Borel reductions will be sharpened by considering continuous reductions, which correspond in a number of cases to "natural" or "invariant" constructions. Second, the techniques of model theory will be applied to settle certain problems in combinatorics, notably the problem of the existence of universal graphs subject to one forbidden subgraph, and the classification of the metrically homogeneous graphs, first proposed in 1998 by Cameron and subsequently seen to have connections with topological dyamics in work of Kechris, Pestov, and Todorcevic. In both cases, new examples of homogeneous structures will be identified (in an appropriate refinement of the category of graphs), and then the relation to topological dynamics will involve the further study of structural Ramsey theory in the associated classes of finite structures. The proposal includes a visitors' program based on extended visits by Saharon Shelah, one of the world's preeminent mathematical logicians, and a variety of his collaborators, who will collaborate on aspects of the proposed research, both on the side of descriptive set theory and in the applications to combinatorics.The thrust of the research is interdisciplinary. The notion of classification is fundamental to many branches of mathematical research, and techniques of mathematical logic, in combination with more specialized tools from other branches of mathematics, some the result of very recent breakthroughs, make it possible to analyze the scope and limitations of this method, revealing a wealth of structure. The techniques of mathematical logic also provide a very broad perspective on problems of combinatorics, raising issues of computability and providing new and systematic techniques for the resolution of concrete problems regarding the complexity of classes of finite structures, and the problem of the existence of a single infinite limit of such a class with a tractable structure, which provides a way for distinguishing "chaotic" from "highly structured" classes, and the identification of new cases which can be analyzed structurally.