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

该研究涉及发展和应用技术的数理逻辑，以广泛的数学问题。首先，描述集合论的经典主题，结合遍历理论和超刚性理论的技术，被发现在代数和其他地方的分类问题的分析中有广泛的应用。特别是主题的博雷尔等价关系将适用于问题的几何群论，围绕中心概念的准等距。 现在经典理论的博雷尔减少将锐化考虑连续减少，这对应于在许多情况下，以“自然”或“不变”的建设。其次，模型论的技术将被应用于解决组合学中的某些问题，特别是受一个禁止子图约束的泛图的存在性问题，以及度量齐性图的分类，该分类首先由卡梅隆于1998年提出，随后被认为与Kechris，Pestov和Todorcevic的工作中的拓扑动力学有关。在这两种情况下，新的同质结构的例子将被确定（在一个适当的改进类别的图），然后拓扑动力学的关系将涉及进一步研究的结构拉姆齐理论在相关类别的有限结构。 该提案包括一个基于世界杰出的数学逻辑学家之一Saharon Shelah及其各种合作者的长期访问的访问计划，他们将在描述集合论和组合学应用方面合作进行拟议研究的各个方面。研究的重点是跨学科的。分类的概念是数学研究的许多分支的基础，而数理逻辑的技术，与其他数学分支的更专业的工具相结合，一些是最近突破的结果，使得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。数理逻辑的技术也为组合学问题提供了非常广阔的视角，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，以及具有易处理结构的此类的单个无限极限的存在问题，它提供了一种区分“混沌”和“高度结构化”类别以及识别可以进行结构分析的新情况的方法。

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.