Interactions of Logic with Group Theory and Combinatorics

谢林和托马斯将探索逻辑技术与代数和组合学中的问题之间的互动。Cherlin将使用大量以有限群论为模型的方法来处理有限Morley秩群，特别是针对与坏场的存在兼容的奇特征例的方法，以及图论中易受模型论分析影响的问题(泛图和wqo问题)。托马斯将致力于Borel等价关系，特别是与代数中自然分类问题相关的等价关系，这可能很好地提供解决目前完全通用的一些公开问题所需的例子，以及提供关于代数问题的相对难度(根据非常稳健的测量系统)的信息，一些非常经典和开放的问题，现在可以看到是基本原因。托马斯还将利用集合论技术继续他在自同构塔问题上的工作。数理逻辑提供了可应用于数学各个领域的极具普遍性的工具。在组合上下文中，模型理论观点提供了可用于非常统一地处理特定问题的方法，而不是在文献中有时遇到的个案基础上。描述性集合论提供了分析代数中已解决和未解决问题的相对难度的方法，特别是提供了具体信息，说明人们在分类问题中可以有效地寻求多详细的答案，从而有可能在先验的基础上区分死胡同和富有成效的问题。

Cherlin and Thomas will pursue interactions of the techniques of logic with problems in algebra and combinatorics. Cherlin will work with groups of finite Morley rank using methods modeled heavily on finite group theory, aiming particularly at an approach to the odd characteristic case compatible with the existence of bad fields, and on problems in graph theory susceptible to model theoretic analysis (universal graphs and problems of wqo). Thomas will work on Borel equivalence relations, particularly with those associated with natural classification problems in algebra, which may well provide the examples needed to settle some problems presently open in full generality, as well as providing information on the relative difficulty (according to a very robust system of measurement) of the algebraic problems, some very classical and open, for what can now seen to be essential reasons. Thomas will also pursue his work on the automorphism tower problem, using set theoretic techniques.Mathematical logic provides tools of great generality which can be applied to various areas of mathematics. In combinatorial contexts the model theoretic point of view provides methods that can be used to handle specific problems very uniformly, rather than on the case by case basis sometimes encountered in the literature. Descriptive set theory provides methods for analyzing the relative difficulty of both solved and unsolved problems in algebra, and in particular provides concrete information as to how detailed an answer one may usefully seek in a classification problem, making it possible to distinguish dead ends from fruitful lines of inquiry on an a priori basis.