Model-Theoretic Classification, Graph Combinatorics, and Topological Dynamics

模型理论研究数学对象的类别可以用某种有限的形式语言定义的方式，以及这些可定义性假设所隐含的结构性质。这种研究方法起源于围绕数学哲学和基础的问题，但近年来，它在经典数学和计算机科学的一些中心对象的研究中发现了惊人的应用。这个项目进一步研究了这些联系，特别是在图（描述网络和相关系统的数学方法）和群体行为（描述空间对称性集合的数学方法）的背景下。这项研究有可能为应用强大的无限模型论机制来解决有限图组合学中的问题开辟一条道路，反过来，也为应用组合学中的深度结果来解决模型论中的问题开辟一条道路。受Morley关于一阶理论的不可数模型可能数量的猜想的启发，Shelah分离了几类重要的“驯服”理论，并开发了一套丰富的机制来分析其中一些理论的模型和可定义集，特别是稳定理论。许多研究人员后来的工作证明，广义稳定性的概念和方法反映了其他数学领域的重要现象。本项目将研究两个这样的联系。1) Fox等人得到了半代数图的改进ramsey型界和强正则引理，Tao等人得到了大有限域代数图的改进ramsey型界和强正则引理，在相应领域有大量应用。这些结果可以很自然地被看作是关于在适合分类图的某些结构中可定义的图的结果。2)可定义群作用的研究与拓扑动力学中的某些问题密切相关，特别是围绕着Glasner等人研究的弱概周期动力系统和驯服系统。研究者将致力于进一步发展广义稳定性的方法，并将其应用于各种“驯服”结构（稳定，o-极小，远端，依赖）中可定义图的组合问题，并研究这些结构中可定义群作用的动力学性质。

Model theory studies the ways in which classes of mathematical objects can be defined in some restricted formal language, and what structural properties are implied by these definability assumptions. This method of study originated in questions around the philosophy and foundations of mathematics, but in recent years it has found striking applications in the study of some central objects of classical mathematics and computer science. This project investigates further these connections, in particular in the context of graphs (mathematical ways of describing networks and related systems) and group actions (mathematical ways of describing collections of symmetries of a space). This study has the potential to open up a route for applications of the powerful infinitary model-theoretic machinery to open questions in finite graph combinatorics, and conversely for applications of deep results in combinatorics to open questions in model theory.Motivated by Morley's conjecture on the possible number of uncountable models of first-order theories, Shelah isolated several important classes of "tame" theories and developed a rich machinery for analyzing models and definable sets for some of those classes, particularly for stable theories. Later work by many researchers demonstrated that notions and methods of generalized stability reflect important phenomena in other areas of mathematics. This project will investigate two such connections. 1) Improved Ramsey-type bounds and strong regularity lemmas were obtained for semi-algebraic graphs by Fox et al. and for algebraic graphs in large finite fields by Tao, with numerous applications in the corresponding areas. These results can naturally be viewed as results about graphs definable in certain structures fitting into the classification picture. 2) Study of definable group actions turns out to be closely related to certain questions in topological dynamics, especially around weakly almost periodic dynamical systems and tame systems studied by Glasner and others. The investigator will work on developing further methods of generalized stability and applying them to questions in combinatorics of graphs definable in various "tame" structures (stable, o-minimal, distal, dependent) and to study dynamical properties of definable group actions in those structures.