CAREER: The Geography of Tame Ordered Structures

逻辑和模型理论的核心在于观察到，在数学中，有一些对象必须被认为是驯服的，而另一些则必须被认为是狂野的。从20世纪上半叶开始，许多著名的逻辑结果都与被认为是野生的物体的存在有关。Gödel对算术的不可判定性的证明，从逻辑的观点来看，这种结构是复杂的（或狂野的）。这样的结果在精神上是否定的，因为它们指出了数学推理的局限性。然而，在上世纪下半叶，焦点发生了变化。模型理论家发现了大量驯服的数学结构，它们没有表现出这种野性，并且由于非常不同的原因，它们通常适用于模型理论方法。这种识别和分析模型理论可以理解的结构的温和类别的程序被称为温和数学的地理学，并以其各种形式在过去三十年中主导了模型理论。虽然它是作为一个具有基础重要性的程序出现的，但它已经导致了模型理论在其他数学领域的惊人应用，最近的应用是数论中的andr<s:1> - oort猜想。由于这个项目是为了探索数学领域，这些联系根本不是巧合。这种交互的发展已经一次又一次地证明，在逻辑之外有广泛的应用，这是以前无法想象的。本项目在真实线扩展的背景下继续这条研究线。它旨在解决模型论中重要的开放性问题，但也自然地发展了模型论与逻辑中的其他领域之间的新联系，如新稳定性和描述性集合论，以及逻辑之外的几何测度论和几何群论。研究者将在有序结构中可定义集合的几何研究中，对驯服和野性行为之间的分界线进行大规模的研究。在早期进展的基础上，研究者将确定各种逻辑驯服条件对可定义集的拓扑和度量驯服的深远影响。此外，该项目在被认为是驯服的结构分类方面提出了新的挑战。这项职业补助金的教育部分将研究者的研究与他的教学和推广工作联系在一起。该项目涉及研究者的研究项目中的本科生、研究生和年轻研究人员，加强了模型理论与其他数学分支之间的联系，并继续研究者在厄巴纳-香槟地区的推广工作。

At the heart of logic and model theory lies the observation that within mathematics there are certain objects that have to be considered tame, and others that have to be considered wild. Many celebrated results in logic from the first half of the 20th century concerned the existence of objects considered wild. Gödel's proof of the undecidability of arithmetic established that this structure is complicated (or wild) from a logical viewpoint. Such results are negative in spirit as they point to the limitations of mathematical reasoning. However, in the second half of the last century the focus changed. Model theorists found a vast number of tame mathematical structures that exhibit no such wildness, and for often very different reasons are amenable to model-theoretic methods. This program of identifying and analyzing tame classes of structures whose model theory can be understood came to be known as the geography of tame mathematics, and in its various forms has dominated model theory throughout the last thirty years. Although arising as a program of foundational importance, it has led to striking applications of model theory to other areas of mathematics, most recently to the André-Oort conjecture in number theory. Since this program is to explore areas of tame mathematics, these connections are not coincidental at all. The development of such interactions has proven again and again to have far reaching applications outside of logic that could not have been envisioned beforehand.This project continues this line of research in the setting of expansions of the real line. It aims to settle important open questions within model theory, but also naturally develops new connections between model theory and other areas in logic such as neostability and descriptive set theory, and outside of logic such as geometric measure theory and geometric group theory. The investigator will lead a large scale investigation of dividing lines between tame and wild behavior arising in the study of the geometry of definable sets in ordered structures. Building on early advances, the investigator will determine far reaching consequences of various logical tameness conditions on the topological and metric tameness of definable sets. Furthermore, the project comprises new challenges in the classification of classes of structures considered tame. The educational component of this CAREER grant ties together the investigator's research with his teaching and outreach efforts. This project involves undergraduate and graduate students and young researchers in the investigator's research program, strengthens the ties between model theory and other branches of mathematics, and continues the investigator's outreach efforts in the Urbana-Champaign area.