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CAREER: The Geography of Tame Ordered Structures

职业：驯服有序结构的地理

基本信息

批准号：
1654725
负责人：
Timur Oikhberg
金额：
$ 40万
依托单位：
University of Illinois at Urbana-Champaign
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1654725&HistoricalAwards=false
关键词：
CAREER Geography Tame Ordered Structures

项目摘要

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.

逻辑和模型理论的核心在于这样的观察：在数学中，某些对象必须被认为是驯服的，而另一些对象必须被认为是狂野的。 20 世纪上半叶许多著名的逻辑成果都与野生物体的存在有关。哥德尔对算术不可判定性的证明表明，从逻辑的角度来看，这种结构是复杂的（或疯狂的）。这些结果在精神上是消极的，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。模型理论学家发现大量温和的数学结构没有表现出这种野性，并且通常出于非常不同的原因都适用于模型理论方法。这种识别和分析模型理论可以被理解的简单结构类别的程序被称为“简单数学地理学”，并以各种形式在过去三十年中主导了模型理论。尽管它是作为一个具有基础重要性的程序而出现的，但它导致了模型论在数学其他领域的惊人应用，最近是数论中的安德烈-奥尔特猜想。由于这个项目是探索温和的数学领域，这些联系根本不是巧合。这种相互作用的发展已经一次又一次地证明其在逻辑之外具有深远的应用，这是事先无法想象的。这个项目在实线扩展的背景下继续了这一研究方向。它旨在解决模型理论中重要的开放性问题，但也自然地在模型理论与逻辑中的其他领域（例如新稳定性和描述性集合论）以及逻辑之外的其他领域（例如几何测度论和几何群论）之间建立新的联系。研究者将领导对有序结构中可定义集合的几何形状研究中出现的驯服行为和野生行为之间的分界线进行大规模调查。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，该项目还面临着对被认为驯服的结构类别进行分类的新挑战。这项职业补助金的教育部分将研究者的研究与其教学和推广工作联系在一起。该项目让本科生、研究生和年轻研究人员参与研究者的研究计划，加强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。