Model theory with applications to algebra, geometry and number theory

模型理论及其在代数、几何和数论中的应用

• 批准号: RGPIN-2021-02474
• 负责人: Moosa, Rahim
• 金额: $ 1.97万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759273
• 关键词: Model; theory; applications; algebra; geometry

Mathematical logic is the study of the kind of reasoning that mathematicians use. One of the major intellectual achievements of the 20th century was the understanding that this metamathematical pursuit has dramatic consequences for mathematics itself. I have in mind here the celebrated work of Goedel which used mathematical logic to expose the inherent and insurmountable limits of mathematics. But another less widely known discovery of the 20th century is that the study of mathematical reasoning can also contribute positively to mathematics; it can produce concrete developments in particular areas of core mathematics such as geometry, algebra, and number theory. My own expertise, and this proposal, are informed by this latter positive tendency in the application of mathematical logic, which finds its home in a branch called model theory. Roughly speaking, the model theoretic approach to mathematics is to fix beforehand the particular mathematical structure suited to the subject under consideration, and then to study those sets that can be defined using formal expressions that refer only to this structure and that are bound syntactically by the rules of first-order logic. These self-imposed restraints on the syntax, which is characteristic of logic, is the source of model theory's effectiveness. The particular applications of model theory to mathematics that I propose to study have to do with geometry and algebra. Algebraic differential equations are at once a classical and currently active subject in mathematics arising from the study of the natural world. Often differential equations are considered part of applied mathematics. But there is a beautiful pure-mathematical aspect to the subject, and it is here that my work lies. In particular, I approach such equations by considering the solutions space as a geometric object. Not only does model theory contribute to differential-algebraic geometry, but in a kind of reverse process, these contributions themselves inspire the development of new techniques in pure model theory itself. In turn, these techniques can be re-applied to other geometric contexts. The research I am proposing will further this dynamical two-way interaction of model theory with geometry and algebra. The long term goal of the proposed research is to advance the field of model theory, through the study of applications to geometry and algebra. This will be achieved through the pursuit of the following complementary research themes: 1) the specific context of Differential-algebraic Geometry, 2) the pure model theory developments in Geometric Stability Theory inspired by (1) as well as other settings, and finally 3) the Dixmier-Moeglin Equivalence as a particular programme of applications illustrating the power of the model-theoretic approach.

数学逻辑是对数学家使用的推理的研究。 20世纪的主要知识成就之一是理解，这种变质的追求对数学本身产生了巨大的后果。我在这里记住了Goedel的著名作品，该作品使用数学逻辑来揭示数学的固有和无法克服的限制。但是，20世纪的另一个鲜为人知的发现是，对数学推理的研究也可以对数学产生积极贡献。它可以在核心数学的特定领域产生具体的发展，例如几何，代数和数理论。我自己的专业知识和该提案是通过在数学逻辑的应用中的积极倾向来告知的，该逻辑的应用趋势是在称为模型理论的分支机构中找到其房屋的。粗略地说，数学的模型理论方法是事先修复适用于所考虑的主题的特定数学结构，然后研究可以使用仅指定该结构的正式表达式来定义的集合，这些结构仅指该结构，并由一阶逻辑规则在语法上绑定。这些对语法的自我限制是逻辑的特征，是模型理论有效性的来源。我建议研究的模型理论在数学上的特殊应用与几何和代数有关。代数微分方程立即是由自然界研究引起的数学上的古典且目前活跃的主题。通常，微分方程被认为是应用数学的一部分。但是这个主题有一个美丽的纯粹数学方面，而我的作品在这里。特别是，我通过将解决方案空间视为几何对象来处理此类方程。模型理论不仅有助于不同的代数几何形状，而且在一种反向过程中，这些贡献本身会激发纯模型理论本身中新技术的发展。反过来，这些技术可以重新应用于其他几何环境。我提出的研究将进一步发展模型理论与几何和代数的动态双向相互作用。拟议研究的长期目标是通过研究对几何和代数的应用来推进模型理论领域。这将通过追求以下互补研究主题来实现：1）差异 - 代码几何的几何形状的特定背景，2）几何稳定理论中的纯模型理论发展，受（1）以及其他环境启发，最后3）dixmier-moeglin等价性作为一种特定的应用程序，以示例模型方法的应用程序。