Model theory with applications to algebra, geometry and number theory

数理逻辑是研究数学家使用的推理的一门学科。20世纪的一个主要智力成就是认识到这种元数学的追求对数学本身有着戏剧性的影响。我在这里想到的是著名的工作哥德尔使用数学逻辑暴露固有的和不可逾越的限制数学。但是，世纪的另一个不太为人所知的发现是，数学推理的研究也可以对数学做出积极的贡献;它可以在几何、代数和数论等核心数学的特定领域产生具体的发展。我自己的专业知识，以及这个建议，都是由数理逻辑应用中的后一种积极趋势所告知的，这种积极趋势在一个称为模型论的分支中找到了归宿。粗略地说，数学的模型论方法是预先确定适合于所考虑的主题的特定数学结构，然后研究那些可以使用形式表达式定义的集合，这些形式表达式只涉及这种结构，并且在语法上受到一阶逻辑规则的约束。这些逻辑所特有的句法约束是模型理论有效性的源泉。模型论在数学中的具体应用，我打算研究的是几何和代数。代数微分方程是一个经典的和目前活跃的数学学科产生的研究自然世界。常微分方程被认为是应用数学的一部分。但这个问题有一个美丽的纯数学方面，这就是我的工作所在。特别是，我接近这样的方程，考虑作为一个几何对象的解决方案空间。模型论不仅对微分代数几何有贡献，而且在一种相反的过程中，这些贡献本身激发了纯模型论本身新技术的发展。反过来，这些技术可以重新应用到其他几何上下文。我所提出的研究将进一步推动模型论与几何和代数的动态双向互动。拟议的研究的长期目标是推进模型理论领域，通过几何和代数的应用研究。这将通过追求以下互补的研究主题来实现：1）微分代数几何的具体背景，2）受（1）以及其他设置启发的几何稳定性理论中的纯模型理论发展，最后3）Dixmier-Moeglin等价性作为一个特定的应用程序，说明了模型理论方法的力量。

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.