Model theory with applications to algebra, geometry and number theory

数理逻辑是研究数学家使用的那种推理的学科。20世纪的主要智力成就之一是认识到这种对元数学的追求对数学本身具有戏剧性的影响。这里我想到的是歌德尔的著名著作，它用数学逻辑揭示了数学固有的和不可逾越的局限性。但20世纪另一个不那么广为人知的发现是，数学推理的研究也可以对数学做出积极的贡献；它可以在核心数学的特定领域产生具体的发展，如几何、代数和数论。我自己的专业知识和这项建议，都是从数理逻辑应用中的后一种积极趋势中得到启发的，数理逻辑在一个被称为模型理论的分支中找到了自己的家。粗略地说，数学的模型理论方法是预先确定适合于所考虑的主题的特定的数学结构，然后研究那些可以使用仅涉及该结构的形式表达式来定义的集合，并且这些集合在句法上受一阶逻辑规则的约束。这些对具有逻辑特征的句法的自我约束，是模型理论有效性的源泉。我建议学习的模型理论在数学中的具体应用与几何和代数有关。代数微分方程式是数学中一门经典而又活跃的学科，源于对自然界的研究。通常，微分方程被认为是应用数学的一部分。但这门学科有一个美丽的纯数学方面，我的工作就在这里。特别是，我通过将解空间视为几何对象来处理此类方程。模型理论不仅对微分代数几何作出了贡献，而且在一种相反的过程中，这些贡献本身也激励了纯模型理论本身的新技术的发展。反过来，这些技术可以重新应用于其他几何环境。我提议的研究将进一步推动模型理论与几何和代数之间的这种动态双向互动。这项拟议研究的长期目标是通过研究几何和代数的应用，推动模型理论领域的发展。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)几何稳定性理论中的纯模型理论的发展(1)以及其他环境，以及最后3)作为一个特定的应用程序的迪克斯米尔-莫格林等价，以说明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.