Model theory of fields and compact complex manifolds

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世纪另一个鲜为人知的发现是，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，数学的模型论方法是预先确定与所考虑的数学领域相适应的特定数学结构，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。这些自我强加于句法的限制是逻辑的特征，是模型理论有效性的来源。*我建议追求的模型理论在数学中的特殊应用与几何有关。复流形是形与形概念的数学抽象，其研究是当代主流数学的一个活跃领域。模型理论的参与是相对较新的，大约有15年的历史，一直是我工作的中心。模型理论不仅对复杂流形的几何学做出了贡献，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发挥模型理论在识别、形式化和促进不同几何背景之间的类比方面所发挥的特殊作用。

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 area of mathematics 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 pursue have to do with geometry. Complex manifolds are mathematical abstractions of the idea of shape and form, and their study is an active area of mainstream contemporary mathematics. The involvement of model theory is relatively new, approximately fifteen years old, and has been at the centre of my work. Not only does model theory contribute to the geometry of complex manifolds, but in a kind of reverse process, these contributions have themselves inspired developments in the techniques of pure model theory itself. In turn, these techniques can be re-applied to other geometric contexts. The research I am proposing will further this special role that model theory plays of recognising, formalising and facilitating analogies between different geometric contexts.**