Model theory of fields and compact complex manifolds

场模型论和紧复流形

• 批准号: RGPIN-2015-04155
• 负责人: Moosa, Rahim
• 金额: $ 1.46万
• 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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=612804
• 关键词: Model; theory; fields; compact; complex

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.

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继承和无法克服的限制。但是，另一个鲜为人知的是20世纪的发现是，数学推理的研究可以在核心数学的特定领域（例如几何，代数和数字理论）产生具体的发展。我自己的专业知识和该提案在数学逻辑的应用中以后的积极趋势所启示，该逻辑的应用在一个称为模型理论的分支机构中找到了自己的家。粗略地说，数学的模型理论方法是事先修复适用于正在考虑的数学领域的特定数学结构，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，这些集合仅指该结构，这些结构是由一阶逻辑规则在语法上绑定的。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。 我建议要提出的模型理论在数学上的特殊应用与几何学有关。复杂的流形是形状和形式概念的数学抽象，它们的研究是主流当代数学的活跃领域。模型理论的参与是相对较新的，大约十五年了，并且一直是我工作的中心。模型理论不仅有助于复杂流形的几何形状，而且在某种反向过程中，这些贡献本身启发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。我提出的研究将进一步推进模型理论扮演识别，形式化和支持不同几何环境之间类比的特殊作用。