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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=677220
• 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世纪最重要的智力成就之一是认识到这种对元数学的追求对数学本身有着戏剧性的影响。我在这里指的是哥德尔的著名著作，他用数学逻辑揭示了数学固有的和不可逾越的局限性。但20世纪另一个不太为人所知的发现是，对数学推理的研究也可以对数学做出积极贡献；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个建议，是由数理逻辑应用中的后一种积极趋势所启发的，它在一个叫做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所考虑的数学领域的特定数学结构，然后研究那些可以使用仅引用该结构的形式表达式定义的集合，并且这些集合在语法上受一阶逻辑规则的约束。这些对句法的自我约束是逻辑的特征，是模型理论有效性的来源。模型理论在数学中的特殊应用，我打算研究的与几何有关。复流形是形状和形式概念的数学抽象，对其的研究是当代主流数学的活跃领域。模型理论的介入是相对较新的，大约有15年的历史，并且一直是我工作的中心。模型理论不仅对复杂流形的几何有贡献，而且在一种相反的过程中，这些贡献本身也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到其他几何环境中。我所提出的研究将进一步推动模型理论在识别、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