Algebraicity, Transcendence, and Decidability in Arithmetic and Geometry through Model Theory

通过模型理论研究算术和几何中的代数性、超越性和可判定性

• 批准号: 2201045
• 负责人: Thomas Scanlon
• 金额: $ 48万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2201045&HistoricalAwards=false
• 关键词: Algebraicity; Transcendence; Decidability; Arithmetic; Geometry

The PI will study fundamental mathematical structures from the point of view of model theory. More specifically, the project breaks into three parts having to do with connections between differential and difference algebra and equations satisfied by special functions, a study of decidability for the theory of rational functions, and an explanation of certain transfer principles through mathematical logic. The project will provide research training opportunities for graduate students.More concretely, the PI will employ the model theory of differential and difference fields to analyze transcendence and algebraicity problems. Specifically, the model theory of differential fields will be used to elucidate transcendence and algebraicity for variations of Hodge structure. The model theory of difference fields will be used to analyze functional transcendence of Mahler functions and to classify invariant varieties for triangular dynamical systems. The tilt/untilt construction in the theory of perfectoids will be given a rigorous account in terms of bi-interpretation in the sense of continuous logic and further equivalences will follow. Definability within the field of rational functions over the complex numbers will be studied in depth. Scanlon will follow a strategy to establish the decidability of its existential theory using the geometry of rational curves on algebraic varieties.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

PI将从模型理论的角度研究基本的数学结构。更具体地说，该项目分为三个部分，分别涉及微分代数和差分代数与特殊函数满足的方程之间的联系，有理函数理论的可决性研究，以及通过数学逻辑解释某些传递原理。该项目将为研究生提供研究培训机会。更具体地说，PI将运用微分和差分领域的模型理论来分析超越和代数问题。具体而言，微分场的模型理论将用于阐明霍奇结构变化的超越性和代数性。差分场模型理论将用于分析马勒函数的泛函超越性，并对三角动力系统的不变变量进行分类。本文将从连续逻辑的双重解释的角度，对完美曲面理论中的倾斜/直到结构进行严格的解释，并进行进一步的等价。本文将深入研究复数上有理数函数域内的可定义性。斯坎伦将遵循一种策略，利用代数变量上有理曲线的几何来建立其存在论的可决性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。