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数学结构。 更具体地说，该项目分为三个部分，这与差分和差异代数之间的连接以及特殊功能满足的方程，对理性函数理论的可定义性的研究以及通过数学逻辑对某些传输原理的解释。该项目将为研究生提供研究培训机会。更具体地说，PI将采用差异和差异领域的模型理论来分析超越和代数问题。具体而言，差异场的模型理论将用于阐明霍奇结构变化的超越和代数。差异字段的模型理论将用于分析Mahler函数的功能性超越性，并对三角形动力学系统的不变品种进行分类。完美素理论中的倾斜/直到构造将在持续逻辑的意义上为双解剖学而进行严格的描述，并随后进行进一步的等价。将深入研究合理函数领域内的确定性。 Scanlon将遵循一项策略，以使用代数品种的理性曲线几何形状来确定其存在理论的确定性。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛影响的评估标准来通过评估来支持的。