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

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，涉及微分和差分代数以及特殊函数满足的方程之间的联系，有理函数理论的可判断性研究，以及通过数理逻辑解释某些转移原理。该项目将为研究生提供研究培训机会。更具体地说，PI将使用微分和差分场的模型理论来分析超越和代数性问题。具体地说，微分场模型理论将被用来阐明霍奇结构变化的超越性和代数性。利用差分场模型理论分析了马勒函数的函数超越性，并对三角动力系统的不变簇进行了分类。完成体理论中的倾斜/直到倾斜结构将在连续逻辑意义下的双向解释方面得到严格的解释，随后将进行进一步的等价。我们将深入研究复数上有理函数的可定义性。Scanlon将遵循一种策略，使用代数变量上的有理曲线几何来建立其存在主义理论的可判断性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.