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

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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.