Model Theory: Connecting Algebraic, Analytic, and Diophantine Geometry Through Definability

重要的数学结构通常可以从非常不同的角度来理解，要么通过使用代数或解析公式进行分析，要么通过从几何角度来考虑它们。这种双重的代数/几何观点被应用于数学的许多分支，特别是研究涉及数字的代数方程(以丢番图几何的名义)，研究多项式方程的解(以代数几何的名义)，或研究微分方程组或差分方程组的解之间可能的代数关系(分别以微分代数几何或差分代数几何的名义)，等等。在实践中，这些领域中考虑的一些问题由于与数论或甚至更复杂的领域的联系而受到极端复杂性的影响。数理逻辑的工作阐明了那些复杂的理论和那些允许温和的几何理论之间的界限。这项研究项目旨在扩展可以建立驯服几何的理论类别，并使用这些来自数理逻辑的结果来回答来自目标领域的问题。本研究项目通过适当理论中可定义性的模型理论镜头，研究各种几何之间的联系，包括代数几何、微分几何、丢番图几何和解析几何。本课程将学习各种数学理论，如偏微分方程式、差分方程式、完美拟空间、形式几何学、代数动力学和齐次动力学。在技术上，包括应用于微分闭域的几何稳定性理论、o-极小性以及取值微分场和解析差环的量词消去，将被用于回答模型理论内部的问题(例如证明或反驳具有几个交换导数的微分闭域中的正则类型的三分性原理)，以及应用于函数超越、动力学和丢番图几何中的问题。预计这项工作将应用于代数微分方程的结构、动力系统的算法、数学物理以及其他领域。这一奖项反映了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。

Important mathematical structures can often be understood from very different perspectives, either by analyzing them with algebraic or analytic formulas or by regarding them geometrically. This dual algebraic/geometric view is applied in many branches of mathematics, especially for the study of algebraic equations involving numbers (under the name of Diophantine geometry), the study of solutions to polynomial equations (under the name of algebraic geometry), or the study of the possible algebraic relations among solutions to systems of differential or difference equations (under the names of differential algebraic geometry or difference algebraic geometry, respectively), among others. In practice, some of the questions considered in these areas suffer from extreme complexity inherited from their connection to number theory or to even more complicated domains. Work in mathematical logic has elucidated the boundary between those complicated theories and those admitting a tame, geometric theory. This research project aims to extend the class of theories for which a tame geometry can be established and to use these results from mathematical logic to answer questions from the target domains. This research project studies the connections between geometries of various kinds, including algebraic, differential, Diophantine, and analytic geometries, through the model-theoretic lens of definability in suitable theories. Mathematical theories as diverse as those of partial differential equations, difference equations, perfectoid spaces, formal geometry, algebraic dynamics, and homogenous dynamics will be studied. Technically, methods including geometric stability theory as applied to differentially closed fields, o-minimality, and quantifier elimination for valued differential fields and analytic difference rings will be applied for the purpose of answering questions internal to model theory (such as proving or refuting the trichotomy principle for regular types in differentially closed fields with several commuting derivations) and for applications to problems in functional transcendence, dynamics, and Diophantine geometry. It is anticipated that the work will have applications to the structure of algebraic differential equations, the arithmetic of dynamical systems, and mathematical physics, as well as in other areas.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.