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，或者用代数或解析公式来分析它们，或者用几何方法来看待它们。这种对偶代数/几何观点应用于数学的许多分支，特别是研究涉及数字的代数方程（称为丢芬图几何），研究多项式方程的解（称为代数几何），或研究微分或差分方程系统的解之间可能的代数关系（分别称为微分代数几何或差分代数几何）。等等。在实践中，在这些领域中考虑的一些问题由于与数论或更复杂的领域的联系而具有极端的复杂性。数理逻辑方面的工作已经阐明了那些复杂的理论和那些承认平淡无奇的几何理论之间的界限。该研究项目旨在扩展可以建立驯服几何的理论类别，并利用这些数学逻辑的结果来回答目标领域的问题。本研究计划透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、丢芬图和解析几何。数学理论的多样性，如偏微分方程，差分方程，完美空间，形式几何，代数动力学和齐次动力学将被研究。从技术上讲，包括应用于微分闭域的几何稳定性理论、o极小性、有值微分域和解析差分环的量词消除等方法将用于回答模型理论内部的问题（例如用几个交换导数证明或驳斥微分闭域中正则型的三分原理），以及应用于功能超越、动力学、和丢番图几何学。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。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。

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.