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，要么通过代数或解析公式来分析它们，要么通过几何来考虑它们。 这种代数/几何对偶观在数学的许多分支中都有应用，特别是对于涉及数的代数方程的研究（在丢番图几何的名称下），研究多项式方程的解（以代数几何的名义），研究微分或差分方程系统解之间可能的代数关系的学科（分别以微分代数几何或差代数几何的名义）等。在实践中，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。数理逻辑的工作已经阐明了那些复杂理论和那些接受驯服的几何理论之间的界限。本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目标领域的问题。本研究计画透过适当理论中可定义性的模型论透镜，研究各种几何学之间的关联，包括代数几何、微分几何、丢番图几何与解析几何。 数学理论作为不同的偏微分方程，差分方程，perfectoid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，方法包括几何稳定性理论，适用于差分封闭领域，o-最小，值微分域和解析差环的量词消去将被应用于回答模型论内部的问题（例如证明或反驳微分闭域中正则型的可交换性原理，其中有几个可交换导子）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、动力系统的算术和数学物理以及其他领域。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.