Model Theory of Generalized Differential Equations and Diophantine Geometry

这个项目包括从数理逻辑意义上通过模型理论的视角研究微分差分方程及其推广，用多模型理论的思想攻击算术动力学中的困难问题，以及几何和算术中可判断性的基础研究。在许多应用中，微分方程组和差分方程组描述了复杂系统的演化和动力学。理解和求解这些方程通常采用各种方法，例如计算方法和分析方法。在这个项目中，来自代数和逻辑的思想，特别是与结构的可定义性和温和性原则有关的思想，将被用来理解微分方程式和差分方程式。由于微分方程式和差分方程式在数学应用到科学中的重要性，这种基础研究应该在数学和科学方面产生广泛的影响。具体地说，对于微分方程组，本项目将把微分场的基本模型理论结果推广到更广义的D-场类。其次，该项目涉及对D-域中可定义性和依赖性的精细结构的研究。特别是，该项目将把Zilber三分法(或者，至少，类似的结果)扩展到无限维类型，从而专门用于经典理论，用于低定差分--微分方程组、偏微分方程组和哈斯微分方程组。此外，该项目的一个目标是明确地刻画具有平凡分叉几何的集合上的诱导结构。第三，该项目将通过值D-域的模型理论来发展D-环的专门化理论。第四，该项目包括基于多模型理论视角的一般D-Galois理论的发展。最后，关于D-域，该项目包括D-域理论的一个雄心勃勃的应用，这可能具有变革性的结果；即，实现Borger的F-1几何作为相对于D-环的特定理论的有限维可定义集的研究。在丢番图几何方面，该项目将通过差分场模型理论和o-极小方法来解决张的密集轨道猜想和动力学Mordell-lang。该项目包括一个建立C(T)理论的复杂片段的可判断性的程序，C(T)是复数上的有理函数域。

This project consists of a study of differential and difference equations and their generalizations through the lens of model theory in the sense of mathematical logic, attacks on difficult problems in arithmetic dynamics using multiple model theoretic ideas, and a fundamental investigation into decidability in geometry and arithmetic. In their many applications, differential and difference equations describe the evolution and dynamics of complex systems. Various methods, computational and analytic, for example, are commonly employed to understand and solve these equations. With this project, ideas from algebra and logic, especially related to the principles of definability and tameness of structure, will be used to understand differential and difference equations. This basic research should have consequences in mathematics and the sciences broadly due to the importance of differential and difference equations in the applications of mathematics to the sciences. Concretely, with regards to differential equations, this project will extend the fundamental model theoretic results about differential fields to the class of D-fields in a more expansive sense. Secondly, the project involves a study of the fine structure of definability and dependence in D-fields. In particular, the project will extend the Zilber trichotomy (or, at least, results of a similar flavor) to infinite dimensional types, and, thus, specializing to classical theories, to underdetermined difference-differential, partial differential, and Hasse differential equations. Moreover, a goal of the project is an explicit characterization the induced structure on sets with trivial forking geometry. Thirdly, the project will develop a theory of specializations of D-rings through a model theory of valued D-fields. Fourthly, the project includes a development of general D-Galois theories based on multiple model theoretic perspectives. Finally in connection to D-fields, the project includes an ambitious application of the theory of D-fields which may have transformative consequences; that is, to realize Borger's F-1-geometry as the study of finite dimensional definable sets relative to a certain theory of D-rings. With regards to diophantine geometry, the project will address Zhang's dense orbit conjecture and the dynamical Mordell-Lang through the model theory of difference fields and methods from o-minimality. The project includes a program establish the decidability of complicated fragments of the theory of C(t), the field of rational functions over the complex numbers.