Model Theory of Generalized Differential Equations and Diophantine Geometry

该项目包括通过数学逻辑意义上的模型论的透镜研究微分和差分方程及其概括，使用多模型理论思想攻击算术动力学中的难题，以及对几何和算术中的可判定性进行基本调查。 在许多应用中，微分方程和差分方程描述了复杂系统的演化和动力学。 各种方法，计算和分析，例如，通常用于理解和解决这些方程。通过这个项目，代数和逻辑的思想，特别是与结构的可定义性和驯服性原则有关的思想，将用于理解微分和差分方程。 由于微分方程和差分方程在数学应用于科学中的重要性，这种基础研究应该在数学和科学中产生广泛的影响。具体地说，在微分方程方面，本项目将把关于微分场的基本模型理论结果在更广泛的意义上推广到D-域类。其次，该项目涉及D-域的可定义性和依赖性的精细结构的研究。特别是，该项目将扩大Zilber剖分（或，至少，类似的味道的结果）无限维类型，并因此，专门研究经典理论，欠定差分微分，偏微分和哈塞微分方程。此外，该项目的一个目标是显式表征的诱导结构集平凡分叉几何。第三，该项目将通过有值D-域的模型理论发展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-伽罗瓦理论的发展。最后在连接到D-领域，该项目包括一个雄心勃勃的应用理论的D-领域可能有变革性的后果，即实现博尔格的F-1-几何的研究有限维可定义集相对于一定的理论的D-环。关于丢番图几何，该项目将通过差分场的模型理论和o-极小性方法解决张的稠密轨道猜想和动力学Mordell-Lang。 该项目包括一个程序，建立C（t）理论的复杂片段的可判定性，即复数上的有理函数领域。

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.