Model Theory of Generalized Differential Equations and Diophantine Geometry

广义微分方程模型论与丢番图几何

• 批准号: 1363372
• 负责人: Thomas Scanlon
• 金额: $ 46万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1363372&HistoricalAwards=false
• 关键词: Model; Theory; Generalized; Differential; Equations

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.

该项目包括通过数理逻辑意义上的模型理论视角研究微分方程和差分方程及其推广，利用多模型理论思想解决算术动力学中的难题，并对几何和算术中的可判定性进行基础研究。 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。 例如，通常采用各种方法（例如计算方法和分析方法）来理解和求解这些方程。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理相关的思想，将被用来理解微分方程和差分方程。 由于微分方程和差分方程在数学科学应用中的重要性，这项基础研究应该对数学和科学产生广泛的影响。具体来说，对于微分方程，本项目将微分场的基本模型理论结果扩展到更广泛意义上的D场类。其次，该项目涉及 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（或者至少是类似风格的结果）扩展到无限维类型，从而专门研究经典理论、欠定差分微分方程、偏微分方程和哈斯微分方程。此外，该项目的目标是明确表征具有简单分叉几何形状的集合上的诱导结构。第三，该项目将通过有价值的 D 场模型理论开发 D 环专业化理论。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 场，该项目包括对 D 场理论的雄心勃勃的应用，这可能会产生变革性的后果；也就是说，将博格的 F-1 几何实现为相对于某种 D 环理论的有限维可定义集合的研究。在丢番图几何方面，该项目将通过差分场模型理论和o-极小性方法来解决张的稠密轨道猜想和动力学Mordell-Lang。 该项目包括一个建立 C(t) 理论（复数有理函数域）复杂片段的可判定性的程序。