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.

该项目包括一项研究，从数学逻辑意义上讲，通过模型理论的范围对差异方程式及其概括，使用多个模型理论思想对算术动力学中的困难问题的攻击以及对几何学和算术中的几何学的基本研究。 在许多应用中，差分方程描述了复杂系统的演变和动力学。 例如，通常使用各种计算和分析方法来理解和解决这些方程。在这个项目中，代数和逻辑的想法，尤其是与结构的可确定性和驯服性有关的原则，将用于理解差异方程式和差异方程。 这项基础研究应在数学和科学方面产生影响，因为差异方程式在数学对科学的应用中的重要性。具体而言，在微分方程方面，该项目将在更广泛的意义上将有关差分字段的基本模型理论结果扩展到D-Fields类。其次，该项目涉及对D-Fields确定性和依赖性的精细结构的研究。特别是，该项目将将Zilber三分法（或至少是类似风味的结果）扩展到无限的尺寸类型，因此将专门研究经典理论延伸到不确定的差异差异差异，部分差分和hasse微分方程。此外，该项目的目标是明确表征带有微不足道的几何形状集合的诱导结构。第三，该项目将通过有价值的D-Fields的模型理论发展D形环的专业理论。第四，该项目包括基于多种模型理论观点的一般D-Galois理论的发展。最后，与D场相关联，该项目包括对D场理论的雄心勃勃的应用，这些理论可能会带来变革性的后果。也就是说，将Borger的F-1几何体现为对某些D形环理论的有限维定集合的研究。关于Diophantine几何形状，该项目将通过差异场的模型理论和O-最低点的模型理论来解决Zhang的密集轨道猜想和动态Mordell-lang。 该项目包括一个程序，建立了C（t）理论复杂片段的可决定性，即复数上的理性函数领域。