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.

该项目包括通过数学逻辑意义上的模型理论来研究微分方程和差分方程及其推广，使用多种模型理论思想来研究算术动力学中的难题，以及对几何和算术中的可决性进行基本研究。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。不同的方法，计算和分析，例如，通常被用来理解和解决这些方程。在这个项目中，代数和逻辑的思想，特别是与结构的可定义性和驯服性有关的原则，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应用于科学方面的重要性，这一基础研究应该对数学和科学产生广泛的影响。具体而言，对于微分方程，本项目将微分场的基本模型理论结果在更广泛的意义上推广到d -场的范畴。其次，研究了d场中可定义性和依赖性的精细结构。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，并且，因此，专门研究经典理论，欠定微分方程，偏微分方程和Hasse微分方程。此外，项目的一个目标是明确表征具有平凡分叉几何的集合上的诱导结构。第三，本项目将通过有值d场的模型理论发展d环的专门化理论。第四，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d场方面，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；即将Borger的f -1几何实现为相对于d环的某一理论的有限维可定义集的研究。在丢番图几何方面，本项目将通过差分场模型理论和极小方法解决张的密集轨道猜想和动力学的莫德尔朗。该项目包括一个程序，建立C(t)理论的复杂片段的可判决性，复数上的有理函数域。