Problems in Model Theory, Diophantine Geometry, and Functional Transcendence

模型论、丢番图几何和函数超越中的问题

• 批准号: 1789662
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2016
• 资助国家: 英国
• 起止时间: 2016 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1789662
• 关键词: Problems; Model; Theory; Diophantine; Geometry

The project will study the interaction of Schanuel-type conjectures and related Diophantine conjectures with model theory.The context to this project will be the Zilber-Pink conjecture in diophantine geometry and its connection with model theory and Schanuel-type conjectures. Schanuel's conjecture is a conjecture of transcendental number theory relating to the transcendence properties of the exponential function. It encapsulates the classical results of Lindemann-Weierstrass and Hilbert's 7th Problem. Though Schanuel's conjecture remains widely viewed as inaccessible using currnet methods, certain model theoretic approaches, such as applications of O-minimality and the study of exponential fields, are some of the more recent developments in the area. The Zilber-Pink conjecture forms a generalization of the Andre-Oort conjecture, both remaining open in the general form, though methods from O-minimal geometry have been applied to the Andre-Oort conjecture to provide results for certain cases. Certain examples of recent work in classical and model theoretic terms in this area are investigations of the model theoretic properties of, and Schanuel-type analogues for, the j-function and its analogues for abelian varieties of higher dimension, the relation between exponentiation and pseudo-exponentiation, and the model-theoretic properties of pseudo-exponential fields in first-order and infinitary logics, and the Pila-Wilkie counting theorem.More specifically, the project will involve studying specific problems within the Zilber-Pink conjecture in either classical or model-theoretic settings, with the primary aim being to in some way increase the tractability of, or investigate certain special cases of, the Zilber-Pink conjecture and related Schanuel-type conjectures.This will involve combining methods from O-minimality or other parts of model theory, arithmetic, and differential algebra. With regards to O-minimality in particular, the recent developments in its application to Diophantine problems and in particular the Andre-Oort Conjecture have developed essentially into a general strategy of tackling varied cases of the Zilber-Pink conjecture which has achieved significant results which are likely not yet exhausted. Another area of model theory that has been applied successfully to Diophantine problems by Hrushovski and others recently has been the field of geometric stability theory.This project falls within the EPSRC Logic and Combinatorics research area

该项目将研究Schanuel型拓扑结构和相关丢番图拓扑结构与模型论的相互作用。该项目的背景将是丢番图几何中的Zilber-Pink猜想及其与模型论和Schanuel型拓扑结构的联系。Schanuel猜想是超越数论中关于指数函数超越性质的一个猜想。它概括了Lindemann-Weierstrass和Hilbert第七问题的经典结果。虽然Schanuel猜想仍然被广泛认为是无法使用currnet方法，某些模型理论的方法，如O-极小的应用和指数场的研究，是该领域的一些最新发展。齐伯-平克猜想是安德烈-奥尔特猜想的推广，两者都在一般形式上保持开放，尽管O-极小几何的方法已经被应用于安德烈-奥尔特猜想，以提供某些情况下的结果。最近在这一领域的经典和模型论方面的工作的某些例子是对j函数及其高维阿贝尔簇的类似物的模型论性质和Schanuel型类似物的研究，指数化和伪指数化之间的关系，以及一阶和无穷逻辑中伪指数场的模型论性质，和Pila-Wilkie计数定理。更具体地说，该项目将涉及在经典或模型理论环境中研究Zilber-Pink猜想中的特定问题，其主要目的是以某种方式增加易处理性，或调查某些特殊情况，Zilber-Pink猜想和相关的Schanuel型猜想。这将涉及O-极小或模型论，算术和微分代数的其他部分的组合方法。特别是关于O-极小性，它在丢番图问题中的应用，特别是安德烈-奥尔特猜想的最新发展，基本上已经发展成为一种处理齐尔伯-平克猜想各种情况的一般策略，已经取得了重要的结果，可能还没有穷尽。模型理论的另一个领域是几何稳定性理论，最近Hrushovski等人成功地将其应用于丢番图问题，该项目属于EPSRC逻辑和组合学研究领域的福尔斯