Problems in Model Theory, Diophantine Geometry, and Functional Transcendence

这个项目将研究Schanuel-型猜想和相关的丢番图猜想与模型理论的相互作用。本项目的背景是丢番图几何中的Zilber-Pink猜想及其与模型理论和Schanuel-型猜想的联系。Schanuel猜想是关于指数函数的超越性质的超越数论的一个猜想。它概括了林德曼-魏尔斯特拉斯和希尔伯特第七问题的经典结果。虽然Schanuel的猜想仍然被广泛认为是无法用Currnet方法实现的，但某些模型理论方法，如O-极小值的应用和指数场的研究，是该领域的一些最新发展。Zilber-Pink猜想形成了Andre-Oort猜想的推广，两者在一般形式下都是开放的，尽管O-极小几何的方法已被应用于Andre-Oort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理论方面的工作的某些例子是研究高维阿贝尔变种的j-函数及其类似物的模型理论性质和Schanuel类型类似物，指数和伪幂之间的关系，一阶和无限逻辑中伪指数场的模型理论性质，以及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下研究Zilber-Pink猜想中的具体问题，主要目的是以某种方式增加或调查某些特殊情况，Zilber-Pink猜想和相关的Schanuel-type猜想。这将涉及到结合O-极小性或模型理论、算术和微分代数的其他部分的方法。特别是关于O-极小性，最近在应用于丢番图问题，特别是Andre-Oort猜想方面的发展基本上已经发展成处理Zilber-Pink猜想的各种情况的一般策略，该猜想已经取得了可能尚未用尽的重要结果。最近Hrushovski和其他人成功地将模型理论应用于丢番图问题的另一个领域是几何稳定性理论。这个项目属于EPSRC逻辑和组合学研究领域

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