Model theory, Diophantine geometry, and automorphic functions

模型论、丢番图几何和自守函数

• 批准号: EP/X009823/1
• 负责人: Vahagn Aslanyan
• 金额: $ 112.94万
• 依托单位: University of Manchester
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX009823%2F1
• 关键词: Model; theory; Diophantine; geometry; automorphic

My research is in model theory, a branch of mathematical logic with striking applications in many fields ranging from number theory to machine learning. The problems I study lie at the intersection of several mathematical disciplines such as model theory, number theory, algebra, and geometry. The model-theoretic perspective reveals new links between those disciplines providing a deeper insight into the problems.The aim of this fellowship is to develop novel model-theoretic tools to answer some fundamental questions about certain classical functions of a complex variable. A renowned example of such a function is the so-called j-function. It arises naturally in number theory and has applications in cryptography and coding theory. The j-function and other similar (more general) functions, known as automorphic functions, are the main objects of study in this proposal.The proposal focuses on two main problems. The first is to understand when systems of equations in several variables involving addition, multiplication, and automorphic functions have solutions in the complex numbers. This is a natural question in complex geometry and is the automorphic variant of the well-known problem of solvability of systems of polynomial equations in several variables, settled by Hilbert's Nullstellensatz (German for "theorem of zeroes"). I have proposed a conjectural analogue of the Nullstellensatz for automorphic functions (the Existential Closedness conjecture) and aim to prove it.The second main problem is to describe the sets of integral, rational or "special" solutions of polynomial equations in several variables (Diophantine equations) related to automorphic functions. Diophantine problems date back to the third century mathematician Diophantus of Alexandria and are among the oldest and hardest problems in mathematics. Modern methods of studying Diophantine equations are based on the idea that polynomial equations can be replaced by geometric objects (curves, surfaces, etc.), namely the sets of points satisfying these equations. Then geometric tools can be used to study rational points on these objects. One of the mainstream problems in Diophantine geometry is related to "unlikely intersections". For example, in a 3-dimensional space two randomly chosen lines are not likely to intersect, and when they do, it is an unlikely intersection. On the other hand, in a plane two such lines are likely to intersect. A famous open problem in the theory of unlikely intersections is the Zilber-Pink conjecture stating roughly that unlikely intersections of a variety with certain "special" varieties are controlled by finitely many special varieties. It is a far-reaching generalisation of some renowned Diophantine statements such as Mordell-Lang and André-Oort. Making progress on the Zilber-Pink conjecture for automorphic functions is a major aim of the proposal.One of the new ideas of the proposal is to consider Existential Closedness and Zilber-Pink for automorphic functions together with their derivatives. This results in more general and harder problems, but it also gives a deeper insight into the conjectures and into the full model-theoretic picture. Moreover, in that generality I have established new links between Existential Closedness and Zilber-Pink which opened the way to a powerful strategy of using the former to attack the latter. I have applied that strategy to obtain the first Zilber-Pink type theorems "with derivatives", with a lot more to explore. I intend to extend and exploit these links to tackle both problems in parallel. Investigating several other related questions, aimed at understanding the geometry of automorphic functions using model-theoretic techniques, are also among the goals of the fellowship. Making progress towards these questions would improve our understanding of the rich mathematical theory around automorphic functions including a notoriously hard open problem known as the Schanuel conjecture.

我的研究方向是模型理论，这是数理逻辑的一个分支，在从数论到机器学习的许多领域都有惊人的应用。我研究的问题是几个数学学科的交集，如模型论、数论、代数和几何。模型理论的观点揭示了这些学科之间的新联系，提供了对问题的更深层次的洞察。该奖学金的目的是开发新的模型理论工具来回答一些关于复变量的某些经典函数的基本问题。这种函数的一个著名的例子是所谓的j函数。它自然产生于数论，在密码学和编码学中有应用。J-函数和其他类似的(更一般的)函数，称为自同构函数，是本方案的主要研究对象。第一个是了解涉及加法、乘法和自同构函数的多变量方程系统何时有复数解。这是复杂几何中的一个自然问题，也是著名的多变量多项式方程组的可解性问题的自同构变体，由Hilbert的Nullstellensatz(德语中的“零点定理”)解决。第二个主要问题是描述与自同构函数有关的多元多项式方程(丢番图方程)的积分、有理或“特殊”解的集合。丢番图问题可以追溯到三世纪亚历山大的数学家丢番图，是数学中最古老和最困难的问题之一。现代研究丢番图方程的方法是基于这样的思想，即多项式方程可以被几何对象(曲线、曲面等)代替，即满足这些方程的点集。然后，可以使用几何工具来研究这些对象上的有理点。丢番图几何中的一个主流问题与“不太可能的交点”有关。例如，在三维空间中，随机选择的两条线不太可能相交，当它们相交时，它就不太可能相交。另一方面，在平面上，两条这样的直线很可能相交。不可能交集理论中一个著名的悬而未决的问题是Zilber-Pink猜想，粗略地说，一个品种与某些“特殊”品种的不可能交集由有限多个特殊品种控制。它是莫德尔-朗和安德烈-奥尔特等一些著名的丢番图声明的深远概括。改进自同构函数的Zilber-Pink猜想是该方案的主要目的之一，其中一个新的思想是考虑自同构函数及其导数的存在闭性和Zilber-Pink猜想。这导致了更一般和更困难的问题，但它也给了我们对猜想和整个模型理论图景的更深层次的洞察。此外，在这个概括性中，我在存在主义的封闭性和Zilber-Pink之间建立了新的联系，Zilber-Pink开启了一种强大的战略，利用前者来攻击后者。我应用这一策略得到了第一个“带导数”的Zilber-Pink型定理，还有更多有待探索的地方。我打算扩大和利用这些联系，以同时处理这两个问题。研究其他几个相关问题，旨在使用模型理论技术理解自同构函数的几何，也是该奖学金的目标之一。在这些问题上取得进展将提高我们对围绕自同构函数的丰富数学理论的理解，包括一个众所周知的困难的公开问题，即Schanuel猜想。