Model theory, Diophantine geometry, and automorphic functions

我的研究方向是模型论，这是数学逻辑的一个分支，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叉点，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，为问题提供了更深入的见解。这个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杂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。它在数论中自然出现，在密码学和编码理论中也有应用。j函数和其他类似的（更一般的）函数，称为自同构函数，是本文的主要研究对象。该提案主要关注两个问题。首先是要理解包含加法、乘法和自同构函数的多个变量方程组何时具有复数解。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可解性问题的自同构变体，该问题由希尔伯特的Nullstellensatz（德语为“零定理”）解决。我提出了自同构函数的零性猜想（存在性闭性猜想）的一个推测性类比，并旨在证明它。第二个主要问题是描述与自同构函数相关的多变量多项式方程（丢芬图方程）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难的问题之一。研究丢番图方程的现代方法是基于多项式方程可以被几何对象（曲线、曲面等）取代的思想，即满足这些方程的点的集合。然后可以使用几何工具来研究这些对象上的有理点。丢芬图几何中的一个主流问题与“不可能的相交”有关。例如，在三维空间中，两条随机选择的线不太可能相交，即使它们相交，也不太可能相交。另一方面，在一个平面上，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开放问题是Zilber-Pink猜想，它粗略地陈述了具有某些“特殊”变种的变种的不可能交集是由有限多个特殊变种控制的。这是对莫德尔-朗和安德里姆-奥尔特等一些著名的吐番图言论的深远概括。在自同构函数的Zilber-Pink猜想上取得进展是本文的主要目的。提出的新思想之一是考虑自同构函数及其导数的存在紧密性和Zilber-Pink。这导致了更一般和更难的问题，但它也提供了对猜想和完整的模型理论图景的更深入的了解。此外，在这种普遍性中，我建立了存在主义封闭性和Zilber-Pink之间的新联系，这为使用前者攻击后者的强大策略开辟了道路。我已经应用了这个策略来获得第一个“带导数”的Zilber-Pink类型定理，还有很多需要探索的。我打算扩展和利用这些联系来并行解决这两个问题。研究其他几个相关问题，旨在利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有关自同构函数的丰富数学理论的理解，包括一个众所周知的难解问题，即Schanuel猜想。

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.