Some problems on unlikely intersections

数论是数学的一个分支，它的一个经典问题是描述多项式方程（即只涉及加法和乘法的方程）的整数或有理解的集合。这类方程被称为丢番图方程，以世纪亚历山大的数学家丢番图命名。了解丢番图方程解集的结构是现代数学中最困难的问题之一。从几何的观点研究丢番图方程往往是方便的。多项式方程被它们的复数解集所取代，这些复数解集被视为几何对象（例如曲线和曲面），然后问题是理解这些几何对象中具有整数或有理坐标的点集。这种方法导致了丢番图几何的发展，这是数论的一个分支，使用几何工具来研究丢番图方程。研究高维丢番图方程的类似物是很自然的。例如，人们可以问一个给定的曲面何时包含特殊类型的点或曲线，或者一个给定的曲面与一个特殊曲线的交点是什么样子，而不是问一个丢番图方程何时有“特殊”类型的解（即具有特殊算术性质）。这些问题通常用“不太可能的交叉点”来表达。当两个几何对象在不期望的情况下相交时，会发生不太可能的相交。例如，如果我们在三维空间中随机选取两条直线，那么它们不太可能相交。然而，它们仍然可以相交，在这种情况下，我们有一个不太可能的交集。本项目旨在探讨不太可能交集理论中的一些问题。这个领域的主要开放问题是所谓的Zilber-Pink猜想。学生将使用文献中的一些已建立的工具和技术，并将其用于证明该猜想的一些新实例。使用的主要方法是皮拉-赞尼尔策略，其中的工具来自o-极小（模型论的一个分支，它本身是数理逻辑的一个分支），微分代数（微分方程的代数理论）和算术（数论的一个分支）。在博士学位的后期阶段，学生应该能够开发相对新颖的技术并解决进一步的问题。学生也可以研究其他相关问题，如相关结果的有效性。

A classical problem in number theory, a branch of mathematics, seeks to describe the set of integral or rational solutions of polynomial equations (i.e. equations involving only addition and multiplication). Equations of this kind are known as Diophantine equations, named after the third century mathematician Diophantus of Alexandria. Understanding the structure of the set of solutions of Diophantine equations is one of the hardest problems in modern mathematics. It is often convenient to study Diophantine equations from a geometric point of view. Polynomial equations are replaced by their sets of solutions in complex numbers, which are treated as geometric objects (for instance, curves and surfaces), and then the question is to understand the set of points with integral or rational coordinates in these geometric objects. This approach led to the development of Diophantine geometry, a branch of number theory using geometric tools to investigate Diophantine equations. It is natural to study the analogues of Diophantine equations in higher dimensions. For instance, instead of asking when a Diophantine equation has a solution of "special" type (i.e. with special arithmetic properties), one may ask when a given surface contains points or curves of special type, or what the intersection of a given surface with a special curve looks like. These questions are often phrased in terms of "unlikely intersections". An unlikely intersection occurs when two geometric objects intersect when they are not expected to. For example, if we pick two random straight lines in a three-dimensional space then they are not likely to intersect. They can still intersect though, in which case we have an unlikely intersection.This project aims to explore some problems in the theory of unlikely intersections. The main open problem in this area is the so-called Zilber-Pink conjecture. The student is expected to use some established tools and techniques from the literature and adapt them to prove some new instances of this conjecture. The main approach to be used is the Pila-Zannier strategy, where the tools come from o-minimality (a branch of model theory, which is itself a branch of mathematical logic), differential algebra (algebraic theory of differential equations) and arithmetic (a branch of number theory). At later stages of their PhD, the student should be able to develop relatively novel techniques and tackle further problems. Other related problems, such as effectivity of the relevant results, may also be studied by the student.