Some problems on unlikely intersections

点击翻译按钮获取中文摘要

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.