Quantitative reduction theory and Diophantine geometry

自古以来，数学家们就试图理解多项式方程何时有整数解。这类问题很容易提出，但令人惊讶地难以解决-一个著名的例子是安德鲁怀尔斯的证明费马大定理，这是开放了350年，直到它在20世纪90年代被解决。答案往往是密切相关的几何形状所定义的方程。许多最深刻的问题都可以用“不太可能的交集”来表达：几何学告诉我们，方程不太可能有某种特定类型的解;如果有很多这种不太可能的解，那么我们就寻找某种隐藏的特殊结构来解释它们。不可能相交的研究涉及到数学的许多领域：数论、几何、遍历理论、数理逻辑。用于解决不可能相交问题的一种工具是归约理论。归约理论是一种构造“瓦片”的方法，这样我们就可以使用瓦片的移动副本来填充几何对象，就像填充一张坐标纸的正方形一样。博雷尔和哈里什-钱德拉发现了一种构造瓦片的方法，只要允许的位移由一个称为算术群的对象给出。这种构造在数论、群论和动力系统中有着广泛的应用。Borel和Harish-Chandra的瓦片是通过将几块瓦片粘在一起来构造的--但对需要多少块瓦片没有控制。该项目的第一部分旨在回答这个问题：我们将多少块瓷砖粘合在一起来制作每块瓷砖？这将给我们关于约化理论应用的定量信息。在项目的第二部分，我们将回答数论中关于伽罗瓦轨道边界的深层次问题。结合定量约化理论，这将使我们能够证明新的情况下，中央猜想不太可能的交叉点，齐伯粉红猜想。

Since antiquity, mathematicians have sought to understand when polynomial equations have solutions in whole numbers. Such questions are easy to ask, but surprisingly difficult to solve - a famous example being Andrew Wiles's proof of Fermat's Last Theorem which was open for 350 years until it was solved in the 1990s.The answer is often closely related to the geometry of the shape defined by the equations. Many of the deepest questions can be posed in terms of "unlikely intersections": the geometry tells us that equations are unlikely to have solutions of a particular type; if there are lots of these unlikely solutions, then we look for some hidden special structure to explain them. The study of unlikely intersections draws on a remarkable range of fields of mathematics: number theory, geometry, ergodic theory, mathematical logic.One tool used to solve questions of unlikely intersections is reduction theory. Reduction theory is a method of constructing "tiles" so that we can fill up a geometric object using shifted copies of the tiles, like the squares which fill a sheet of graph paper. Borel and Harish-Chandra discovered a recipe for constructing tiles whenever the permitted shifts are given by an object called an arithmetic group. This construction has numerous applications in number theory, group theory and dynamical systems.Borel and Harish-Chandra's tiles are constructed by gluing together several pieces -- but there is no control over how many pieces are needed. The first part of this project seeks to answer the question: How many pieces do we glue together to make each tile? This will give us quantitative information about the applications of reduction theory.In the second part of the project, we will answer deep questions from number theory about bounds for Galois orbits. Combined with quantitative reduction theory, this will enable us to prove new cases of the central conjecture on unlikely intersections, the Zilber-Pink conjecture.