Quantitative reduction theory and Diophantine geometry

自古以来，数学家一直试图了解多项式方程何时有整数解。这样的问题很容易问，但令人惊讶的是很难解决--一个著名的例子是安德鲁·怀尔斯的费马大定理的证明，这一证明一直持续了350年，直到20世纪90年代才被解决。答案往往与方程所定义的形状的几何密切相关。许多最深层次的问题可以用“不太可能的交集”来提出：几何学告诉我们，方程不太可能有特定类型的解；如果有很多这样的不太可能的解，那么我们就寻找一些隐藏的特殊结构来解释它们。不可能交叉口的研究涉及到大量的数学领域：数论、几何、遍历理论、数理逻辑。一个用来解决不可能交叉口问题的工具是归约理论。归约理论是一种构造“瓷砖”的方法，这样我们就可以用瓷砖的平移副本来填充几何对象，就像填充一张图表纸的正方形一样。Borel和Harish-Chandra发现了一种每当允许的移位由一个称为算术组的对象给出时，构造瓷砖的方法。这种结构在数论、群论和动力系统中有许多应用。Borel和Harish-Chandra的瓷砖是通过将几块粘合在一起而构成的--但没有控制需要多少块。这个项目的第一部分试图回答这个问题：我们将多少块胶水粘在一起来制作每一块瓷砖？这将为我们提供有关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华轨的界的深层次问题。结合定量还原理论，这将使我们能够证明中心猜想在不太可能的交叉点上的新情况，即Zilber-Pink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.