Quantitative reduction theory and Diophantine geometry

自古以来，数学家们就试图弄清楚多项式方程何时有整数解。这样的问题很容易提出，但令人惊讶的是很难解决——一个著名的例子是安德鲁·怀尔斯对费马大定理的证明，该定理公开了350年，直到20世纪90年代才被解决。答案通常与方程所定义的几何形状密切相关。许多最深刻的问题可以用“不可能的交集”来提出：几何告诉我们方程不可能有特定类型的解；如果有很多这样的不可能的解决方案，那么我们寻找一些隐藏的特殊结构来解释它们。对不可能交集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。用于解决不太可能的交集问题的一个工具是约简理论。约简理论是一种构造“瓦片”的方法，这样我们就可以用瓦片的移动拷贝来填充一个几何对象，就像填充一张方格纸一样。Borel和Harish-Chandra发现了一种构造瓷砖的方法，只要允许的移动是由一个被称为算术群的对象给出的。这种构造在数论、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瓷砖粘合在一起制成的，但无法控制需要多少块瓷砖。这个项目的第一部分试图回答这个问题：我们用多少块胶水粘合在一起来制作每块瓷砖？这将给我们关于约简理论应用的定量信息。在项目的第二部分，我们将回答关于伽罗瓦轨道边界的数论的深层次问题。结合定量还原理论，这将使我们能够证明关于不可能交集的中心猜想的新情况，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.