Quantitative reduction theory and Diophantine geometry

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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.