Billiards within confocal quadrics and beyond

该项目旨在分析共聚焦圆锥范围内的数学台球。数学台球在任何涉及碰撞和反射的情况下都有应用，任何包含反射和碰撞的现象都可以使用数学台球进行建模。本项目旨在通过探索这些台球与多边形台球的关系来彻底改变由几个共焦二次曲面所限定的区域内台球的分析，并在共焦二次曲面内的高维推广及其与多面体内台球的关系方面取得研究进展。该项目将连接几个重要的科学工作领域，包括多边形台球、经典可积系统、Teichmuller空间和相对论。项目成果将影响数学的各个领域，如几何、代数几何和动力系统。

This project aims to analyse mathematical billiards within domains bounded by confocal conics. Mathematical billiards have applications in any situation that involves collisions and reflections, and any phenomenon that includes reflections and collisions can be modelled using mathematical billiards. This project aims to revolutionise the analysis of billiards within domains bounded by several confocal conics by exploring the relations of such billiards with polygonal billiards, and making research advances with the higher-dimensional generalisations within confocal quadrics and their relations with billiards within polyhedra. The project will link several significant areas of scientific work including polygonal billiards, classical integrable systems, Teichmuller spaces, and relativity theory. The project outcomes will have impact across areas of mathematics such as geometry, algebraic geometry, and dynamical systems.