Reflection Groups and Discrete Dynamical Systems

该项目旨在解决物理学特别感兴趣的离散动力系统中的长期问题，通过使用反射群来揭示意想不到的几何见解。数学有能力从复杂的观察中抽象出关键的模式。真实的世界中常见的对称性，如蜂窝的六边形图案，出现在高维系统的复杂结构中。通过揭示可积系统中空间填充多面体的结构，该项目试图找到高维离散模型到二维的追求。预期的产出包括新的减少离散Painlevé方程，新的圆形图案有用的计算机图形和离散全纯函数。

This project aims to solve long-standing problems in discrete dynamical systems that are of particular interest to physics, by using reflection groups to reveal unexpected geometric insights. Mathematics has the power to abstract crucial patterns from complex observations. Symmetries familiar in the real world, like the hexagonal patterns of honeycombs, arise inside convoluted structures in high-dimensional systems. By revealing the structure of space-filling polytopes in integrable systems, the project seeks to find sought-after reductions of high-dimensional discrete models to two dimensions. Expected outputs include new reductions to discrete Painlevé equations, new circle patterns useful for computer graphics and discrete holomorphic functions.