Dynamics of Large Group Actions, Rigidity, and Diophantine geometry

在我们的项目中，我们打算探索多项式方程有理/积分解集的结构与相关动力系统的性质之间的深刻联系。虽然丢番图方程理论是数学中最古老的学科之一，但它仍然是一个基本上未被探索的前沿，有许多广泛开放的基本问题和猜想。有许多重要的多项式方程，它们的解集具有自然群作用。在这些情况下，人们可以构建对解集合进行编码的动态系统。这里所说的动力系统，我们指的是配备了一组变换的空间。这些变换的轨道结构通常非常复杂。特别是，人们可以观察到密密麻麻地填满空间的轨道或聚集在分形集上的轨道。尽管如此，我们开发了新的工具来分析轨道结构，特别是得出关于轨道分布和重现的信息。利用有理/积分解集和相应动力系统的轨道结构之间的相互作用，我们期望揭示动力系统理论和丢番图几何的新见解。为了解决丢番图几何中的问题，我们需要在遍历理论中开发新的工具，这些工具位于经典框架之外，也将对遍历理论家感兴趣。我们研究了动力系统理论中的基本现象，如轨道的分布和递归，以及动力系统在大群作用下的各种刚性性质。

In our project we intend to explore profound connections between the structure of the sets of rational/integral solutions of polynomial equations and the properties of related dynamical systems. Although the theory of Diophantine equations is one of the oldest subjects in mathematics, it is still a mostly unexplored frontier with many widely open fundamental problems and conjectures. There are many important classes of polynomial equations where the sets of solutions are equipped with natural group actions. In these cases one can build dynamical systems that encode the sets of solutions. By a dynamical system here we mean a space equipped with a group of transformations. The orbit structure of these transformations is typically very complicated. In particular, one may observe orbits that fill the space densely or accumulate on a fractal set. Nonetheless, we develop new tools to analyse the orbit structure and, in particular, to derive information about distribution and recurrence of orbits. Utilising the interplay between the sets of rational/integral solutions and the orbit structure of the corresponding dynamical systems, we expect to uncover new insights into both the theory of dynamical systems and Diophantine geometry. In order to address the problems in Diophantine geometry we require to develop new tools in ergodic theory that lie outside the classical framework and will be of interest to ergodic theorists as well. We investigate such fundamental phenomena in the theory of dynamical systems as distribution and recurrence of orbits and various rigidity properties of dynamical systems for actions of large groups.