Dynamics of Large Group Actions, Rigidity, and Diophantine geometry

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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.