Arithmetic sparsity in mixed Hodge settings

The study of equations and their rational solutions is one of the oldest and hardest subjects in mathematics. Landmark results in this area were the finiteness theorems by Faltings. For higher dimensional varieties of general type, non-density or sparsity are expected, and finiteness is expected only in very specific cases. We will first give a brief overview of previous results and conjectures on finiteness, non-density, and sparsity of integral/rational points, especially those on moduli spaces. After a review of the notion of variation of mixed Hodge structures, we will present new sparsity results on integral points in the mixed Hodge setting, and discuss possible future directions.