Dynamical Cubes and a criteria for systems having product extensions

For a minimal$ Z^2-$topological dynamical systems, we introduce a cube structure and a generalization of the regionally proximal relation, which allow us to characterize product systems and their factors. We also introduce the concept of topological magic systems, which is the topological counterpart of measure theoretic magic systems introduced by Host in his study of multiple averages for commuting transformations. We give various applications of these structures, including the construction of some special factors in topological dynamics, and a computation of the automorphism group of tiling systems. This is joint work with Wenbo Sun.