Abstract: We introduce two novel applied topology techniques to address real-world problems in different fields. The first, used in information theory for the study of complex systems, is a principled compression method of higher order signals obtained from a combinatorial version of the Laplace-Beltrami operator. The second, used for the study of conformation spaces of molecules, is an enhancement of the traditional persistent homology pipeline obtained from recent developments in the effective construction of cohomology operations.