The methods developed in this work can be extend to other scientific domains because manifolds and graphs can represent many different materials and chemistry simulation datasets, permitting analysis via topology and the Euler characteristic. For example, density functional theory (DFT) calculations can provide direct graphical representations of molecular bonds. The topology of these molecular graphs provides insight into physical and chemical characteristics. Topology and the EC allow us to summarize these complex molecular graphs for tasks such as high-throughput screening. For example, topological analysis can improve the high-throughput screening of nanoporous materials used for gas storage and separation, like zeolites and metal organic frameworks (MOFs). DFT and molecular dynamics simulations also provide powerful manifold rep-resentations of data through molecular surfaces. Molecular surfaces are one way to capture complex interactions between molecules or molecules and their environment. These molecular surfaces are manifolds with associated functions. Thus, topological methods can quantify and summarize molecular surface data. Topology also provides complementary information that is not captured in statistical summaries (e.g., sigma profiles). We envision that such topological analysis can thus a wide range of systems and processes involving analysis of molecular surfaces, including in studies of protein folding and docking, nanoparticles, membranes, pharmaceuticals, and catalysts. In future work, we will address some of the existing challenges faced by the Euler characteristic in its application to the study of physical and chemical systems. For example, we aim to form deeper relationships between the Euler characteristic and the thermodynamics of chemical systems through connections between topology and geometry that are found in stochastic geometry. We will also further explore statistical frameworks such as hypothesis testing in the analysis of the Euler characteristic, similar to those found in the study of random fields. This information will aid in providing comparisons between the Euler characteristic and other topological measures such as those found in persistence homology, and when either method can be efficiently applied. These advancements will continue to establish the Euler characteristic as a physically meaningful measure for the study of other chemical systems such as colloidal polymers, gels, surfactants, multi-phase systems, and proteins that demonstrate complex topology and geometry.