Using Algebraic and Geometric Topology to Characterize Hierarchical Organization in Complex Solutions and Their Interfaces

Complex, multicomponent, solutions are often characterized by multiple length and timescale correlations that challenge chemical intuition. Recently developed tools are leveraging graph-theoretical interpretations of the intermolecular networks of interactions in such systems, supporting topological data analysis as a means to characterize organizational patterns, from the identification of molecular species and solvation environments, to new interpretations of fluid phase transformation. One may interpret these topological descriptors as high-dimensional order parameters that can also be used to enhance sampling of the energy landscape. Complementary geometric topological methods of liquid surfaces have also emerged as a powerful tool for the identification of interfacial structures responsible for transport. Ongoing work is combining both the algebraic and geometric topology formalism to create a holistic approach that relates hierarchical organizational patterns to structure and even function within complex solutions and their interfaces.