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.