The simulation of complex chemical, materials and biophysical systems is increasingly reliant upon the transfer of information across length and timescales to both accelerate simulation time as well as interpret collective phenomena that derive from many body interactions beyond the scale of electrons. Ultimately, it is desirable to extend many-body theories beyond traditional domains of electronic structure theory and into complex condensed matter systems, where propagating system states in time must be able to self-consistently account for many-body effects (described by different granularities of information). This presents a unique grand challenge for the computational chemistry community and will require interdisciplinary collaboration with the mathematics and computer science communities – where multidimensional data analysis inspired by machine learning and AI is rapidly advancing the mathematical languages associated with the shape of data, specifically the mathematical languages of graph theory and computational topology.

The focus of this CECAM workshop will be to bring together computational chemists, materials scientists and biophysicists with leaders in the mathematics of graphs and computational topology. The workshop will educate these communities, foster collaboration and inspire development of both applied and fundamental computational methods in chemistry. We will focus upon each scale of information relevant to computational chemistry (from electrons to the mesoscale) and identify opportunities where graph theory and topology can help in method development and information transfer to accelerate interdisciplinary innovation.

For example, at the smallest scale of electrons, continued development is needed for reduced-complexity electronic structure methods (e.g., through effective Hamiltonians). There, fundamental questions remain about how to achieve the best parametrization, what optimization methods to employ, how electronic data is represented, and perhaps most importantly – how to maintain physical transparency through step-by-step coarse-graining that may use nonphysics neural network structures to represent the data from electronic structure calculations.  As chemical complexity grows via molecular degrees of freedom, chemical composition, or diversity of intermolecular interactions, the breadth of configuration ensembles can increase significantly - reflecting an increasingly rugged underlying potential energy landscape. Thus, there is need to characterize and predict spatial heterogeneities, collective dynamics and the relation to energy landscape topology. This is important not only in sampling, but also for predictive models that seek to understand the relationships between physicochemical properties and the configurational phase space. Fundamentally, such chemical insight can be greatly accelerated through mathematical notions of distance (e.g., distances to compare the precise graph combinatorial structure). This may include summaries of the spectral structures of graph representations of the molecular system in combination with the energy landscape - such as the spectra of the graph Laplace operator or the diffusion operator associated with input graphs. Recent advancements in topological data analysis also provide new ways to compare graph representation of high-dimensional data via persistent homology, that have the potential to be adapted for chemical systems.

Through an organizational structure that combines use-case scenarios, grand-challenge talks and methodological and software tutorials (alongside poster presentations and roundtable discussions), this CECAM workshop will set the stage for increased collaboration between the applied math and computational sciences.