FRG: Collaborative Research: Stability of Structures Large and Small

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This award supports collaborative research efforts in the area of materials science by an interdisciplinary team comprising pure mathematicians, applied mathematicians, computer scientists, and physicists. Answers to natural questions about the stability and rigidity of material structures involve understanding the geometry of their components. Recent advances in materials synthesis have emphasized the need for a deeper understanding of the geometric stability of physical structures at the atomic scale and the need for insight at all scales, from atomic to macroscopic. Key mathematical tools for this analysis come from the area of "rigidity theory," which studies the mathematical properties of discrete sets of points with the distances between certain pairs of points held fixed or constrained by distance inequalities. Rigidity theory lies at the nexus of discrete geometry, graph theory, and algorithms, and it has deep connections to semidefinite programming and convex geometry. This project aims to deepen understanding of the stability of material structures. One goal of this project is to develop a mechanistic explanation of tunneling between asymmetric stable configurations of two-dimensional disordered materials, such as glass. Construction of accurate mechanistic and computational models requires deep mathematical analysis and development of appropriate algorithms. A second goal is to develop methods for predicting the stability, configurational entropy, and kinetics of small short-ranged-potential systems in three dimensions. Examples of such distance-constraint systems include small molecular structures, as well as colloidal clusters, containing a few particles bound together by reversible attractive interactions, modeled as sticky spheres. What kinds of rigid configurations are there, and what are computationally feasible tests for their rigidity? How do these particles move and the structures deform? There is a tight link between rigidity theory and the general convexity and duality properties of the positive semidefinite cone, a central concept in numerical optimization. Finding a recursive decomposition of a generically rigid framework into rigid subsystems is a longstanding problem. Additionally, matroid theory, important in rigidity theory, has made the characterization of rigid systems more approachable and more algorithmically efficient. Rigidity theory could have implications for algorithms for low-rank matrix completion as well. These connections, questions, and implications will be explored in this project.