Meera Sitharam, Ph.D., a CISE associate professor, was awarded $299,243 of a $1.05M grant funded by the National Science Foundation.
Grant Abstract: This grant supports the collaborative research efforts of an interdisciplinary team comprising pure mathematics, applied mathematics, computer science, and physics on some problems in materials science. The world is full of many varied structures, and those that last, do not fall apart. One natural way of understanding the stability and rigidity of such structures is by understanding their geometry. How each part is connected to the other parts and the particular way they are positioned provides insight and understanding, whether it is at the scale of atoms bonding to other atoms in glass, a crystal, or granular materials, or whether it is the way the soaring tensegrities of the artist Kenneth Snelson majestically fill space. The rigidity of such structures is inherent and unavoidable. Recent advances in materials synthesis have emphasized the need for a deeper understanding of the geometric stability of physical structures at the atomic level and the need for understanding at all levels. Key mathematical tools for this analysis come from the area of “rigidity theory”, which studies the behavior 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.
One goal of this project is a mechanistic explanation of tunneling between asymmetric stable configurations of two-dimensional disordered materials, such as glass. Accurate mechanistic and computational models require formalizing and proving deep mathematical conjectures and developing 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. Another example of such distance-constraint systems includes 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. Rigidity theory could even have implications for algorithms for low-rank matrix completion. Finding a recursive decomposition of a generically rigid framework into rigid subsystems is a problem that arguably has been open since Maxwell in the 19th century. Additionally, matroid theory, a large part of rigidity theory, has made the characterization of rigid systems more approachable and more algorithmically efficient. These connections, problems, and implications will be explored in this project.