Algorithms and Analysis for Models in Materials Science, Fluids, and Probability

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

The project will span applications to problems in physics, chemistry, materials science, and data science; ideas from several mathematical disciplines, including partial differential equations, microlocal analysis, and harmonic analysis, will be used. Mathematics from these areas has made great strides recently on a variety of applications related to molecular structure in quantum mechanics, dynamics of atoms depositing on a surface, partitioning networks into clustered groups, stability of complex phenomena in fluid dynamics, and more. The research here is based largely on newly-developed theories from partial differential equations and probability, though several components of the work will be devoted to implementation of computational algorithms and analysis. Projects will involve training researchers at all levels, including undergraduates, graduate students, and postdoctoral scholars, as well as collaboration with researchers from statistics, physics, computational chemistry, scientific computing, and network science. This research project is aimed at the study of several important interdisciplinary topics ranging from fundamental questions about molecular structure to explorations of algorithms in data analysis. For instance, the principal investigator will continue work on bifurcation theory in models from density functional theory, as well as on finding numerical and analytic tools for studying nonlinear bifurcations of topological lattice models arising in quantum optics. A substantial part of the project will be done in collaboration with students and colleagues developing numerical methods for fast, accurate computation of fluid flows in complicated geometries, with the intention of making experimental predictions. Special attention will be paid to highly nonlinear models and applications of tools from spectral theory. Beyond physical applications, spectral theory can also play important roles in data analysis, specifically in spectral clustering. Along these lines, using ideas from probability, optimization and harmonic analysis, the investigator aims to develop algorithms for sub-graph detection in networks, and more generally Markov chain partitioning. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.