the analysis of the peridynanmic nonlocal continuum theory, including mechanical and mathematical analyses. This includes a probabilistic interpretation for the nonlocal diffusion equation, a special case of the peridynamic energy equation; and
Markov chains on directed graphs and their statistical analyses; and
the numerical solution of large-scale eigenvalue problems and differential equations. I am a co-author of ARPACK and Anasazi, open source software for computing eigenvalues of large-scale matrices. I have collaborated with scientists to use ARPACK in applications in Quantum physics, CFD, spectral graph theory, structural dynamics and radiation transport.