Pakzad's research concerns nonconvex calculus of variations and geometric analysis. He has worked on harmonic mappings into sphere, on Sobolev spaces between manifolds and Sobolev isometric immersions. This includes study of regularity and rigidity properties of isometric immersions from a an analytical perspective. Most recently he works on problems in nonlinear theory of elasticity and shell and plate theories, and also on metric-driven shape formation and non-Euclidean elasticity. Study of geometric PDEs from various new perspectives is a major ingredient in advancing these projects.