My research is in geometric analysis with a focus on fully nonlinear second and fourth-order elliptic partial differential equations that arise naturally in differential geometry. I primarily study geometric variational problems applying tools from minimal surface theory, Lagrangian geometry, Kähler geometry, geometric measure theory, and the theory of elliptic equations. Area minimization problems amongst Lagrangian surfaces lead to the study of certain fourth order PDEs that I work on. Geometrically motivated variational problems for the volume functional, in the Lagrangian setting, give rise to nonlinear fourth-order elliptic equations. A key tool for the study of second-order elliptic equations is Schauder theory, which is by now well-developed with equations in divergence form occupying an important place. In fourth order, an inherent analog is an equation in linear or nonlinear form that has a double divergence structure. Equations such as bi-harmonic functions, extremal Kähler metrics, the Willmore surface, the Hamiltonian stationary equations, share this structure.