My main research interest is nonlinear dispersive PDE. I am currently working on existence and uniqueness problems for nonlinear parabolic equations, such as the porous medium equation, in rough domains. I am also working on problems regarding global dynamics for nonlinear Schrödinger equations in the energy critical and subcritical settings. I am additionally interested in electromagnetic nonlinear Schrödinger equations, calculus of variations, and isoperimetric inequalities.