Staffilani Contributed to the study of the theory of non-linear partial differential equations of dispersive type. Her publications include papers with Kirkpatrick and Schlein on Bose-Einstein condensation, with Tataru on Strichartz estimates on curved space, with Christianson and Hur on dispersion for the water wave equation and with Rodnianski and Rubinstein on 1-D Schrödinger. She is one of the gang of five, with Colliander, Keel, Takaoka, and Tao, who produced three ground-breaking achievements: their proof of global well-posedness for the quintic defocusing non-linear Schrödinger equation, their result on frequency cascades for the cubic non-linear Schrödinger equation on the two-torus, and their proof of symplectic non-squeezing for the periodic KdV equation. These highly cited results settled celebrated, long outstanding conjectures, and also introduced revolutionary new techniques such as almost conserved energies and the interaction Morawetz inequality. They also contributed in solidifying an infinite dimensional generalization of the Gromov symplectic non-squeezing lemma into non-linear PDE theory, originally started by Kuksin and Bourgain. More recently Staffilani has been working on questions concerning Gibbs measures and almost sure well-posedness results.
Staffilani has been actively involved in making the mathematical community more diverse by mentoring students from under represented groups. She is the Diversity Representative of the Department of Mathematics at MIT.